Calculus Word Problems

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Calculus word problems give you both the question and the information needed to solve the question using text rather than numbers and equations. You’ll find a variety of solved word problems on this site, with step by step examples. Some have short videos. Click next to the type of question you want to see a solution for, and you’ll be taken to an article with a step be step solution:

Question Example	Step by Step Solution Link
Minimum profits for company x, whose profit function is f(t) = 100t2 – 50t + 9,	Minimum Profits
Find the profit equation of a business with a revenue equation of 2000x – 10×2 and a cost equation of 2000 + 500x.,	Maximum Profits
Volume of the Largest Rectangular Box	Volume of Largest Box
Find the maximum area of a rectangle whose perimeter is 100 meters.	Optimization
A ball rolls along with a position of x(t) = 0.000015t5– 0.004t3 + 0.4t. What is the instantaneous velocity?	Instantaneous Velocity
Find the volume of an egg	Volume of an Egg
A textbook falls from a window. The book will dent your car if it’s going more than 100 feet per second. Will your car get dented?	Velocity of a Falling Object
A rock is dropped into the center of a circular pond. The ripple moved outward at 4 m/s. How fast does the area change, with respect to time, when the ripple is 3m from the center?	Related Rates
The length of a rectangular drainage pond is changing at a rate of 8 ft/hr and the perimeter of the pond is changing at a rate of 24 ft/hr. At what rate is the width changing?	Related Rates

Don’t see your question listed? Go to our calculus problem solving index. You can also Use the search box at the top left of the page. We have hundreds of solved questions on the site, so yours might be there!

Calculus Word Problems: General Tips.

Step 1:Think about what you learned the previous week in class. Integration? Derivatives? Related rates? Your homework is going almost always be based on what you learned in class the week before, so skim through your notes and/or textbook and look for any relevant formulas.

Step 2: Read through the word problem carefully. Don’t skim or skip over phrases and sentences that may seem unimportant. You may miss details that change the entire meaning of the passage. Once you’ve read through the problem once, write down the answer that the question is asking for. Note the format that you need to give the answer in. For example, you might have to give certain units.

Step 3: Break the word problem down into individual statements about the scenario described in the word problem. Record any numbers or equations given in these statements. Note the units of measurement.

Step 4: Either extract the main equation from the word problem or build the equation using the information presented in the statements. Fill in as much of the equation as you can using the information from the word problem.

Step 5: Solve the equation to produce any answer needed to satisfy calculus word problems.

Step 6: Double-check your calculations and consider whether your answer makes sense in the context of the word problem.

Tips:

• Be sure to state the answer in the format in which it was requested.
• If the equation remains unsolvable, you may need to extract or build more equations using the rest of the available but unused information to answer the empty spaces in your main equation. This is one of the more difficult parts of solving calculus word problems. It’s also one of the most important. Having a solid understanding of calculus, particularly the fact that derivatives represent the rate of change of the equation, will help you when creating the necessary equations.