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High school statistics, ap®︎/college statistics, precalculus, differential calculus, integral calculus, ap®︎/college calculus ab, ap®︎/college calculus bc, multivariable calculus, differential equations, linear algebra.

• Addition and subtraction
• Place value (tens and hundreds)
• Addition and subtraction within 20
• Addition and subtraction within 100
• Addition and subtraction within 1000
• Measurement and data
• Counting and place value
• Measurement and geometry
• Place value
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• Add and subtract within 20
• Add and subtract within 100
• Add and subtract within 1,000
• Money and time
• Measurement
• Intro to multiplication
• 1-digit multiplication
• Addition, subtraction, and estimation
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• Understand fractions
• Equivalent fractions and comparing fractions
• More with multiplication and division
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• Multiply by 1-digit numbers
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• Factors, multiples and patterns
• Add and subtract fractions
• Multiply fractions
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• Plane figures
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• Add decimals
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• Divide fractions
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• Divide decimals
• Powers of ten
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• Algebraic thinking
• Converting units of measure
• Properties of shapes
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• Negative numbers: addition and subtraction
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• Algebraic expressions
• Linear equations and inequalities
• Graphing lines and slope
• Expressions with exponents
• Quadratics and polynomials
• Equations and geometry
• Algebra foundations
• Solving equations & inequalities
• Working with units
• Linear equations & graphs
• Forms of linear equations
• Inequalities (systems & graphs)
• Absolute value & piecewise functions
• Exponents & radicals
• Exponential growth & decay
• Quadratics: Multiplying & factoring
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• Transformation properties and proofs
• Right triangles & trigonometry
• Non-right triangles & trigonometry (Advanced)
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• Modeling data distributions
• Exploring bivariate numerical data
• Study design
• Probability
• Counting, permutations, and combinations
• Random variables
• Sampling distributions
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• Two-sample inference for the difference between groups
• Inference for categorical data (chi-square tests)
• Advanced regression (inference and transforming)
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• Scatterplots
• Data distributions
• Two-way tables
• Binomial probability
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• Displaying and describing quantitative data
• Inference comparing two groups or populations
• Chi-square tests for categorical data
• More on regression
• Prepare for the 2020 AP®︎ Statistics Exam
• AP®︎ Statistics Standards mappings
• Polynomials
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• Derivatives: definition and basic rules
• Derivatives: chain rule and other advanced topics
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• Applications of integrals
• Differentiation: definition and basic derivative rules
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• Contextual applications of differentiation
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• Integration and accumulation of change
• Applications of integration
• AP Calculus AB solved free response questions from past exams
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• Integrals review
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• Integrating multivariable functions
• Green’s, Stokes’, and the divergence theorems
• First order differential equations
• Second order linear equations
• Laplace transform
• Vectors and spaces
• Matrix transformations
• Alternate coordinate systems (bases)

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Problems & Exercises

1.2 physical quantities and units.

The speed limit on some interstate highways is roughly 100 km/h. (a) What is this in meters per second? (b) How many miles per hour is this?

A car is traveling at a speed of 33 m/s 33 m/s size 12{"33"" m/s"} {} . (a) What is its speed in kilometers per hour? (b) Is it exceeding the 90 km/h 90 km/h size 12{"90"" km/h"} {} speed limit?

Show that 1 . 0 m/s = 3 . 6 km/h 1 . 0 m/s = 3 . 6 km/h size 12{1 "." 0`"m/s"=3 "." "6 km/h"} {} . Hint: Show the explicit steps involved in converting 1 . 0 m/s = 3 . 6 km/h. 1 . 0 m/s = 3 . 6 km/h. size 12{1 "." 0`"m/s"=3 "." "6 km/h"} {}

American football is played on a 100-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1 meter equals 3.281 feet.)

Soccer fields vary in size. A large soccer field is 115 m long and 85 m wide. What are its dimensions in feet and inches? (Assume that 1 meter equals 3.281 feet.)

What is the height in meters of a person who is 6 ft 1.0 in. tall? (Assume that 1 meter equals 39.37 in.)

Mount Everest, at 29,028 feet, is the tallest mountain on the Earth. What is its height in kilometers? (Assume that 1 kilometer equals 3,281 feet.)

The speed of sound is measured to be 342 m/s 342 m/s size 12{"342"" m/s"} {} on a certain day. What is this in km/h?

Tectonic plates are large segments of the Earth’s crust that move slowly. Suppose that one such plate has an average speed of 4.0 cm/year. (a) What distance does it move in 1 s at this speed? (b) What is its speed in kilometers per million years?

(a) Refer to Table 1.3 to determine the average distance between the Earth and the Sun. Then calculate the average speed of the Earth in its orbit in kilometers per second. (b) What is this in meters per second?

1.3 Accuracy, Precision, and Significant Figures

Express your answers to problems in this section to the correct number of significant figures and proper units.

Suppose that your bathroom scale reads your mass as 65 kg with a 3% uncertainty. What is the uncertainty in your mass (in kilograms)?

A good-quality measuring tape can be off by 0.50 cm over a distance of 20 m. What is its percent uncertainty?

(a) A car speedometer has a 5.0 % 5.0 % size 12{5.0%} {} uncertainty. What is the range of possible speeds when it reads 90 km/h 90 km/h size 12{"90"" km/h"} {} ? (b) Convert this range to miles per hour. 1 km = 0.6214 mi 1 km = 0.6214 mi size 12{"1 km" "=" "0.6214 mi"} {}

An infant’s pulse rate is measured to be 130 ± 5 130 ± 5 size 12{"130" +- 5} {} beats/min. What is the percent uncertainty in this measurement?

(a) Suppose that a person has an average heart rate of 72.0 beats/min. How many beats does he or she have in 2.0 y? (b) In 2.00 y? (c) In 2.000 y?

A can contains 375 mL of soda. How much is left after 308 mL is removed?

State how many significant figures are proper in the results of the following calculations: (a) 106 . 7 98 . 2 / 46 . 210 1 . 01 106 . 7 98 . 2 / 46 . 210 1 . 01 size 12{ left ("106" "." 7 right ) left ("98" "." 2 right )/ left ("46" "." "210" right ) left (1 "." "01" right )} {} (b) 18 . 7 2 18 . 7 2 size 12{ left ("18" "." 7 right ) rSup { size 8{2} } } {} (c) 1 . 60 × 10 − 19 3712 1 . 60 × 10 − 19 3712 size 12{ left (1 "." "60" times "10" rSup { size 8{ - "19"} } right ) left ("3712" right )} {} .

(a) How many significant figures are in the numbers 99 and 100? (b) If the uncertainty in each number is 1, what is the percent uncertainty in each? (c) Which is a more meaningful way to express the accuracy of these two numbers, significant figures or percent uncertainties?

(a) If your speedometer has an uncertainty of 2 . 0 km/h 2 . 0 km/h size 12{2 "." 0" km/h"} {} at a speed of 90 km/h 90 km/h size 12{"90"" km/h"} {} , what is the percent uncertainty? (b) If it has the same percent uncertainty when it reads 60 km/h 60 km/h size 12{"60"" km/h"} {} , what is the range of speeds you could be going?

(a) A person’s blood pressure is measured to be 120 ± 2 mm Hg 120 ± 2 mm Hg size 12{"120" +- 2" mm Hg"} {} . What is its percent uncertainty? (b) Assuming the same percent uncertainty, what is the uncertainty in a blood pressure measurement of 80 mm Hg 80 mm Hg size 12{"80"" mm Hg"} {} ?

A person measures his or her heart rate by counting the number of beats in 30 s 30 s size 12{"30"" s"} {} . If 40 ± 1 40 ± 1 size 12{"40" +- 1} {} beats are counted in 30 . 0 ± 0 . 5 s 30 . 0 ± 0 . 5 s size 12{"30" "." 0 +- 0 "." 5" s"} {} , what is the heart rate and its uncertainty in beats per minute?

What is the area of a circle 3 . 102 cm 3 . 102 cm size 12{3 "." "102"" cm"} {} in diameter?

If a marathon runner averages 9.5 mi/h, how long does it take him or her to run a 26.22-mi marathon?

A marathon runner completes a 42 . 188 -km 42 . 188 -km size 12{"42" "." "188""-km"} {} course in 2 h 2 h size 12{2" h"} {} , 30 min, and 12 s 12 s size 12{"12"" s"} {} . There is an uncertainty of 25 m 25 m size 12{"25"" m"} {} in the distance traveled and an uncertainty of 1 s in the elapsed time. (a) Calculate the percent uncertainty in the distance. (b) Calculate the uncertainty in the elapsed time. (c) What is the average speed in meters per second? (d) What is the uncertainty in the average speed?

The sides of a small rectangular box are measured to be 1 . 80 ± 0 . 01 cm 1 . 80 ± 0 . 01 cm size 12{1 "." "80" +- 0 "." "01"" cm"} {} , {} 2 . 05 ± 0 . 02 cm, and 3 . 1 ± 0 . 1 cm 2 . 05 ± 0 . 02 cm, and 3 . 1 ± 0 . 1 cm size 12{2 "." "05" +- 0 "." "02"" cm, and 3" "." 1 +- 0 "." "1 cm"} {} long. Calculate its volume and uncertainty in cubic centimeters.

When non-metric units were used in the United Kingdom, a unit of mass called the pound-mass (lbm) was employed, where 1 lbm = 0 . 4539 kg 1 lbm = 0 . 4539 kg size 12{1" lbm"=0 "." "4539"`"kg"} {} . (a) If there is an uncertainty of 0 . 0001 kg 0 . 0001 kg size 12{0 "." "0001"`"kg"} {} in the pound-mass unit, what is its percent uncertainty? (b) Based on that percent uncertainty, what mass in pound-mass has an uncertainty of 1 kg when converted to kilograms?

The length and width of a rectangular room are measured to be 3 . 955 ± 0 . 005 m 3 . 955 ± 0 . 005 m size 12{3 "." "955" +- 0 "." "005"" m"} {} and 3 . 050 ± 0 . 005 m 3 . 050 ± 0 . 005 m size 12{3 "." "050" +- 0 "." "005"" m"} {} . Calculate the area of the room and its uncertainty in square meters.

A car engine moves a piston with a circular cross section of 7 . 500 ± 0 . 002 cm 7 . 500 ± 0 . 002 cm size 12{7 "." "500" +- 0 "." "002"`"cm"} {} diameter a distance of 3 . 250 ± 0 . 001 cm 3 . 250 ± 0 . 001 cm size 12{3 "." "250" +- 0 "." "001"`"cm"} {} to compress the gas in the cylinder. (a) By what amount is the gas decreased in volume in cubic centimeters? (b) Find the uncertainty in this volume.

1.4 Approximation

How many heartbeats are there in a lifetime?

A generation is about one-third of a lifetime. Approximately how many generations have passed since the year 0 AD?

How many times longer than the mean life of an extremely unstable atomic nucleus is the lifetime of a human? (Hint: The lifetime of an unstable atomic nucleus is on the order of 10 − 22  s 10 − 22  s size 12{"10" rSup { size 8{ - "22"} } " s"} {} .)

Calculate the approximate number of atoms in a bacterium. Assume that the average mass of an atom in the bacterium is ten times the mass of a hydrogen atom. (Hint: The mass of a hydrogen atom is on the order of 10 − 27  kg 10 − 27  kg size 12{"10" rSup { size 8{ - "27"} } " kg"} {} and the mass of a bacterium is on the order of 10 − 15  kg. 10 − 15  kg. size 12{"10" rSup { size 8{ - "15"} } "kg"} {} )

Approximately how many atoms thick is a cell membrane, assuming all atoms there average about twice the size of a hydrogen atom?

(a) What fraction of Earth’s diameter is the greatest ocean depth? (b) The greatest mountain height?

(a) Calculate the number of cells in a hummingbird assuming the mass of an average cell is ten times the mass of a bacterium. (b) Making the same assumption, how many cells are there in a human?

Assuming one nerve impulse must end before another can begin, what is the maximum firing rate of a nerve in impulses per second?

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7 Types of Homework for Students

There are seven types of homework. These are practice, preparation, extension, integration, research, application, and flipped homework.

Each type of homework has its own role for students learning. The important task for teachers is to select homework that will best provide holistic support to a student.

This doesn’t mean just supporting students for the upcoming exams but also ensuring students are not overwhelmed by homework and that they can still live a balanced life outside of school.

Types of Homework

1. practice.

Practice is the most common task students have when they are given homework. It entails attempting to reinforce information learned in school so that students will commit it to long-term memory.

An example of practice homework is math workbooks. Usually, a teacher will complete the math task in class so students know how to do it, then give them a workbook of 20 – 50 tasks to complete overnight to reinforce the task.

The benefit of practice homework is that it can certainly help students commit what they learned in class to memory. This is especially the case if teachers used spaced repetition. This is a strategy whereby the teacher re-introduces things learned in previous weeks and months during homework revision so the information is not lost over time.

The downside of practice homework is that the child is learning on their own during this period. This means that, if the child hasn’t sufficiently learned the content in class, they don’t have anyone to help them during homework time. This can lead to a great deal of frustration and despair for students staring at a piece of paper feeling lost and confused.

Related: Homework Statistics

2. Preparation

Preparation homework is given to students before a lesson so they have the important information at hand before class.

Commonly, this occurs when teachers provide reading materials for students to look over between class. They then come to class having read the materials so they are ready to discuss and debate the topics.

It’s also commonly used in language learning, where it’s called pre-teaching of vocabulary. The teacher provides vocabulary to learn before class so the students can come to class and practice it together.

The benefit of preparation homework is that it bunches a lot of the didactic (non-interactive) parts of learning into the pre-class time so students can spend most of their time in class interacting with the teacher and peers.

The downside of preparation homework is that it rarely works as expected. Teachers often spend the first 10 to 15 minutes of class re-teaching what was supposed to be done for homework either because students found it too hard or they didn’t do the tasks.

3. Extension

Extension homework involves providing students with tasks that are based upon what was learned in class, but goes over and above those tasks.

It is the equivalent of teaching a student to swim then letting them out into the ocean. The students have to apply their knowledge in a new context without the support or ‘ scaffolding ’ of a teacher hovering over their shoulders.

The benefit of extension homework is it can give students a lot of confidence. They can go home and feel as if they’re excelling at their work when they find that they’re doing it without the support of a teacher.

The downside of extension homework is that it can be risky: by definition, students are extending themselves beyond what was learned in class, meaning they will come across new information and new contexts where they might need help that’s not available at the time.

Related: Homework Pros and Cons

4. Integration

Integration homework requires students to bring together, or ‘integrate’, knowledge from various subjects and knowledge areas into one project.

The homework may involve the integration of history lessons with writing lessons to create a book report, or the integration of math with business studies to create a business plan.

Integration is great for students to draw connections between things they have learned in various different classes at school. However, it also involves a lot of complexity that may make students feel confused or overwhelmed.

5. Research

Research homework involves using your time after school and on weekends to gather data that will be discussed in class.

Often, this can involve interviewing family members, taking photos around your community, or looking up information on the internet.

The benefit of research homework is that students often come to class with exciting things to share. They will also come with questions to help stimulate conversation in class.

However, it’s important for teachers to be aware that not all students have access to high-speed internet and other resources to conduct research. In fact, in my homework statistics article, I discussed how a staggering 24% of low-income American teens regularly fail to complete homework due to lack of access to technology.

6. Application

Application homework involves taking knowledge learned in class and applying it to real-world settings.

This is common in immersive language learning settings, for example, when teachers ask students to go out into the streets and practice new vocabulary when ordering a drink or buying food.

The benefit of application homework is that it gives real-world context to what was learned in class. By applying your knowledge, you can more effectively commit it to memory than simply rote learning it at your kitchen table.

Application in a real-world setting provides a context and a story that you can easily pull from your memory in the future, helping you to retain information long-term.

7. Flipped Homework

The flipped learning movement is a movement that involves making students learn at home on their own then come to class to practice it.

For teachers, this means thinking about school as the space for ‘traditional’ homework and home as the space for instruction.

This is increasingly popular with technology and hybrid learning environments. For example, a teacher might assign a YouTube video to watch as homework before class. Then, the class involves dissecting the video and critiquing it.

The benefit of flipped learning is that it maximizes time for constructivist and hands-on active learning in the classroom. The downside is students can get confused during preparation, they may not complete the preparation work, or technology may fail on them.

What are the Functions of Homework?

Homework serves multiple purposes. These include:

• Getting Ahead: Many schools and parents give their children homework (including private tutoring help so their children can have an advantage in life over other children.
• Keeping Up: On a societal level, many governments encourage homework so that students can keep up with other societies in the ‘education race’ of the 21st Century. Some societies, like South Korea, heavily emphasize homework, and others feel they need to keep up.
• Catching Up: Many students are assigned homework to catch up with the rest of their class. If you find a topic very difficult, you may find you need to spend a lot more time on your own working through the information than your peers.
• Retention: Homework helps you to repeat and retain information. With extra practice, you can commit information to memory.
• Excitement: Some well-made homework can get students excited to come to class to discuss what they did.
• Application: Situation-based learning, where you’re applying what you learned in class to real-world situations, can help progress students’ knowledge in ways that you can’t in school.
• Home-School Relationships: Homework also serves the purpose of ensuring parents know what’s going on in school. Many parents like to see what children are learning so they can monitor their children’s progress and keep teachers accountable.

Homework comes in all shapes and sizes. While many teachers stick to the old fashioned repetition homework mode, I’d encourage you to try out a range of other types of homework that can keep students engaged, encourage more discussion and collaboration in class, and ensure that the difficult work where students need the teacher’s help takes place in the classroom. This will provide maximum support for your students, help propel them forward, and prevent arguments for why homework should be banned .

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Choose Your Test

Sat / act prep online guides and tips, the 5 best homework help websites (free and paid).

Other High School , General Education

Listen: we know homework isn’t fun, but it is a good way to reinforce the ideas and concepts you’ve learned in class. But what if you’re really struggling with your homework assignments?

If you’ve looked online for a little extra help with your take-home assignments, you’ve probably stumbled across websites claiming to provide the homework help and answers students need to succeed . But can homework help sites really make a difference? And if so, which are the best homework help websites you can use? 

Below, we answer these questions and more about homework help websites–free and paid. We’ll go over: 

• The basics of homework help websites
• The cost of homework help websites 
• The five best homework websites out there 
• The pros and cons of using these websites for homework help 
• The line between “learning” and “cheating” when using online homework help 
• Tips for getting the most out of a homework help website

So let’s get started! 

The Basics About Homework Help Websites–Free and Paid

Homework help websites are designed to help you complete your homework assignments, plain and simple. 

What Makes a Homework Help Site Worth Using

Most of the best sites allow users to ask questions and then provide an answer (or multiple possible answers) and explanation in seconds. In some instances, you can even send a photo of a particular assignment or problem instead of typing the whole thing out! 

Homework help sites also offer more than just help answering homework questions. Common services provided are Q&A with experts, educational videos, lectures, practice tests and quizzes, learning modules, math solving tools, and proofreading help. Homework help sites can also provide textbook solutions (i.e. answers to problems in tons of different textbooks your school might be using), one-on-one tutoring, and peer-to-peer platforms that allow you to discuss subjects you’re learning about with your fellow students. 

And best of all, nearly all of them offer their services 24/7, including tutoring! 

What You Should Should Look Out For

When it comes to homework help, there are lots–and we mean lots –of scam sites out there willing to prey on desperate students. Before you sign up for any service, make sure you read reviews to ensure you’re working with a legitimate company. 

A word to the wise: the more a company advertises help that veers into the territory of cheating, the more likely it is to be a scam. The best homework help websites are going to help you learn the concepts you’ll need to successfully complete your homework on your own. (We’ll go over the difference between “homework help” and “cheating” a little later!) 

You don't need a golden piggy bank to use homework help websites. Some provide low or no cost help for students like you!

How Expensive Are the Best Homework Help Websites?

First of all, just because a homework help site costs money doesn’t mean it’s a good service. Likewise, just because a homework help website is free doesn’t mean the help isn’t high quality. To find the best websites, you have to take a close look at the quality and types of information they provide! 

When it comes to paid homework help services, the prices vary pretty widely depending on the amount of services you want to subscribe to. Subscriptions can cost anywhere from $2 to $150 dollars per month, with the most expensive services offering several hours of one-on-one tutoring with a subject expert per month.

The 5 Best Homework Help Websites 

So, what is the best homework help website you can use? The answer is that it depends on what you need help with. 

The best homework help websites are the ones that are reliable and help you learn the material. They don’t just provide answers to homework questions–they actually help you learn the material. 

That’s why we’ve broken down our favorite websites into categories based on who they’re best for . For instance, the best website for people struggling with math might not work for someone who needs a little extra help with science, and vice versa. 

Keep reading to find the best homework help website for you! 

Best Free Homework Help Site: Khan Academy

• Price: Free!
• Best for: Practicing tough material 

Not only is Khan Academy free, but it’s full of information and can be personalized to suit your needs. When you set up your account , you choose which courses you need to study, and Khan Academy sets up a personal dashboard of instructional videos, practice exercises, and quizzes –with both correct and incorrect answer explanations–so you can learn at your own pace. 

As an added bonus, it covers more course topics than many other homework help sites, including several AP classes.

Runner Up: Brainly.com offers a free service that allows you to type in questions and get answers and explanations from experts. The downside is that you’re limited to two answers per question and have to watch ads. 

Best Paid Homework Help Site: Chegg

• Price: $14.95 to $19.95 per month
• Best for: 24/7 homework assistance  

This service has three main parts . The first is Chegg Study, which includes textbook solutions, Q&A with subject experts, flashcards, video explanations, a math solver, and writing help. The resources are thorough, and reviewers state that Chegg answers homework questions quickly and accurately no matter when you submit them.  

Chegg also offers textbook rentals for students who need access to textbooks outside of their classroom. Finally, Chegg offers Internship and Career Advice for students who are preparing to graduate and may need a little extra help with the transition out of high school. 

Another great feature Chegg provides is a selection of free articles geared towards helping with general life skills, like coping with stress and saving money. Chegg’s learning modules are comprehensive, and they feature solutions to the problems in tons of different textbooks in a wide variety of subjects. 

Runner Up: Bartleby offers basically the same services as Chegg for $14.99 per month. The reason it didn’t rank as the best is based on customer reviews that say user questions aren’t answered quite as quickly on this site as on Chegg. Otherwise, this is also a solid choice!

Best Site for Math Homework Help: Photomath

• Price: Free (or $59.99 per year for premium services) 
• Best for: Explaining solutions to math problems

This site allows you to t ake a picture of a math problem, and instantly pulls up a step-by-step solution, as well as a detailed explanation of the concept. Photomath also includes animated videos that break down mathematical concepts to help you better understand and remember them. 

The basic service is free, but for an additional fee you can get extra study tools and learn additional strategies for solving common math problems.

Runner Up: KhanAcademy offers in-depth tutorials that cover complex math topics for free, but you won’t get the same tailored help (and answers!) that Photomath offers. 

Best Site for English Homework Help: Princeton Review Academic Tutoring

• Price: $40 to $153 per month, depending on how many hours of tutoring you want 
• Best for: Comprehensive and personalized reading and writing help 

While sites like Grammarly and Sparknotes help you by either proofreading what you write via an algorithm or providing book summaries, Princeton Review’s tutors provide in-depth help with vocabulary, literature, essay writing and development, proofreading, and reading comprehension. And unlike other services, you’ll have the chance to work with a real person to get help. 

The best part is that you can get on-demand English (and ESL) tutoring from experts 24/7. That means you can get help whenever you need it, even if you’re pulling an all-nighter! 

This is by far the most expensive homework site on this list, so you’ll need to really think about what you need out of a homework help website before you commit. One added benefit is that the subscription covers over 80 other subjects, including AP classes, which can make it a good value if you need lots of help!  

Best Site for STEM Homework Help: Studypool

• Best for: Science homework help
• Price: Varies; you’ll pay for each question you submit

When it comes to science homework help, there aren’t a ton of great resources out there. The best of the bunch is Studypool, and while it has great reviews, there are some downsides as well. 

Let’s start with the good stuff. Studypool offers an interesting twist on the homework help formula. After you create a free account, you can submit your homework help questions, and tutors will submit bids to answer your questions. You’ll be able to select the tutor–and price point–that works for you, then you’ll pay to have your homework question answered. You can also pay a small fee to access notes, lectures, and other documents that top tutors have uploaded. 

The downside to Studypool is that the pricing is not transparent . There’s no way to plan for how much your homework help will cost, especially if you have lots of questions! Additionally, it’s not clear how tutors are selected, so you’ll need to be cautious when you choose who you’d like to answer your homework questions.  

What Are the Pros and Cons of Using Homework Help Sites?

Homework help websites can be a great resource if you’re struggling in a subject, or even if you just want to make sure that you’re really learning and understanding topics and ideas that you’re interested in. But, there are some possible drawbacks if you don’t use these sites responsibly. 

We’ll go over the good–and the not-so-good–aspects of getting online homework help below. 

3 Pros of Using Homework Help Websites 

First, let’s take a look at the benefits. 

#1: Better Grades Beyond Homework

This is a big one! Getting outside help with your studies can improve your understanding of concepts that you’re learning, which translates into better grades when you take tests or write essays. 

Remember: homework is designed to help reinforce the concepts you learned in class. If you just get easy answers without learning the material behind the problems, you may not have the tools you need to be successful on your class exams…or even standardized tests you’ll need to take for college. 

#2: Convenience

One of the main reasons that online homework help is appealing is because it’s flexible and convenient. You don’t have to go to a specific tutoring center while they’re open or stay after school to speak with your teacher. Instead, you can access helpful resources wherever you can access the internet, whenever you need them.

This is especially true if you tend to study at off hours because of your extracurriculars, work schedule, or family obligations. Sites that offer 24/7 tutoring can give you the extra help you need if you can’t access the free resources that are available at your school. 

#3: Variety

Not everyone learns the same way. Maybe you’re more of a visual learner, but your teacher mostly does lectures. Or maybe you learn best by listening and taking notes, but you’re expected to learn something just from reading the textbook . 

One of the best things about online homework help is that it comes in a variety of forms. The best homework help sites offer resources for all types of learners, including videos, practice activities, and even one-on-one discussions with real-life experts. 

This variety can also be a good thing if you just don’t really resonate with the way a concept is being explained (looking at you, math textbooks!).

Not so fast. There are cons to homework help websites, too. Get to know them below!

3 Cons of Using Homework Help Websites 

Now, let’s take a look at the drawbacks of online homework help. 

#1: Unreliable Info

This can be a real problem. In addition to all the really good homework help sites, there are a whole lot of disreputable or unreliable sites out there. The fact of the matter is that some homework help sites don’t necessarily hire people who are experts in the subjects they’re talking about. In those cases, you may not be getting the accurate, up-to-date, and thorough information you need.

Additionally, even the great sites may not be able to answer all of your homework questions. This is especially true if the site uses an algorithm or chatbot to help students…or if you’re enrolled in an advanced or college-level course. In these cases, working with your teacher or school-provided tutors are probably your best option. 

#2: No Clarification

This depends on the service you use, of course. But the majority of them provide free or low-cost help through pre-recorded videos. Watching videos or reading info online can definitely help you with your homework… but you can’t ask questions or get immediate feedback if you need it .

#3: Potential For Scamming 

Like we mentioned earlier, there are a lot of homework help websites out there, and lots of them are scams. The review comments we read covered everything from outdated or wrong information, to misleading claims about the help provided, to not allowing people to cancel their service after signing up. 

No matter which site you choose to use, make sure you research and read reviews before you sign up–especially if it’s a paid service! 

When Does “Help” Become “Cheating”?

Admittedly, whether using homework help websites constitutes cheating is a bit of a grey area. For instance, is it “help” when a friend reads your essay for history class and corrects your grammar, or is it “cheating”? The truth is, not everyone agrees on when “help” crosses the line into “cheating .” When in doubt, it can be a good idea to check with your teacher to see what they think about a particular type of help you want to get. 

That said, a general rule of thumb to keep in mind is to make sure that the assignment you turn in for credit is authentically yours . It needs to demonstrate your own thoughts and your own current abilities. Remember: the point of every homework assignment is to 1) help you learn something, and 2) show what you’ve learned. 

So if a service answers questions or writes essays for you, there’s a good chance using it constitutes cheating. 

Here’s an example that might help clarify the difference for you. Brainstorming essay ideas with others or looking online for inspiration is “help” as long as you write the essay yourself. Having someone read it and give you feedback about what you need to change is also help, provided you’re the one that makes the changes later. 

But copying all or part of an essay you find online or having someone write (or rewrite) the whole thing for you would be “cheating.” The same is true for other subjects. Ultimately, if you’re not generating your own work or your own answers, it’s probably cheating.

5 Tips for Finding the Best Homework Help Websites for You

Now that you know some of our favorite homework help websites, free and paid, you can start doing some additional research on your own to decide which services might work best for you! Here are some top tips for choosing a homework help website. 

Tip 1: Decide How You Learn Best 

Before you decide which site or sites you’re going to use for homework help, y ou should figure out what kind of learning style works for you the most. Are you a visual learner? Then choose a site that uses lots of videos to help explain concepts. If you know you learn best by actually doing tasks, choose a site that provides lots of practice exercises.

Tip 2: Determine Which Subjects You Need Help With

Just because a homework help site is good overall doesn’t mean that it’s equally good for every subject. If you only need help in math, choose a site that specializes in that area. But if history is where you’re struggling, a site that specializes in math won’t be much help. So make sure to choose a site that you know provides high-quality help in the areas you need it most. 

Tip 3: Decide How Much One-On-One Help You Need 

This is really about cost-effectiveness. If you learn well on your own by reading and watching videos, a free site like Khan Academy is a good choice. But if you need actual tutoring, or to be able to ask questions and get personalized answers from experts, a paid site that provides that kind of service may be a better option.

Tip 4: Set a Budget

If you decide you want to go with a paid homework help website, set a budget first . The prices for sites vary wildly, and the cost to use them can add up quick. 

Tip 5: Read the Reviews

Finally, it’s always a good idea to read actual reviews written by the people using these homework sites. You’ll learn the good, the bad, and the ugly of what the users’ experiences have been. This is especially true if you intend to subscribe to a paid service. You’ll want to make sure that users think it’s worth the price overall!

What’s Next?

If you want to get good grades on your homework, it’s a good idea to learn how to tackle it strategically. Our expert tips will help you get the most out of each assignment…and boost your grades in the process.

Doing well on homework assignments is just one part of getting good grades. We’ll teach you everything you need to know about getting great grades in high school in this article.

Of course, test grades can make or break your GPA, too. Here are 17 expert tips that’ll help you get the most out of your study prep before you take an exam.

Ashley Sufflé Robinson has a Ph.D. in 19th Century English Literature. As a content writer for PrepScholar, Ashley is passionate about giving college-bound students the in-depth information they need to get into the school of their dreams.

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• Finding Elasticity of Demand
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• Find the Mean Absolute Deviation
• Finding the Mid-Range (Mid-Extreme)
• Finding the Interquartile Range (H-Spread)
• Finding the Midhinge
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8.E: Differential Equations (Exercises)

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8.1: Basics of Differential Equations

In exercises 1 - 7, determine the order of each differential equation.

1) \( y′+y=3y^2\)

2) \( (y′)^2=y′+2y\)

3) \( y'''+y''y′=3x^2\)

4) \( y′=y''+3t^2\)

5) \( \dfrac{dy}{dt}=t\)

6) \( \dfrac{dy}{dx}+\dfrac{d^2y}{dx^2}=3x^4\)

7) \(\left(\dfrac{dy}{dt}\right)^2+8\dfrac{dy}{dt}+3y=4t\)

In exercises 8 - 17, verify that the given function is a solution to the given differential equation.

8) \( y=\dfrac{x^3}{3}\quad\) solves \(\quad y′=x^2\)

9) \( y=2e^{−x}+x−1\quad\) solves \(\quad y′=x−y\)

10) \( y=e^{3x}−\dfrac{e^x}{2}\quad\) solves \(\quad y′=3y+e^x\)

11) \( y=\dfrac{1}{1−x}\quad\) solves \(\quad y′=y^2\)

12) \( y=e^{x^2}/2\quad\) solves \(\quad y′=xy\)

13) \( y=4+\ln x\quad\) solves \(\quad xy′=1\)

14) \( y=3−x+x\ln x\quad\) solves \(\quad y′=\ln x\)

15) \( y=2e^x−x−1\quad\) solves \(\quad y′=y+x\)

16) \( y=e^x+\dfrac{\sin x}{2}−\dfrac{\cos x}{2}\quad\) solves \(\quad y′=\cos x+y\)

17) \( y=πe^{−\cos x}\quad\) solves \(\quad y′=y\sin x\)

In exercises 18 - 27, verify the given general solution and find the particular solution.

18) Find the particular solution to the differential equation \( y′=4x^2\) that passes through \( (−3,−30)\), given that \( y=C+\dfrac{4x^3}{3}\) is a general solution.

19) Find the particular solution to the differential equation \( y′=3x^3\) that passes through \( (1,4.75)\), given that \( y=C+\dfrac{3x^4}{4}\) is a general solution.

20) Find the particular solution to the differential equation \( y′=3x^2y\) that passes through \( (0,12)\), given that \( y=Ce^{x^3}\) is a general solution.

21) Find the particular solution to the differential equation \( y′=2xy\) that passes through \( (0,\frac{1}{2})\), given that \( y=Ce^{x^2}\) is a general solution.

22) Find the particular solution to the differential equation \( y′=(2xy)^2\) that passes through \( (1,−\frac{1}{2})\), given that \( y=−\dfrac{3}{C+4x^3}\) is a general solution.

23) Find the particular solution to the differential equation \( y′x^2=y\) that passes through \( (1,\frac{2}{e})\), given that \( y=Ce^{−1/x}\) is a general solution.

24) Find the particular solution to the differential equation \( 8\dfrac{dx}{dt}=−2\cos(2t)−\cos(4t)\) that passes through \( (π,π)\), given that \( x=C−\frac{1}{8}\sin(2t)−\frac{1}{32}\sin(4t)\) is a general solution.

25) Find the particular solution to the differential equation \( \dfrac{du}{dt}=\tan u\) that passes through \( (1,\frac{π}{2})\), given that \( u=\sin^{−1}(e^{C+t})\) is a general solution.

26) Find the particular solution to the differential equation \( \dfrac{dy}{dt}=e^{(t+y)}\) that passes through \( (1,0)\), given that \( y=−\ln(C−e^t)\) is a general solution.

27) Find the particular solution to the differential equation \( y′(1−x^2)=1+y\) that passes through \( (0,−2),\) given that \( y=C\dfrac{\sqrt{x+1}}{\sqrt{1−x}}−1\) is a general solution.

In exercises 28 - 37, find the general solution to the differential equation.

28) \( y′=3x+e^x\)

29) \( y′=\ln x+\tan x\)

30) \( y′=\sin x e^{\cos x}\)

31) \( y′=4^x\)

32) \( y′=\sin^{−1}(2x)\)

33) \( y′=2t\sqrt{t^2+16}\)

34) \( x′=\coth t+\ln t+3t^2\)

35) \( x′=t\sqrt{4+t}\)

36) \( y′=y\)

37) \( y′=\dfrac{y}{x}\)

In exercises 38 - 42, solve the initial-value problems starting from \( y(t=0)=1\) and \( y(t=0)=−1.\) Draw both solutions on the same graph.

38) \( \dfrac{dy}{dt}=2t\)

39) \( \dfrac{dy}{dt}=−t\)

40) \( \dfrac{dy}{dt}=2y\)

41) \( \dfrac{dy}{dt}=−y\)

42) \( \dfrac{dy}{dt}=2\)

In exercises 43 - 47, solve the initial-value problems starting from \( y_0=10\). At what time does \(y\) increase to \(100\) or drop to \(1\)?

43) \( \dfrac{dy}{dt}=4t\)

44) \( \dfrac{dy}{dt}=4y\)

45) \( \dfrac{dy}{dt}=−2y\)

46) \( \dfrac{dy}{dt}=e^{4t}\)

47) \( \dfrac{dy}{dt}=e^{−4t}\)

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For exercises 48 - 52, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from \( y(t=0)=−10\) to \( y(t=0)=10\) increasing by \( 2\). Is there some critical point where the behavior of the solution begins to change?

48) [T] \( y′=y(x)\)

49) [T] \( xy′=y\)

50) [T] \( y′=t^3\)

51) [T] \( y′=x+y\) (Hint: \( y=Ce^x−x−1\) is the general solution)

52) [T] \( y′=x\ln x+\sin x\)

53) Find the general solution to describe the velocity of a ball of mass \( 1\) lb that is thrown upward at a rate of \( a\) ft/sec.

54) In the preceding problem, if the initial velocity of the ball thrown into the air is \( a=25\) ft/s, write the particular solution to the velocity of the ball. Solve to find the time when the ball hits the ground.

55) You throw two objects with differing masses \( m_1\) and \( m_2\) upward into the air with the same initial velocity of \( a\) ft/s. What is the difference in their velocity after \( 1\) second?

56) [T] You throw a ball of mass \( 1\) kilogram upward with a velocity of \( a=25\) m/s on Mars, where the force of gravity is \( g=−3.711\) m/s 2 . Use your calculator to approximate how much longer the ball is in the air on Mars.

57) [T] For the previous problem, use your calculator to approximate how much higher the ball went on Mars.

58) [T] A car on the freeway accelerates according to \( a=15\cos(πt),\) where \( t\) is measured in hours. Set up and solve the differential equation to determine the velocity of the car if it has an initial speed of \( 51\) mph. After \( 40\) minutes of driving, what is the driver’s velocity?

59) [T] For the car in the preceding problem, find the expression for the distance the car has traveled in time \( t\), assuming an initial distance of \( 0\). How long does it take the car to travel \( 100\) miles? Round your answer to hours and minutes.

60) [T] For the previous problem, find the total distance traveled in the first hour.

61) Substitute \( y=Be^{3t}\) into \( y′−y=8e^{3t}\) to find a particular solution.

62) Substitute \( y=a\cos(2t)+b\sin(2t)\) into \( y′+y=4\sin(2t)\) to find a particular solution.

63) Substitute \( y=a+bt+ct^2\) into \( y′+y=1+t^2\) to find a particular solution.

64) Substitute \( y=ae^t\cos t+be^t\sin t\) into \( y′=2e^t\cos t\) to find a particular solution.

65) Solve \( y′=e^{kt}\) with the initial condition \( y(0)=0\) and solve \( y′=1\) with the same initial condition. As \( k\) approaches \( 0\), what do you notice?

8.2: Direction Fields and Numerical Methods

For the following problems, use the direction field below from the differential equation \(\displaystyle y'=−2y.\) Sketch the graph of the solution for the given initial conditions.

1) \(\displaystyle y(0)=1\)

2) \(\displaystyle y(0)=0\)

3) \(\displaystyle y(0)=−1\)

4) Are there any equilibria? What are their stabilities?

Solution: \(\displaystyle y=0\) is a stable equilibrium

For the following problems, use the direction field below from the differential equation \(\displaystyle y'=y^2−2y\). Sketch the graph of the solution for the given initial conditions.

5) \(\displaystyle y(0)=3\)

6) \(\displaystyle y(0)=1\)

7) \(\displaystyle y(0)=−1\)

8) Are there any equilibria? What are their stabilities?

Solution: \(\displaystyle y=0\) is a stable equilibrium and \(\displaystyle y=2\) is unstable

Draw the direction field for the following differential equations, then solve the differential equation. Draw your solution on top of the direction field. Does your solution follow along the arrows on your direction field?

9) \(\displaystyle y'=t^3\)

10) \(\displaystyle y'=e^t\)

11) \(\displaystyle \frac{dy}{dx}=x^2cosx\)

12) \(\displaystyle \frac{dy}{dt}=te^t\)

13) \(\displaystyle \frac{dx}{dt}=cosh(t)\)

Draw the directional field for the following differential equations. What can you say about the behavior of the solution? Are there equilibria? What stability do these equilibria have?

14) \(\displaystyle y'=y^2−1\)

15) \(\displaystyle y'=y−x\)

16) \(\displaystyle y'=1−y^2−x^2\)

17) \(\displaystyle y'=t^2siny\)

18) \(\displaystyle y'=3y+xy\)

Match the direction field with the given differential equations. Explain your selections.

19) \(\displaystyle y'=−3y\)

20) \(\displaystyle y'=−3t\)

Solution: \(\displaystyle E\)

21) \(\displaystyle y'=e^t\)

22) \(\displaystyle y'=\frac{1}{2}y+t\)

Solution: \(\displaystyle A\)

23) \(\displaystyle y'=−ty\)

24) \(\displaystyle y'=tsiny\)

Solution: \(\displaystyle B\)

25) \(\displaystyle y'=−tcosy\)

26) \(\displaystyle y'=ttany\)

27) \(\displaystyle y'=sin^2y\)

28) \(\displaystyle y'=y^2t^3\)

Solution: \(\displaystyle C\)

Estimate the following solutions using Euler’s method with \(\displaystyle n=5\) steps over the interval \(\displaystyle t=[0,1].\) If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method?

29) \(\displaystyle y'=−3y,y(0)=1\)

30) \(\displaystyle y'=t^2\)

Solution: \(\displaystyle 2.24,\) exact: \(\displaystyle 3\)

31) \(\displaystyle y′=3t−y,y(0)=1.\) Exact solution is \(\displaystyle y=3t+4e^{−t}−3\)

32) \(\displaystyle y′=y+t^2,y(0)=3.\) Exact solution is \(\displaystyle y=5e^t−2−t^2−2t\)

Solution: \(\displaystyle 7.739364,\) exact: \(\displaystyle 5(e−1)\)

33) \(\displaystyle y′=2t,y(0)=0\)

34) [T] \(\displaystyle y'=e^{(x+y)},y(0)=−1.\) Exact solution is \(\displaystyle y=−ln(e+1−e^x)\)

Solution: \(\displaystyle −0.2535\) exact: \(\displaystyle 0\)

35) \(\displaystyle y′=y^2ln(x+1),y(0)=1.\) Exact solution is \(\displaystyle y=−\frac{1}{(x+1)(ln(x+1)−1)}\)

36) \(\displaystyle y′=2^x,y(0)=0,\) Exact solution is \(\displaystyle y=\frac{2^x−1}{ln(2)}\)

Solution: \(\displaystyle 1.345,\) exact: \(\displaystyle \frac{1}{ln(2)}\)

37) \(\displaystyle y′=y,y(0)=−1.\) Exact solution is \(\displaystyle y=−e^x\).

38) \(\displaystyle y′=−5t,y(0)=−2.\) Exact solution is \(\displaystyle y=−\frac{5}{2}t^2−2\)

Solution: \(\displaystyle −4,\) exact: \(\displaystyle −1/2\)

Differential equations can be used to model disease epidemics. In the next set of problems, we examine the change of size of two sub-populations of people living in a city: individuals who are infected and individuals who are susceptible to infection. \(\displaystyle S\) represents the size of the susceptible population, and \(\displaystyle I\) represents the size of the infected population. We assume that if a susceptible person interacts with an infected person, there is a probability \(\displaystyle c\) that the susceptible person will become infected. Each infected person recovers from the infection at a rate \(\displaystyle r\) and becomes susceptible again. We consider the case of influenza, where we assume that no one dies from the disease, so we assume that the total population size of the two sub-populations is a constant number, \(\displaystyle N\). The differential equations that model these population sizes are

\(\displaystyle S'=rI−cSI\) and \(\displaystyle I'=cSI−rI.\)

Here \(\displaystyle c\) represents the contact rate and \(\displaystyle r\) is the recovery rate.

39) Show that, by our assumption that the total population size is constant \(\displaystyle (S+I=N),\) you can reduce the system to a single differential equation in \(\displaystyle I:I'=c(N−I)I−rI.\)

40) Assuming the parameters are \(\displaystyle c=0.5,N=5,\) and \(\displaystyle r=0.5\), draw the resulting directional field.

41) [T] Use computational software or a calculator to compute the solution to the initial-value problem \(\displaystyle y'=ty,y(0)=2\) using Euler’s Method with the given step size \(\displaystyle h\). Find the solution at \(\displaystyle t=1\). For a hint, here is “pseudo-code” for how to write a computer program to perform Euler’s Method for \(\displaystyle y'=f(t,y),y(0)=2:\)

Create function \(\displaystyle f(t,y)\)

Define parameters \(\displaystyle y(1)=y_0,t(0)=0,\) step size \(\displaystyle h\), and total number of steps, \(\displaystyle N\)

Write a for loop:

for \(\displaystyle k=1\) to \(\displaystyle N\)

\(\displaystyle fn=f(t(k),y(k))\)

\(\displaystyle y(k+1)=y(k)+h*fn\)

\(\displaystyle t(k+1)=t(k)+h\)

42) Solve the initial-value problem for the exact solution.

Solution: \(\displaystyle y'=2e^{t^2/2}\)

43) Draw the directional field

44) \(\displaystyle h=1\)

Solution: \(\displaystyle 2\)

45) [T] \(\displaystyle h=10\)

46) [T] \(\displaystyle h=100\)

Solution: \(\displaystyle 3.2756\)

47) [T] \(\displaystyle h=1000\)

48) [T] Evaluate the exact solution at \(\displaystyle t=1\). Make a table of errors for the relative error between the Euler’s method solution and the exact solution. How much does the error change? Can you explain?

Solution: \(\displaystyle 2\sqrt{e}\)

Consider the initial-value problem \(\displaystyle y'=−2y,y(0)=2.\)

49) Show that \(\displaystyle y=2e^{−2x}\) solves this initial-value problem.

50) Draw the directional field of this differential equation.

51) [T] By hand or by calculator or computer, approximate the solution using Euler’s Method at \(\displaystyle t=10\) using \(\displaystyle h=5\).

52) [T] By calculator or computer, approximate the solution using Euler’s Method at \(\displaystyle t=10\) using \(\displaystyle h=100.\)

Solution: \(\displaystyle 4.0741e^{−10}\)

53) [T] Plot exact answer and each Euler approximation (for \(\displaystyle h=5\) and \(\displaystyle h=100\)) at each h on the directional field. What do you notice?

8.3: Separable Equations

In exercises 1 - 4, solve the following initial-value problems with the initial condition \( y_0=0\) and graph the solution.

1) \( \dfrac{dy}{dt}=y+1\)

2) \( \dfrac{dy}{dt}=y−1\)

3) \( \dfrac{dy}{dt}=-y+1\)

4) \( \dfrac{dy}{dt}=−y−1\)

In exercises 5 - 14, find the general solution to the differential equation.

5) \( x^2y'=(x+1)y\)

6) \( y'=\tan(y)x\)

7) \( y'=2xy^2\)

8) \( \dfrac{dy}{dt}=y\cos(3t+2)\)

9) \( 2x\dfrac{dy}{dx}=y^2\)

10) \( y'=e^yx^2\)

11) \( (1+x)y'=(x+2)(y−1)\)

12) \( \dfrac{dx}{dt}=3t^2(x^2+4)\)

13) \( t\dfrac{dy}{dt}=\sqrt{1−y^2}\)

14) \( y'=e^xe^y\)

In exercises 15 - 24, find the solution to the initial-value problem.

15) \( y'=e^{y−x}, \quad y(0)=0\)

16) \( y'=y^2(x+1), \quad y(0)=2\)

17) \( \dfrac{dy}{dx}=y^3xe^{x^2}, \quad y(0)=1\)

18) \( \dfrac{dy}{dt}=y^2e^x\sin(3x), \quad y(0)=1\)

19) \( y'=\dfrac{x}{\text{sech}^2y}, \quad y(0)=0\)

20) \( y'=2xy(1+2y), \quad y(0)=−1\)

21) \( \dfrac{dx}{dt}=\ln(t)\sqrt{1−x^2}, \quad x(1)=0\)

22) \( y'=3x^2(y^2+4),\quad y(0)=0\)

23) \( y'=e^y5^x, \quad y(0)=\ln(\ln(5))\)

24) \( y'=−2x\tan(y), \quad y(0)=\dfrac{π}{2}\)

For problems 25 - 29, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution?

25) [T] \( y'=1−2y\)

\( y=Ce^{−2}x+\dfrac{1}{2}\)

26) [T] \( y'=y^2x^3\)

27) [T] \( y'=y^3e^x\)

\( y=\dfrac{1}{\sqrt{2}\sqrt{C−e^x}}\)

28) [T] \( y'=e^y\)

29) [T] \( y'=y\ln(x)\)

\( y=Ce^{−x}x^x\)

30) Most drugs in the bloodstream decay according to the equation \( y'=cy\), where \( y\) is the concentration of the drug in the bloodstream. If the half-life of a drug is \( 2\) hours, what fraction of the initial dose remains after \( 6\) hours?

31) A drug is administered intravenously to a patient at a rate \( r\) mg/h and is cleared from the body at a rate proportional to the amount of drug still present in the body, \( d\) Set up and solve the differential equation, assuming there is no drug initially present in the body.

32) [T] How often should a drug be taken if its dose is \( 3\) mg, it is cleared at a rate \( c=0.1\) mg/h, and \( 1\) mg is required to be in the bloodstream at all times?

33) A tank contains \( 1\) kilogram of salt dissolved in \( 100\) liters of water. A salt solution of \( 0.1\) kg salt/L is pumped into the tank at a rate of \( 2\) L/min and is drained at the same rate. Solve for the salt concentration at time \( t\). Assume the tank is well mixed.

34) A tank containing \( 10\) kilograms of salt dissolved in \( 1000\) liters of water has two salt solutions pumped in. The first solution of \( 0.2\) kg salt/L is pumped in at a rate of \( 20\) L/min and the second solution of \( 0.05\) kg salt/L is pumped in at a rate of \( 5\) L/min. The tank drains at \( 25\) L/min. Assume the tank is well mixed. Solve for the salt concentration at time \( t\).

35) [T] For the preceding problem, find how much salt is in the tank \( 1\) hour after the process begins.

36) Torricelli’s law states that for a water tank with a hole in the bottom that has a cross-section of \( A\) and with a height of water \( h\) above the bottom of the tank, the rate of change of volume of water flowing from the tank is proportional to the square root of the height of water, according to \( \dfrac{dV}{dt}=−A\sqrt{2gh}\), where \( g\) is the acceleration due to gravity. Note that \( \dfrac{dV}{dt}=A\dfrac{dh}{dt}\). Solve the resulting initial-value problem for the height of the water, assuming a tank with a hole of radius \( 2\) ft. The initial height of water is \( 100\) ft.

37) For the preceding problem, determine how long it takes the tank to drain.

For problems 38 - 44, use Newton’s law of cooling.

38) The liquid base of an ice cream has an initial temperature of \( 200°F\) before it is placed in a freezer with a constant temperature of \( 0°F\). After \( 1\) hour, the temperature of the ice-cream base has decreased to \( 140°F\). Formulate and solve the initial-value problem to determine the temperature of the ice cream.

39) [T] The liquid base of an ice cream has an initial temperature of \( 210°F\) before it is placed in a freezer with a constant temperature of \( 20°F\). After \( 2\) hours, the temperature of the ice-cream base has decreased to \( 170°F\). At what time will the ice cream be ready to eat? (Assume \( 30°F\) is the optimal eating temperature.)

40) [T] You are organizing an ice cream social. The outside temperature is \( 80°F\) and the ice cream is at \( 10°F\). After \( 10\) minutes, the ice cream temperature has risen by \( 10°F\). How much longer can you wait before the ice cream melts at \( 40°F\)?

41) You have a cup of coffee at temperature \( 70°C\) and the ambient temperature in the room is \( 20°C\). Assuming a cooling rate \( k\) of \( 0.125,\) write and solve the differential equation to describe the temperature of the coffee with respect to time.

42) [T] You have a cup of coffee at temperature \( 70°C\) that you put outside, where the ambient temperature is \( 0°C.\) After \( 5\) minutes, how much colder is the coffee?

43) You have a cup of coffee at temperature \( 70°C\) and you immediately pour in \( 1\) part milk to \( 5\) parts coffee. The milk is initially at temperature \( 1°C.\) Write and solve the differential equation that governs the temperature of this coffee.

44) You have a cup of coffee at temperature \( 70°C,\) which you let cool \( 10\) minutes before you pour in the same amount of milk at \( 1°C\) as in the preceding problem. How does the temperature compare to the previous cup after \( 10\) minutes?

45) Solve the generic problem \( y'=ay+b\) with initial condition \( y(0)=c.\)

46) Prove the basic continual compounded interest equation. Assuming an initial deposit of \( P_0\) and an interest rate of \( r\), set up and solve an equation for continually compounded interest.

47) Assume an initial nutrient amount of \( I\) kilograms in a tank with \( L\) liters. Assume a concentration of \( c\) kg/L being pumped in at a rate of \( r\) L/min. The tank is well mixed and is drained at a rate of \( r\) L/min. Find the equation describing the amount of nutrient in the tank.

48) Leaves accumulate on the forest floor at a rate of \( 2\) g/cm 2 /yr and also decompose at a rate of \( 90%\) per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time \( 0\) there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?

49) Leaves accumulate on the forest floor at a rate of \( 4\) g/cm 2 /yr. These leaves decompose at a rate of \( 10%\) per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor. Does this amount approach a steady value? What is that value?

8.4: The Logistic Equation

Basic logistic model.

For problems 1 - 11, consider the logistic equation in the form \( P'=CP−P^2.\) Draw the directional field and find the stability of the equilibria.

1) \( C=3\)

2) \( C=0\)

\( P=0\) semi-stable

3) \( C=−3\)

4) Solve the logistic equation for \( C=10\) and an initial condition of \( P(0)=2.\)

5) Solve the logistic equation for \( C=−10\) and an initial condition of \( P(0)=2\).

6) A population of deer inside a park has a carrying capacity of \( 200\) and a growth rate of \( 2%\). If the initial population is \( 50\) deer, what is the population of deer at any given time?

7) A population of frogs in a pond has a growth rate of \( 5%.\) If the initial population is \( 1000\) frogs and the carrying capacity is \( 6000\), what is the population of frogs at any given time?

8) [T] Bacteria grow at a rate of \( 20%\) per hour in a petri dish. If there is initially one bacterium and a carrying capacity of \( 1\) million cells, how long does it take to reach \( 500,000\) cells?

9) [T] Rabbits in a park have an initial population of \( 10\) and grow at a rate of \( 4%\) per year. If the carrying capacity is \( 500\), at what time does the population reach \( 100\) rabbits?

10) [T] Two monkeys are placed on an island. After \( 5\) years, there are \( 8\) monkeys, and the estimated carrying capacity is \( 25\) monkeys. When does the population of monkeys reach \( 16\) monkeys?

11) [T] A butterfly sanctuary is built that can hold \( 2000\) butterflies, and \( 400\) butterflies are initially moved in. If after \( 2\) months there are now \( 800\) butterflies, when does the population get to \( 1500\) butterflies?

Logistic Population Model with Depletion

The following problems consider the logistic equation with an added term for depletion, either through death or emigration.

12) [T] The population of trout in a pond is given by \( P'=0.4P\left(1−\dfrac{P}{10000}\right)−400\), where \( 400\) trout are caught per year. Use your calculator or computer software to draw a directional field and draw a few sample solutions. What do you expect for the behavior?

13) In the preceding problem, what are the stabilities of the equilibria \( 0<P_1<P_2\)?

14) [T] For the preceding problem, use software to generate a directional field for the value \( f=400\). What are the stabilities of the equilibria?

\( P_1\) semi-stable

15) [T] For the preceding problems, use software to generate a directional field for the value \( f=600.\) What are the stabilities of the equilibria?

16) [T] For the preceding problems, consider the case where a certain number of fish are added to the pond, or \( f=−200.\) What are the nonnegative equilibria and their stabilities?

\( P_2>0\) stable

It is more likely that the amount of fishing is governed by the current number of fish present, so instead of a constant number of fish being caught, the rate is proportional to the current number of fish present, with proportionality constant \( k\), as \( P'=0.4P\left(1−\dfrac{P}{10000}\right)−kP.\)

17) [T] For the previous fishing problem, draw a directional field assuming \( k=0.1\). Draw some solutions that exhibit this behavior. What are the equilibria and what are their stabilities?

18) [T] Use software or a calculator to draw directional fields for \( k=0.4\). What are the nonnegative equilibria and their stabilities?

\( P_1=0\) is semi-stable

19) [T] Use software or a calculator to draw directional fields for \( k=0.6\). What are the equilibria and their stabilities?

20) Solve this equation, assuming a value of \( k=0.05\) and an initial condition of \( 2000\) fish.

21) Solve this equation, assuming a value of \( k=0.05\) and an initial condition of \( 5000\) fish.

Minimal Sustainable Population Thresholds

The following problems add in a minimal threshold value for the species to survive, \( T\), which changes the differential equation to \( P'(t)=rP\left(1−\dfrac{P}{K}\right)\left(1−\dfrac{T}{P}\right)\).

22) Draw the directional field of the threshold logistic equation, assuming \( K=10,r=0.1,T=2\). When does the population survive? When does it go extinct?

23) For the preceding problem, solve the logistic threshold equation, assuming the initial condition \( P(0)=P_0\).

24) Bengal tigers in a conservation park have a carrying capacity of \( 100\) and need a minimum of \( 10\) to survive. If they grow in population at a rate of \( 1%\) per year, with an initial population of \( 15\) tigers, solve for the number of tigers present.

25) A forest containing ring-tailed lemurs in Madagascar has the potential to support \( 5000\) individuals, and the lemur population grows at a rate of \( 5%\) per year. A minimum of 500 individuals is needed for the lemurs to survive. Given an initial population of \( 600\) lemurs, solve for the population of lemurs.

26) The population of mountain lions in Northern Arizona has an estimated carrying capacity of \( 250\) and grows at a rate of \( 0.25%\) per year and there must be \( 25\) for the population to survive. With an initial population of \( 30\) mountain lions, how many years will it take to get the mountain lions off the endangered species list (at least \( 100\))?

The Gompertz Equation

The following questions consider the Gompertz equation , a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells.

27) The Gompertz equation is given by \( P(t)'=α\ln\left(\frac{K}{P(t)}\right)P(t).\) Draw the directional fields for this equation assuming all parameters are positive, and given that \( K=1.\)

28) Assume that for a population, \( K=1000\) and \( α=0.05\). Draw the directional field associated with this differential equation and draw a few solutions. What is the behavior of the population?

29) Solve the Gompertz equation for generic \( α\) and \( K\) and \( P(0)=P_0\).

30) [T] The Gompertz equation has been used to model tumor growth in the human body. Starting from one tumor cell on day \( 1\) and assuming \( α=0.1\) and a carrying capacity of \( 10\) million cells, how long does it take to reach “detection” stage at \( 5\) million cells?

31) [T] It is estimated that the world human population reached \( 3\) billion people in \( 1959\) and \( 6\) billion in \( 1999\). Assuming a carrying capacity of \( 16\) billion humans, write and solve the differential equation for logistic growth, and determine what year the population reached \( 7\) billion.

32) [T] It is estimated that the world human population reached \( 3\) billion people in \( 1959\) and \( 6\) billion in \( 1999\). Assuming a carrying capacity of \( 16\) billion humans, write and solve the differential equation for Gompertz growth, and determine what year the population reached \( 7\) billion. Was logistic growth or Gompertz growth more accurate, considering world population reached \( 7\) billion on October \( 31,2011?\)

33) Show that the population grows fastest when it reaches half the carrying capacity for the logistic equation \( P'=rP\left(1−\dfrac{P}{K}\right)\).

34) When does population increase the fastest in the threshold logistic equation \( P'(t)=rP\left(1−\dfrac{P}{K}\right)\left(1−\dfrac{T}{P}\right)\)?

35) When does population increase the fastest for the Gompertz equation \( P(t)'=α\ln\left(\frac{K}{P(t)}\right)P(t)?\)

Below is a table of the populations of whooping cranes in the wild from \( 1940\) to \( 2000\). The population rebounded from near extinction after conservation efforts began. The following problems consider applying population models to fit the data. Assume a carrying capacity of \( 10,000\) cranes. Fit the data assuming years since \( 1940\) (so your initial population at time \( 0\) would be \( 22\) cranes).

Source: www.savingcranes.org/images/...wc_numbers.pdf

36) Find the equation and parameter \( r\) that best fit the data for the logistic equation.

37) Find the equation and parameters \( r\) and \( T\) that best fit the data for the threshold logistic equation.

38) Find the equation and parameter \( α\) that best fit the data for the Gompertz equation.

39) Graph all three solutions and the data on the same graph. Which model appears to be most accurate?

40) Using the three equations found in the previous problems, estimate the population in \( 2010\) (year \( 70\) after conservation). The real population measured at that time was \( 437\). Which model is most accurate?

8.5: First-order Linear Equations

Are the following differential equations linear? Explain your reasoning.

1) \(\displaystyle \frac{dy}{dx}=x^2y+sinx\)

2) \(\displaystyle \frac{dy}{dt}=ty\)

Solution: \(\displaystyle Yes\)

3) \(\displaystyle \frac{dy}{dt}+y^2=x\)

4) \(\displaystyle y'=x^3+e^x\)

5) \(\displaystyle y'=y+e^y\)

Write the following first-order differential equations in standard form.

6) \(\displaystyle y'=x^3y+sinx\)

Solution: \(\displaystyle y'−x^3y=sinx\)

7) \(\displaystyle y'+3y−lnx=0\)

8) \(\displaystyle −xy'=(3x+2)y+xe^x\)

Solution: \(\displaystyle y'+\frac{(3x+2)}{x}y=−e^x\)

9) \(\displaystyle \frac{dy}{dt}=4y+ty+tant\)

10) \(\displaystyle \frac{dy}{dt}=yx(x+1)\)

Solution: \(\displaystyle \frac{dy}{dt}−yx(x+1)=0\)

What are the integrating factors for the following differential equations?

11) \(\displaystyle y'=xy+3\)

12) \(\displaystyle y'+e^xy=sinx\)

Solution: \(\displaystyle e^x\)

13) \(\displaystyle y'=xln(x)y+3x\)

14) \(\displaystyle \frac{dy}{dx}=tanh(x)y+1\)

Solution: \(\displaystyle −ln(coshx)\)

15) \(\displaystyle \frac{dy}{dt}+3ty=e^ty\)

Solve the following differential equations by using integrating factors.

16) \(\displaystyle y'=3y+2\)

Solution: \(\displaystyle y=Ce^{3x}−\frac{2}{3}\)

17) \(\displaystyle y'=2y−x^2\)

18) \(\displaystyle xy'=3y−6x^2\)

Solution: \(\displaystyle y=Cx^3+6x^2\)

19) \(\displaystyle (x+2)y'=3x+y\)

20) \(\displaystyle y'=3x+xy\)

Solution: \(\displaystyle y=Ce^{x^2/2}−3\)

21) \(\displaystyle xy'=x+y\)

22) \(\displaystyle sin(x)y'=y+2x\)

Solution: \(\displaystyle y=Ctan(\frac{x}{2})−2x+4tan(\frac{x}{2})ln(sin(\frac{x}{2}))\)

23) \(\displaystyle y'=y+e^x\)

24) \(\displaystyle xy'=3y+x^2\)

Solution: \(\displaystyle y=Cx^3−x^2\)

25) \(\displaystyle y'+lnx=\frac{y}{x}\)

Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?

26) [T] \(\displaystyle (x+2)y'=2y−1\)

Solution: \(\displaystyle y=C(x+2)^2+\frac{1}{2}\)

27) [T] \(\displaystyle y'=3e^{t/3}−2y\)

28) [T] \(\displaystyle xy'+\frac{y}{2}=sin(3t)\)

Solution: \(\displaystyle y=\frac{C}{\sqrt{x}}+2sin(3t)\)

29) [T] \(\displaystyle xy'=2\frac{cosx}{x}−3y\)

30) [T] \(\displaystyle (x+1)y'=3y+x^2+2x+1\)

Solution: \(\displaystyle y=C(x+1)^3−x^2−2x−1\)

31) [T] \(\displaystyle sin(x)y'+cos(x)y=2x\)

32) [T] \(\displaystyle \sqrt{x^2+1}y'=y+2\)

Solution: \(\displaystyle y=Ce^{sinh^{−1}x}−2\)

33) [T] \(\displaystyle x^3y'+2x^2y=x+1\)

Solve the following initial-value problems by using integrating factors.

34) \(\displaystyle y'+y=x,y(0)=3\)

Solution: \(\displaystyle y=x+4e^x−1\)

35) \(\displaystyle y'=y+2x^2,y(0)=0\)

36) \(\displaystyle xy'=y−3x^3,y(1)=0\)

Solution: \(\displaystyle y=−\frac{3x}{2}(x^2−1)\)

37) \(\displaystyle x^2y'=xy−lnx,y(1)=1\)

38) \(\displaystyle (1+x^2)y'=y−1,y(0)=0\)

Solution: \(\displaystyle y=1−e^{tan^{−1}x}\)

39) \(\displaystyle xy'=y+2xlnx,y(1)=5\)

40) \(\displaystyle (2+x)y'=y+2+x,y(0)=0\)

Solution: \(\displaystyle y=(x+2)ln(\frac{x+2}{2})\)

41) \(y'=xy+2xe^x,y(0)=2\)

42) \(\displaystyle \sqrt{x}y'=y+2x,y(0)=1\)

Solution: \(\displaystyle y=2e^{2\sqrt{x}}−2x−2\sqrt{x}−1\)

43) \(\displaystyle y'=2y+xe^x,y(0)=−1\)

Using your expression from the preceding problem, what is the terminal velocity? (Hint: Examine the limiting behavior; does the velocity approach a value?)

44) [T] Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall \(\displaystyle 5000\) meters if the mass is \(\displaystyle 100\) kilograms, the acceleration due to gravity is \(\displaystyle 9.8\) m/s 2 and the proportionality constant is \(\displaystyle 4\)?

Solution: \(\displaystyle 40.451\) seconds

45) A more accurate way to describe terminal velocity is that the drag force is proportional to the square of velocity, with a proportionality constant \(\displaystyle k\). Set up the differential equation and solve for the velocity.

46) Using your expression from the preceding problem, what is the terminal velocity? (Hint: Examine the limiting behavior: Does the velocity approach a value?)

Solution: \(\displaystyle \sqrt{\frac{gm}{k}}\)

47) [T] Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall \(\displaystyle 5000\) meters if the mass is \(\displaystyle 100\) kilograms, the acceleration due to gravity is \(\displaystyle 9.8\)m/s 2 and the proportionality constant is \(\displaystyle 4\)? Does it take more or less time than your initial estimate?

For the following problems, determine how parameter \(\displaystyle a\) affects the solution.

48) Solve the generic equation \(\displaystyle y'=ax+y\). How does varying \(\displaystyle a\) change the behavior?

Solution: \(\displaystyle y=Ce^x−a(x+1)\)

49) Solve the generic equation \(\displaystyle y'=ax+y.\) How does varying \(\displaystyle a\) change the behavior?

50) Solve the generic equation \(\displaystyle y'=ax+xy\). How does varying \(\displaystyle a\) change the behavior?

Solution: \(\displaystyle y=Ce^{x^2/2}−a\)

51) Solve the generic equation \(\displaystyle y'=x+axy.\) How does varying \(\displaystyle a\) change the behavior?

52) Solve \(\displaystyle y'−y=e^{kt}\) with the initial condition \(\displaystyle y(0)=0\). As \(\displaystyle k\) approaches \(\displaystyle 1\), what happens to your formula?

Solution: \(\displaystyle y=\frac{e^{kt}−e^t}{k−1}\)

Chapter Review Exercise

True or False? Justify your answer with a proof or a counterexample.

1) The differential equation \(\displaystyle y'=3x^2y−cos(x)y''\) is linear.

2) The differential equation \(\displaystyle y'=x−y\) is separable.

Solution: \(\displaystyle F\)

3) You can explicitly solve all first-order differential equations by separation or by the method of integrating factors.

4) You can determine the behavior of all first-order differential equations using directional fields or Euler’s method.

Solution: \(\displaystyle T\)

For the following problems, find the general solution to the differential equations.

5) \(\displaystyle y′=x^2+3e^x−2x\)

6) \(\displaystyle y'=2^x+cos^{−1}x\)

Solution: \(\displaystyle y(x)=\frac{2^x}{ln(2)}+xcos^{−1}x−\sqrt{1−x^2}+C\)

7) \(\displaystyle y'=y(x^2+1)\)

8) \(\displaystyle y'=e^{−y}sinx\)

Solution: \(\displaystyle y(x)=ln(C−cosx)\)

9) \(\displaystyle y'=3x−2y\)

10) \(\displaystyle y'=ylny\)

Solution: \(\displaystyle y(x)=e^{e^{C+x}}\)

For the following problems, find the solution to the initial value problem.

11) \(\displaystyle y'=8x−lnx−3x^4,y(1)=5\)

12) \(\displaystyle y'=3x−cosx+2,y(0)=4\)

Solution: \(\displaystyle y(x)=4+\frac{3}{2}x^2+2x−sinx\)

13) \(\displaystyle xy'=y(x−2),y(1)=3\)

14) \(\displaystyle y'=3y^2(x+cosx),y(0)=−2\)

Solution: \(\displaystyle y(x)=−\frac{2}{1+3(x^2+2sinx)}\)

15) \(\displaystyle (x−1)y'=y−2,y(0)=0\)

16) \(\displaystyle y'=3y−x+6x^2,y(0)=−1\)

Solution: \(\displaystyle y(x)=−2x^2−2x−\frac{1}{3}−\frac{2}{3}e^{3x}\)

For the following problems, draw the directional field associated with the differential equation, then solve the differential equation. Draw a sample solution on the directional field.

17) \(\displaystyle y'=2y−y^2\)

18) \(\displaystyle y'=\frac{1}{x}+lnx−y,\) for \(\displaystyle x>0\)

Solution: \(\displaystyle y(x)=Ce^{−x}+lnx\)

For the following problems, use Euler’s Method with \(\displaystyle n=5\) steps over the interval \(\displaystyle t=[0,1].\) Then solve the initial-value problem exactly. How close is your Euler’s Method estimate?

19) \(\displaystyle y'=−4yx,y(0)=1\)

20) \(\displaystyle y'=3^x−2y,y(0)=0\)

Solution: Euler: \(\displaystyle 0.6939\), exact solution: \(\displaystyle y(x)=\frac{3^x−e^{−2x}}{2+ln(3)}\)

For the following problems, set up and solve the differential equations.

21) A car drives along a freeway, accelerating according to \(\displaystyle a=5sin(πt),\) where \(\displaystyle t\) represents time in minutes. Find the velocity at any time \(\displaystyle t\), assuming the car starts with an initial speed of \(\displaystyle 60\) mph.

22) You throw a ball of mass \(\displaystyle 2\) kilograms into the air with an upward velocity of \(\displaystyle 8\) m/s. Find exactly the time the ball will remain in the air, assuming that gravity is given by \(\displaystyle g=9.8m/s^2\).

Solution: \(\displaystyle \frac{40}{49}\) second

23) You drop a ball with a mass of \(\displaystyle 5\) kilograms out an airplane window at a height of \(\displaystyle 5000\)m. How long does it take for the ball to reach the ground?

24) You drop the same ball of mass \(\displaystyle 5\) kilograms out of the same airplane window at the same height, except this time you assume a drag force proportional to the ball’s velocity, using a proportionality constant of \(\displaystyle 3\) and the ball reaches terminal velocity. Solve for the distance fallen as a function of time. How long does it take the ball to reach the ground?

Solution: \(\displaystyle x(t)=5000+\frac{245}{9}−\frac{49}{3}t−\frac{245}{9}e^{−5/3t},t=307.8\) seconds

25) A drug is administered to a patient every \(\displaystyle 24\) hours and is cleared at a rate proportional to the amount of drug left in the body, with proportionality constant \(\displaystyle 0.2\). If the patient needs a baseline level of \(\displaystyle 5\) mg to be in the bloodstream at all times, how large should the dose be?

26) A \(\displaystyle 1000\) -liter tank contains pure water and a solution of \(\displaystyle 0.2\) kg salt/L is pumped into the tank at a rate of \(\displaystyle 1\) L/min and is drained at the same rate. Solve for total amount of salt in the tank at time \(\displaystyle t\).

Solution: \(\displaystyle T(t)=200(1−e^{−t/1000})\)

27) You boil water to make tea. When you pour the water into your teapot, the temperature is \(\displaystyle 100°C.\) After \(\displaystyle 5\) minutes in your \(\displaystyle 15°C\) room, the temperature of the tea is \(\displaystyle 85°C\). Solve the equation to determine the temperatures of the tea at time \(\displaystyle t\). How long must you wait until the tea is at a drinkable temperature (\(\displaystyle 72°C\))?

28) The human population (in thousands) of Nevada in \(\displaystyle 1950\) was roughly \(\displaystyle 160\). If the carrying capacity is estimated at \(\displaystyle 10\) million individuals, and assuming a growth rate of \(\displaystyle 2%\) per year, develop a logistic growth model and solve for the population in Nevada at any time (use \(\displaystyle 1950\) as time = 0). What population does your model predict for \(\displaystyle 2000\)? How close is your prediction to the true value of \(\displaystyle 1,998,257\)?

Solution: \(\displaystyle P(t)=\frac{1600000e^{0.02t}}{9840+160e^{0.02t}}\)

Repeat the previous problem but use Gompertz growth model. Which is more accurate?

Contributors and Attributions

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org .

Does Homework Really Help Students Learn?

A conversation with a Wheelock researcher, a BU student, and a fourth-grade teacher

“Quality homework is engaging and relevant to kids’ lives,” says Wheelock’s Janine Bempechat. “It gives them autonomy and engages them in the community and with their families. In some subjects, like math, worksheets can be very helpful. It has to do with the value of practicing over and over.” Photo by iStock/Glenn Cook Photography

Do your homework.

If only it were that simple.

Educators have debated the merits of homework since the late 19th century. In recent years, amid concerns of some parents and teachers that children are being stressed out by too much homework, things have only gotten more fraught.

“Homework is complicated,” says developmental psychologist Janine Bempechat, a Wheelock College of Education & Human Development clinical professor. The author of the essay “ The Case for (Quality) Homework—Why It Improves Learning and How Parents Can Help ” in the winter 2019 issue of Education Next , Bempechat has studied how the debate about homework is influencing teacher preparation, parent and student beliefs about learning, and school policies.

She worries especially about socioeconomically disadvantaged students from low-performing schools who, according to research by Bempechat and others, get little or no homework.

BU Today  sat down with Bempechat and Erin Bruce (Wheelock’17,’18), a new fourth-grade teacher at a suburban Boston school, and future teacher freshman Emma Ardizzone (Wheelock) to talk about what quality homework looks like, how it can help children learn, and how schools can equip teachers to design it, evaluate it, and facilitate parents’ role in it.

BU Today: Parents and educators who are against homework in elementary school say there is no research definitively linking it to academic performance for kids in the early grades. You’ve said that they’re missing the point.

Bempechat : I think teachers assign homework in elementary school as a way to help kids develop skills they’ll need when they’re older—to begin to instill a sense of responsibility and to learn planning and organizational skills. That’s what I think is the greatest value of homework—in cultivating beliefs about learning and skills associated with academic success. If we greatly reduce or eliminate homework in elementary school, we deprive kids and parents of opportunities to instill these important learning habits and skills.

We do know that beginning in late middle school, and continuing through high school, there is a strong and positive correlation between homework completion and academic success.

That’s what I think is the greatest value of homework—in cultivating beliefs about learning and skills associated with academic success.

You talk about the importance of quality homework. What is that?

Quality homework is engaging and relevant to kids’ lives. It gives them autonomy and engages them in the community and with their families. In some subjects, like math, worksheets can be very helpful. It has to do with the value of practicing over and over.

What are your concerns about homework and low-income children?

The argument that some people make—that homework “punishes the poor” because lower-income parents may not be as well-equipped as affluent parents to help their children with homework—is very troubling to me. There are no parents who don’t care about their children’s learning. Parents don’t actually have to help with homework completion in order for kids to do well. They can help in other ways—by helping children organize a study space, providing snacks, being there as a support, helping children work in groups with siblings or friends.

Isn’t the discussion about getting rid of homework happening mostly in affluent communities?

Yes, and the stories we hear of kids being stressed out from too much homework—four or five hours of homework a night—are real. That’s problematic for physical and mental health and overall well-being. But the research shows that higher-income students get a lot more homework than lower-income kids.

Teachers may not have as high expectations for lower-income children. Schools should bear responsibility for providing supports for kids to be able to get their homework done—after-school clubs, community support, peer group support. It does kids a disservice when our expectations are lower for them.

The conversation around homework is to some extent a social class and social justice issue. If we eliminate homework for all children because affluent children have too much, we’re really doing a disservice to low-income children. They need the challenge, and every student can rise to the challenge with enough supports in place.

What did you learn by studying how education schools are preparing future teachers to handle homework?

My colleague, Margarita Jimenez-Silva, at the University of California, Davis, School of Education, and I interviewed faculty members at education schools, as well as supervising teachers, to find out how students are being prepared. And it seemed that they weren’t. There didn’t seem to be any readings on the research, or conversations on what high-quality homework is and how to design it.

Erin, what kind of training did you get in handling homework?

Bruce : I had phenomenal professors at Wheelock, but homework just didn’t come up. I did lots of student teaching. I’ve been in classrooms where the teachers didn’t assign any homework, and I’ve been in rooms where they assigned hours of homework a night. But I never even considered homework as something that was my decision. I just thought it was something I’d pull out of a book and it’d be done.

I started giving homework on the first night of school this year. My first assignment was to go home and draw a picture of the room where you do your homework. I want to know if it’s at a table and if there are chairs around it and if mom’s cooking dinner while you’re doing homework.

The second night I asked them to talk to a grown-up about how are you going to be able to get your homework done during the week. The kids really enjoyed it. There’s a running joke that I’m teaching life skills.

Friday nights, I read all my kids’ responses to me on their homework from the week and it’s wonderful. They pour their hearts out. It’s like we’re having a conversation on my couch Friday night.

It matters to know that the teacher cares about you and that what you think matters to the teacher. Homework is a vehicle to connect home and school…for parents to know teachers are welcoming to them and their families.

Bempechat : I can’t imagine that most new teachers would have the intuition Erin had in designing homework the way she did.

Ardizzone : Conversations with kids about homework, feeling you’re being listened to—that’s such a big part of wanting to do homework….I grew up in Westchester County. It was a pretty demanding school district. My junior year English teacher—I loved her—she would give us feedback, have meetings with all of us. She’d say, “If you have any questions, if you have anything you want to talk about, you can talk to me, here are my office hours.” It felt like she actually cared.

Bempechat : It matters to know that the teacher cares about you and that what you think matters to the teacher. Homework is a vehicle to connect home and school…for parents to know teachers are welcoming to them and their families.

Ardizzone : But can’t it lead to parents being overbearing and too involved in their children’s lives as students?

Bempechat : There’s good help and there’s bad help. The bad help is what you’re describing—when parents hover inappropriately, when they micromanage, when they see their children confused and struggling and tell them what to do.

Good help is when parents recognize there’s a struggle going on and instead ask informative questions: “Where do you think you went wrong?” They give hints, or pointers, rather than saying, “You missed this,” or “You didn’t read that.”

Bruce : I hope something comes of this. I hope BU or Wheelock can think of some way to make this a more pressing issue. As a first-year teacher, it was not something I even thought about on the first day of school—until a kid raised his hand and said, “Do we have homework?” It would have been wonderful if I’d had a plan from day one.

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Senior Contributing Editor

Sara Rimer A journalist for more than three decades, Sara Rimer worked at the Miami Herald , Washington Post and, for 26 years, the New York Times , where she was the New England bureau chief, and a national reporter covering education, aging, immigration, and other social justice issues. Her stories on the death penalty’s inequities were nominated for a Pulitzer Prize and cited in the U.S. Supreme Court’s decision outlawing the execution of people with intellectual disabilities. Her journalism honors include Columbia University’s Meyer Berger award for in-depth human interest reporting. She holds a BA degree in American Studies from the University of Michigan. Profile

She can be reached at [email protected] .

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There are 81 comments on Does Homework Really Help Students Learn?

Insightful! The values about homework in elementary schools are well aligned with my intuition as a parent.

when i finish my work i do my homework and i sometimes forget what to do because i did not get enough sleep

same omg it does not help me it is stressful and if I have it in more than one class I hate it.

Same I think my parent wants to help me but, she doesn’t care if I get bad grades so I just try my best and my grades are great.

I think that last question about Good help from parents is not know to all parents, we do as our parents did or how we best think it can be done, so maybe coaching parents or giving them resources on how to help with homework would be very beneficial for the parent on how to help and for the teacher to have consistency and improve homework results, and of course for the child. I do see how homework helps reaffirm the knowledge obtained in the classroom, I also have the ability to see progress and it is a time I share with my kids

The answer to the headline question is a no-brainer – a more pressing problem is why there is a difference in how students from different cultures succeed. Perfect example is the student population at BU – why is there a majority population of Asian students and only about 3% black students at BU? In fact at some universities there are law suits by Asians to stop discrimination and quotas against admitting Asian students because the real truth is that as a group they are demonstrating better qualifications for admittance, while at the same time there are quotas and reduced requirements for black students to boost their portion of the student population because as a group they do more poorly in meeting admissions standards – and it is not about the Benjamins. The real problem is that in our PC society no one has the gazuntas to explore this issue as it may reveal that all people are not created equal after all. Or is it just environmental cultural differences??????

I get you have a concern about the issue but that is not even what the point of this article is about. If you have an issue please take this to the site we have and only post your opinion about the actual topic

This is not at all what the article is talking about.

This literally has nothing to do with the article brought up. You should really take your opinions somewhere else before you speak about something that doesn’t make sense.

we have the same name

so they have the same name what of it?

lol you tell her

totally agree

What does that have to do with homework, that is not what the article talks about AT ALL.

Yes, I think homework plays an important role in the development of student life. Through homework, students have to face challenges on a daily basis and they try to solve them quickly.I am an intense online tutor at 24x7homeworkhelp and I give homework to my students at that level in which they handle it easily.

More than two-thirds of students said they used alcohol and drugs, primarily marijuana, to cope with stress.

You know what’s funny? I got this assignment to write an argument for homework about homework and this article was really helpful and understandable, and I also agree with this article’s point of view.

I also got the same task as you! I was looking for some good resources and I found this! I really found this article useful and easy to understand, just like you! ^^

i think that homework is the best thing that a child can have on the school because it help them with their thinking and memory.

I am a child myself and i think homework is a terrific pass time because i can’t play video games during the week. It also helps me set goals.

Homework is not harmful ,but it will if there is too much

I feel like, from a minors point of view that we shouldn’t get homework. Not only is the homework stressful, but it takes us away from relaxing and being social. For example, me and my friends was supposed to hang at the mall last week but we had to postpone it since we all had some sort of work to do. Our minds shouldn’t be focused on finishing an assignment that in realty, doesn’t matter. I completely understand that we should have homework. I have to write a paper on the unimportance of homework so thanks.

homework isn’t that bad

Are you a student? if not then i don’t really think you know how much and how severe todays homework really is

i am a student and i do not enjoy homework because i practice my sport 4 out of the five days we have school for 4 hours and that’s not even counting the commute time or the fact i still have to shower and eat dinner when i get home. its draining!

i totally agree with you. these people are such boomers

why just why

they do make a really good point, i think that there should be a limit though. hours and hours of homework can be really stressful, and the extra work isn’t making a difference to our learning, but i do believe homework should be optional and extra credit. that would make it for students to not have the leaning stress of a assignment and if you have a low grade you you can catch up.

Studies show that homework improves student achievement in terms of improved grades, test results, and the likelihood to attend college. Research published in the High School Journal indicates that students who spent between 31 and 90 minutes each day on homework “scored about 40 points higher on the SAT-Mathematics subtest than their peers, who reported spending no time on homework each day, on average.” On both standardized tests and grades, students in classes that were assigned homework outperformed 69% of students who didn’t have homework. A majority of studies on homework’s impact – 64% in one meta-study and 72% in another – showed that take home assignments were effective at improving academic achievement. Research by the Institute for the Study of Labor (IZA) concluded that increased homework led to better GPAs and higher probability of college attendance for high school boys. In fact, boys who attended college did more than three hours of additional homework per week in high school.

So how are your measuring student achievement? That’s the real question. The argument that doing homework is simply a tool for teaching responsibility isn’t enough for me. We can teach responsibility in a number of ways. Also the poor argument that parents don’t need to help with homework, and that students can do it on their own, is wishful thinking at best. It completely ignores neurodiverse students. Students in poverty aren’t magically going to find a space to do homework, a friend’s or siblings to help them do it, and snacks to eat. I feel like the author of this piece has never set foot in a classroom of students.

THIS. This article is pathetic coming from a university. So intellectually dishonest, refusing to address the havoc of capitalism and poverty plays on academic success in life. How can they in one sentence use poor kids in an argument and never once address that poor children have access to damn near 0 of the resources affluent kids have? Draw me a picture and let’s talk about feelings lmao what a joke is that gonna put food in their belly so they can have the calories to burn in order to use their brain to study? What about quiet their 7 other siblings that they share a single bedroom with for hours? Is it gonna force the single mom to magically be at home and at work at the same time to cook food while you study and be there to throw an encouraging word?

Also the “parents don’t need to be a parent and be able to guide their kid at all academically they just need to exist in the next room” is wild. Its one thing if a parent straight up is not equipped but to say kids can just figured it out is…. wow coming from an educator What’s next the teacher doesn’t need to teach cause the kid can just follow the packet and figure it out?

Well then get a tutor right? Oh wait you are poor only affluent kids can afford a tutor for their hours of homework a day were they on average have none of the worries a poor child does. Does this address that poor children are more likely to also suffer abuse and mental illness? Like mentioned what about kids that can’t learn or comprehend the forced standardized way? Just let em fail? These children regularly are not in “special education”(some of those are a joke in their own and full of neglect and abuse) programs cause most aren’t even acknowledged as having disabilities or disorders.

But yes all and all those pesky poor kids just aren’t being worked hard enough lol pretty sure poor children’s existence just in childhood is more work, stress, and responsibility alone than an affluent child’s entire life cycle. Love they never once talked about the quality of education in the classroom being so bad between the poor and affluent it can qualify as segregation, just basically blamed poor people for being lazy, good job capitalism for failing us once again!

why the hell?

you should feel bad for saying this, this article can be helpful for people who has to write a essay about it

This is more of a political rant than it is about homework

I know a teacher who has told his students their homework is to find something they are interested in, pursue it and then come share what they learn. The student responses are quite compelling. One girl taught herself German so she could talk to her grandfather. One boy did a research project on Nelson Mandela because the teacher had mentioned him in class. Another boy, a both on the autism spectrum, fixed his family’s computer. The list goes on. This is fourth grade. I think students are highly motivated to learn, when we step aside and encourage them.

The whole point of homework is to give the students a chance to use the material that they have been presented with in class. If they never have the opportunity to use that information, and discover that it is actually useful, it will be in one ear and out the other. As a science teacher, it is critical that the students are challenged to use the material they have been presented with, which gives them the opportunity to actually think about it rather than regurgitate “facts”. Well designed homework forces the student to think conceptually, as opposed to regurgitation, which is never a pretty sight

Wonderful discussion. and yes, homework helps in learning and building skills in students.

not true it just causes kids to stress

Homework can be both beneficial and unuseful, if you will. There are students who are gifted in all subjects in school and ones with disabilities. Why should the students who are gifted get the lucky break, whereas the people who have disabilities suffer? The people who were born with this “gift” go through school with ease whereas people with disabilities struggle with the work given to them. I speak from experience because I am one of those students: the ones with disabilities. Homework doesn’t benefit “us”, it only tears us down and put us in an abyss of confusion and stress and hopelessness because we can’t learn as fast as others. Or we can’t handle the amount of work given whereas the gifted students go through it with ease. It just brings us down and makes us feel lost; because no mater what, it feels like we are destined to fail. It feels like we weren’t “cut out” for success.

homework does help

here is the thing though, if a child is shoved in the face with a whole ton of homework that isn’t really even considered homework it is assignments, it’s not helpful. the teacher should make homework more of a fun learning experience rather than something that is dreaded

This article was wonderful, I am going to ask my teachers about extra, or at all giving homework.

I agree. Especially when you have homework before an exam. Which is distasteful as you’ll need that time to study. It doesn’t make any sense, nor does us doing homework really matters as It’s just facts thrown at us.

Homework is too severe and is just too much for students, schools need to decrease the amount of homework. When teachers assign homework they forget that the students have other classes that give them the same amount of homework each day. Students need to work on social skills and life skills.

I disagree.

Beyond achievement, proponents of homework argue that it can have many other beneficial effects. They claim it can help students develop good study habits so they are ready to grow as their cognitive capacities mature. It can help students recognize that learning can occur at home as well as at school. Homework can foster independent learning and responsible character traits. And it can give parents an opportunity to see what’s going on at school and let them express positive attitudes toward achievement.

Homework is helpful because homework helps us by teaching us how to learn a specific topic.

As a student myself, I can say that I have almost never gotten the full 9 hours of recommended sleep time, because of homework. (Now I’m writing an essay on it in the middle of the night D=)

I am a 10 year old kid doing a report about “Is homework good or bad” for homework before i was going to do homework is bad but the sources from this site changed my mind!

Homeowkr is god for stusenrs

I agree with hunter because homework can be so stressful especially with this whole covid thing no one has time for homework and every one just wants to get back to there normal lives it is especially stressful when you go on a 2 week vaca 3 weeks into the new school year and and then less then a week after you come back from the vaca you are out for over a month because of covid and you have no way to get the assignment done and turned in

As great as homework is said to be in the is article, I feel like the viewpoint of the students was left out. Every where I go on the internet researching about this topic it almost always has interviews from teachers, professors, and the like. However isn’t that a little biased? Of course teachers are going to be for homework, they’re not the ones that have to stay up past midnight completing the homework from not just one class, but all of them. I just feel like this site is one-sided and you should include what the students of today think of spending four hours every night completing 6-8 classes worth of work.

Are we talking about homework or practice? Those are two very different things and can result in different outcomes.

Homework is a graded assignment. I do not know of research showing the benefits of graded assignments going home.

Practice; however, can be extremely beneficial, especially if there is some sort of feedback (not a grade but feedback). That feedback can come from the teacher, another student or even an automated grading program.

As a former band director, I assigned daily practice. I never once thought it would be appropriate for me to require the students to turn in a recording of their practice for me to grade. Instead, I had in-class assignments/assessments that were graded and directly related to the practice assigned.

I would really like to read articles on “homework” that truly distinguish between the two.

oof i feel bad good luck!

thank you guys for the artical because I have to finish an assingment. yes i did cite it but just thanks

thx for the article guys.

Homework is good

I think homework is helpful AND harmful. Sometimes u can’t get sleep bc of homework but it helps u practice for school too so idk.

I agree with this Article. And does anyone know when this was published. I would like to know.

It was published FEb 19, 2019.

Studies have shown that homework improved student achievement in terms of improved grades, test results, and the likelihood to attend college.

i think homework can help kids but at the same time not help kids

This article is so out of touch with majority of homes it would be laughable if it wasn’t so incredibly sad.

There is no value to homework all it does is add stress to already stressed homes. Parents or adults magically having the time or energy to shepherd kids through homework is dome sort of 1950’s fantasy.

What lala land do these teachers live in?

Homework gives noting to the kid

Homework is Bad

homework is bad.

why do kids even have homework?

Comments are closed.

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200 Wh Question Examples With Answers

To be able to clearly communicate in English and hold a conversation, students must learn how to use “Wh Questions” and “Yes/No Questions”. Asking and answering yes/no questions in a conversation is quite easy as there are only two possible answers: yes and no. Wh questions, on the other hand, can be more tricky for English language learners to master as these are open questions with many possible responses.

In this post, we explain what Wh Questions are and provide you with lots of examples. All of the of the wh questions below also have example answers, and so, by the end of this post, you’ll have a clearer understanding of how to effectively use and respond to Wh questions in real-life situations.

What Are Wh Questions?

Wh Questions, also known as “open questions”, are question words that begin with “Wh” (except for “how”). There are seven Wh questions in English. These are who , what , when , where , why , which, and how .

Wh questions are used when we want to get information from someone. The particular question you use depends on the type of information you want. Let’s look at each of these 7 Wh Questions in more detail with examples.

Wh Question Examples: Who

“Who” is used in questions to ask about people. Here are some examples of wh questions with “who” and some example answers.

Wh Question Examples: What

“What” is used in questions to ask about something. Here are some examples of wh questions with “what” and some example answers.

Wh Question Examples: When

“When” is used in questions to ask about time. Here are some examples of wh questions with “when” and some example answers.

Wh Question Examples: Where

“Where” is used in questions to ask about place. Here are some examples of wh questions with “where” and some example answers.

Wh Question Examples: Why

“Why” is used in questions to ask the reason for something. Here are some examples of wh questions with “why” with answers.

Wh Question Examples: Which

“Which” is used in questions to ask about choices. Here are some examples of wh questions with “which” with answers.

Wh Question Examples: How

“How” is used in questions to ask about various things, including way/manner, condition/quality, and extent/degree. Although “how” doesn’t start with “Wh,” how is still considered a Wh Question word. Here are some examples of wh questions with “how” with answers.

Related Resources

I hope you found these Wh Question examples useful. Before you go, check out these related resources you can use when learning or teaching about Wh Questions in English: Wh Questions PowerPoint / Wh Questions Worksheets

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7 Solution-Focused Therapy Techniques and Worksheets (+PDF)

It has analyzed a person’s problems from where they started and how those problems have an effect on that person’s life.

Out of years of observation of family therapy sessions, the theory and applications of solution-focused therapy developed.

Let’s explore the therapy, along with techniques and applications of the approach.

Before you read on, we thought you might like to download our three Positive Psychology Exercises for free . These science-based exercises will explore fundamental aspects of positive psychology including strengths, values, and self-compassion, and will give you the tools to enhance the wellbeing of your clients, students, or employees.

This Article Contains:

5 solution-focused therapy techniques, handy sft worksheets (pdf), solution-focused therapy interventions, 5 sft questions to ask clients, solution-focused brief therapy (sfbt techniques), 4 activities & exercises, best sft books, a take-home message.

Solution-focused therapy is a type of treatment that highlights a client’s ability to solve problems, rather than why or how the problem was created. It was developed over some time after observations of therapists in a mental health facility in Wisconsin by Steve de Shazer and Insoo Kim Berg and their colleagues.

Like positive psychology, Solution Focused Therapy (SFT) practitioners focus on goal-oriented questioning to assist a client in moving into a future-oriented direction.

Solution-focused therapy has been successfully applied to a wide variety of client concerns due to its broad application. It has been utilized in a wide variety of client groups as well. The approach presupposes that clients have some knowledge of what will improve their lives.

The following areas have utilized SFT with varying success:

• relationship difficulties
• drug and alcohol abuse
• eating disorders
• anger management
• communication difficulties
• crisis intervention
• incarceration recidivism reduction

Goal clarification is an important technique in SFT. A therapist will need to guide a client to envision a future without the problem with which they presented. With coaching and positive questioning, this vision becomes much more clarified.

With any presenting client concern, the main technique in SFT is illuminating the exception. The therapist will guide the client to an area of their life where there is an exception to the problem. The exception is where things worked well, despite the problem. Within the exception, an approach for a solution may be forged.

The ‘miracle question’ is another technique frequently used in SFT. It is a powerful tool that helps clients to move into a solution orientation. This question allows clients to begin small steps toward finding solutions to presenting problems (Santa Rita Jr., 1998). It is asked in a specific way and is outlined later in this article.

Experiment invitation is another way that therapists guide clients into solution orientation. By inviting clients to build on what is already working, clients automatically focus on the positive. In positive psychology, we know that this allows the client’s mind to broaden and build from that orientation.

Utilizing what has been working experimentally allows the client to find what does and doesn’t work in solving the issue at hand. During the second half of a consultation with a client, many SFT therapists take a break to reflect on what they’ve learned during the beginning of the session.

Consultation breaks and invitations for more information from clients allow for both the therapist and client to brainstorm on what might have been missed during the initial conversations. After this break, clients are complemented and given a therapeutic message about the presenting issue. The message is typically stated in the positive so that clients leave with a positive orientation toward their goals.

Here are four handy worksheets for use with solution-focused therapy.

• Miracle worksheet
• Exceptions to the Problem Worksheet
• Scaling Questions Worksheet
• SMART+ Goals Worksheet

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Compliments are frequently used in SFT, to help the client begin to focus on what is working, rather than what is not. Acknowledging that a client has an impact on the movement toward a goal allows hope to become present. Once hope and perspective shift occurs, a client can decide what daily actions they would like to take in attaining a goal.

Higher levels of hope and optimism can predict the following desirable outcomes (Peterson & Seligman, 2004):

• achievement in all sorts of areas
• freedom from anxiety and depression
• improved social relationships
• improved physical well being

Mind mapping is an effective intervention also used to increase hope and optimism. This intervention is often used in life coaching practices. A research study done on solution-focused life coaching (Green, Oades, & Grant, 2006) showed that this type of intervention increases goal striving and hope, in addition to overall well-being.

Though life coaching is not the same as therapy, this study shows the effectiveness of improving positive behavior through solution-focused questioning.

Mind mapping is a visual thinking tool that helps structure information. It helps clients to better analyze, comprehend, and generate new ideas in areas they might not have been automatically self-generated. Having it on paper gives them a reference point for future goal setting as well.

Empathy is vital in the administration of SFBT. A client needs to feel heard and held by the practitioner for any forward movement to occur. Intentionally leaning in to ensure that a client knows that the practitioner is engaged in listening is recommended.

Speaking to strengths and aligning those strengths with goal setting are important interventions in SFT. Recognizing and acknowledging what is already working for the client validates strengths. Self-recognition of these strengths increases self-esteem and in turn, improves forward movement.

The questions asked in Solution-Focused Therapy are positively directed and in a goal-oriented stance. The intention is to allow a perspective shift by guiding clients in the direction of hope and optimism to lead them to a path of positive change. Results and progress come from focusing on the changes that need to be made for goal attainment and increased well being.

1. Miracle Question

Here is a clear example of how to administer the miracle question. It should be delivered deliberately. When done so, it allows the client to imagine the miracle occurring.

“ Now, I want to ask you a strange question. Suppose that while you are sleeping tonight and the entire house is quiet, a miracle happens. The miracle is that the problem which brought you here is solved. However, because you are sleeping, you don’t know that the miracle has happened. So, when you wake up tomorrow morning, what will be different that will tell you that a miracle has happened and the problem which brought you here is solved? ” (de Shazer, 1988)

2. Presupposing change questions

A practitioner of solution-focused therapy asks questions in an approach derived way.

Here are a few examples of presupposing change questions:

“What stopped complete disaster from occurring?” “How did you avoid falling apart.” “What kept you from unraveling?”

3. Exception Questions

Examples of exception questions include:

1. Tell me about times when you don’t get angry. 2. Tell me about times you felt the happiest. 3. When was the last time that you feel you had a better day? 4. Was there ever a time when you felt happy in your relationship? 5. What was it about that day that made it a better day? 6. Can you think of a time when the problem was not present in your life?

4. Scaling Questions

These are questions that allow a client to rate their experience. They also allow for a client to evaluate their motivation to change their experience. Scaling questions allow for a practitioner to add a follow-up question that is in the positive as well.

An example of a scaling question: “On a scale of 1-10, with 10 representing the best it can be and one the worst, where would you say you are today?”

A follow-up question: “ Why a four and not a five?”

Questions like these allow the client to explore the positive, as well as their commitment to the changes that need to occur.

5. Coping Questions

These types of questions open clients up to their resiliency. Clients are experts in their life experience. Helping them see what works, allows them to grow from a place of strength.

“How have you managed so far?” “What have you done to stay afloat?” “What is working?”

3 Scaling questions from Solution Focused Therapy – Uncommon Practitioners

The main idea behind SFBT is that the techniques are positively and solution-focused to allow a brief amount of time for the client to be in therapy. Overall, improving the quality of life for each client, with them at the center and in the driver’s seat of their growth. SFBT typically has an average of 5-8 sessions.

During the sessions, goals are set. Specific experimental actions are explored and deployed into the client’s daily life. By keeping track of what works and where adjustments need to be made, a client is better able to track his or her progress.

A method has developed from the Miracle Question entitled, The Miracle Method . The steps follow below (Miller & Berg, 1996). It was designed for combatting problematic drinking but is useful in all areas of change.

• State your desire for something in your life to be different.
• Envision a miracle happening, and your life IS different.
• Make sure the miracle is important to you.
• Keep the miracle small.
• Define the change with language that is positive, specific, and behavioral.
• State how you will start your journey, rather than how you will end it.
• Be clear about who, where, and when, but not the why.

A short selection of exercises which can be used

1. Solution-focused art therapy/ letter writing

A powerful in-session task is to request a client to draw or write about one of the following, as part of art therapy :

• a picture of their miracle
• something the client does well
• a day when everything went well. What was different about that day?
• a special person in their life

2. Strengths Finders

Have a client focus on a time when they felt their strongest. Ask them to highlight what strengths were present when things were going well. This can be an illuminating activity that helps clients focus on the strengths they already have inside of them.

A variation of this task is to have a client ask people who are important in their lives to tell them how they view the client’s strengths. Collecting strengths from another’s perspective can be very illuminating and helpful in bringing a client into a strength perspective.

3. Solution Mind Mapping

A creative way to guide a client into a brainstorm of solutions is by mind mapping. Have the miracle at the center of the mind map. From the center, have a client create branches of solutions to make that miracle happen. By exploring solution options, a client will self-generate and be more connected to the outcome.

4. Experiment Journals

Encourage clients to do experiments in real-life settings concerning the presenting problem. Have the client keep track of what works from an approach perspective. Reassure the client that a variety of experiments is a helpful approach.

17 Top-Rated Positive Psychology Exercises for Practitioners

Expand your arsenal and impact with these 17 Positive Psychology Exercises [PDF] , scientifically designed to promote human flourishing, meaning, and wellbeing.

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These books are recommended reads for solution-focused therapy.

1. The Miracle Method: A Radically New Approach to Problem Drinking – Insoo Kim Berg and Scott D. Miller Ph.D.

The Miracle Method by Scott D. Miller and Insoo Kim Berg is a book that has helped many clients overcome problematic drinking since the 1990s.

By utilizing the miracle question in the book, those with problematic drinking behaviors are given the ability to envision a future without the problem.

Concrete, obtainable steps in reaching the envisioned future are laid out in this supportive read.

Available on Amazon .

2. Solution Focused Brief Therapy: 100 Key Points and Techniques – Harvey Ratney, Evan George and Chris Iveson

Solution Focused Brief Therapy: 100 Key Points and Techniques is a well-received book on solution-focused therapy. Authors Ratner, George, and Iveson provide a concisely written and easily understandable guide to the approach.

Its accessibility allows for quick and effective change in people’s lives.

The book covers the approach’s history, philosophical underpinnings, techniques, and applications. It can be utilized in organizations, coaching, leadership, school-based work, and even in families.

The work is useful for any practitioner seeking to learn the approach and bring it into practice.

3. Handbook of Solution-Focused Brief Therapy (Jossey-Bass Psychology) – Scott D. Miller, Mark Hubble and Barry L. Duncan

It includes work from 28 of the lead practitioners in the field and how they have integrated the solution-focused approach with the problem-focused approach.

It utilizes research across treatment modalities to better equip new practitioners with as many tools as possible.

4. More Than Miracles: The State of the Art of Solution -Focused Therapy  (Routledge Mental Health Classic Editions) – Steve de Shazer and Yvonne Dolan

It allows the reader to peek into hundreds of hours of observation of psychotherapy.

It highlights what questions work and provides a thoughtful overview of applications to complex problems.

Solution-Focused Therapy is an approach that empowers clients to own their abilities in solving life’s problems. Rather than traditional psychotherapy that focuses on how a problem was derived, SFT allows for a goal-oriented focus to problem-solving. This approach allows for future-oriented, rather than past-oriented discussions to move a client forward toward the resolutions of their present problem.

This approach is used in many different areas, including education, family therapy , and even in office settings. Creating cooperative and collaborative opportunities to problem solve allows mind-broadening capabilities. Illuminating a path of choice is a compelling way to enable people to explore how exactly they want to show up in this world.

Thanks for reading!

We hope you enjoyed reading this article. Don’t forget to download our three Positive Psychology Exercises for free .

• de Shazer, S. (1988). Clues: Investigating solutions in brief therapy. New York, NY: W.W. Norton and Co.
• Green, L. S., Oades, L. G., & Grant, A. M. (2006). Cognitive-behavioral, solution-focused life coaching: Enhancing goal striving, well-being, and hope. The Journal of Positive Psychology, 1 (3), 142-149.
• Miller, S. D., & Berg, I. K. (1996). The miracle method: A radically new approach to problem drinking. New York, NY: W.W. Norton and Co.
• Peterson, C., & Seligman, M. E. P., (2004).  Character strengths and virtues: A handbook and classification (Vol. 1). New York, NY: Oxford University Press.
• Santa Rita Jr, E. (1998). What do you do after asking the miracle question in solution-focused therapy. Family Therapy, 25( 3), 189-195.

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