The Ultimate Guide to Solving Calculus Word Problems

Many math-phobic readers cringe at the sight of calculus word problems. There’s so much information given, so many problem types, and so many possible strategies to apply.

Where do you even start!?

That’s why I’ve put together this ultimate guide to solving calculus word problems. Calculus is sometimes regarded as the most challenging of the math disciplines. But, with this guide, it doesn’t have to be!

Here I will share step-by-step tips and basic approaches to calculus word problems that you can use to solve any problem. I will also walk you through a set of sample problems for each problem type you will encounter in your studies of calculus!

It’s time to demystify these difficult problems!

Step-By-Step Tips and Basic Approaches to Calculus Word Problems

Calculus word problems will look different depending on the course you are in and the overall plan of the program you are studying in. You could be facing limits, continuity, derivatives, integral calculus, differential equations, and more! Calculus is a big complex topic and each curriculum covers it to a different extent.

But, regardless of the calculus word problems you are solving, there are a few basic problem-solving approaches you should consider taking. These step-by-step tips will help you solve any calculus word problems that you encounter!

the best problem solving method read reflect select

Read the problem carefully: Reading (and re-reading) the problem is important. This helps you look for key information, as well as important equations and quantities. Highlight them to help them stand out!
Reflect on any new mathematical principles have you learned: Are you studying differential calculus or integral calculus? Knowing the difference between the two will help you select an appropriate problem solving strategy.
Select a strategy for the problem type: Read the problem again for any prompts that direct you toward a specific strategy or any related theorems. For example, optimization problems ask you to find the maximum of some quantity. Rate of change problems usually ask about how quickly a quantity changes. Sample problems of each of these are provided below!

Calculus Word Problems Sample Problems

As you review this collection of basic sample problems, consider the step-by-step tips outlined above. Focus your thinking on how you can use these tips to help you progress with the problem!

Basic Problem: Rate of Change

When it comes to traditional calculus first-course content, rate of change calculus word problems are quite common. The following sample problem will show you how to apply derivatives to solve a rate of change problem.

A boat is traveling along a straight path on the surface of the water in a lake. The boat’s position at time t is given by the function $s(t)=2t^3−5t^2+3t+10$, where $s(t)$ is measured in meters and t is measured in seconds. Determine the boat’s instantaneous velocity at $t=3$ seconds.

Reading the problem carefully, we can see that we are given a position function and are asked to find instantaneous velocity. The instantaneous velocity at a specific time t is given by the derivative of the position function given. We can apply the power rule to find the derivative:

$$ \begin{split} v(t)=&\frac{ds}{dt} \\ \\=&\frac{d}{dt}(2t^3−5t^2+3t+10) \\ \\ =&6t^2−10t+3 \end{split}$$

To find the instantaneous velocity a time t = 3 seconds, we substitute t = 3 into the derivative function:

$$ \begin{split} v(3)=&6(3)^2−10(3)+3 \\ \\ = &27 \end{split} $$

Therefore, the instantaneous rate of change of the boat on the surface of the water at t = 3 seconds is 27 m/s.

You can learn more about instantaneous rate of change by checking out my instantaneous rate of change guide!

Difficult Calculus Word Problems: Optimization Problems

When I was a math student, I remember some of the most challenging advanced problems in differential calculus being optimization problems. From factoring to applying the first derivative test, there were so many steps that I struggled to even start the problem!

In the video below, I will show you step-by-step solutions that can be applied to solve optimization calculus word problems!

Integral Calculus

One of the tricks to integral calculus problems is recognizing that you are being asked to go backwards from a function given. Remembering that the integral is the opposite of the derivative will help you get started. Consider the following problem:

A particle moves along a straight line. Its velocity at time t is given by the function $v(t)=3t^2 – 2t + 5$, where v(t) is measured in meters per second and t is measured in seconds. Find the displacement of the particle on the interval $[0, 4]$.

Since velocity is the derivative of displacement, we need to integrate the velocity function v(t) in order to find a displacement function for the particle.

$$ \begin{split} s(t)=& \int v(t) dt \\ \\ =&\int 3t^2 – 2t + 5 dt\\ \end{split} $$

From here, we can integrate each term in the expression for v(t) and apply the power rule for integration:

$$ \begin{split} s(t)=& \int 3t^2 dt – \int 2t dt+\int 5 dt \\ \\ =& t^3 -t^2+5t+C\\ \end{split} $$

Lastly, we can find the displacement over the interval [0, 4] by evaluating $s(4) – s(0)$:

$$ \begin{split} &s(4)-s(0) \\ \\ =& (4)^3 -(4)^2+5(4)+C – (0^3 -0^2+5(0)+C) \\ \\ =& 68\end{split} $$

Therefore, the displacement of the particle over the time interval [0, 4] is 68 meters.

Reflecting on Calculus Word Problems

As a math student, I always found word problems to be the hardest mathematical problems. I always felt like something was missing to get me started on a solution.

As a math teacher of many years, I now have my own set of step-by-step tips that I can share with you when you feel the same way! And when you use helpful strategies like these, calculus isn’t really that hard!

Which of these problems do you find the most challenging? I hope these basic approaches help you with solving real-world calculus word problems!

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