There are many uses for derivatives in the natural and social sciences, including physics, chemistry, biology, and economics, just to name a few. You will encounter a lot of word problems dealing with these subjects and how they use derivatives. Most people become weary right after reading the problem because the physics/chemistry/biology/economics wording can make the question and information difficult to understand. Don't let that scare you off, though! You don't need to know these other subjects in order to work these word problems, and I'll show you why.
First, let's talk about the typical format. You will (almost always) be given an equation with variables that represent a numerical value in that field of study. Then, you will be asked a specific rate of change question, either to find a formula for the rate of change, or to find the rate of change at a particular variable value. Sometimes you will be asked to explain what the rate of change means in the particular situation.
That's it! Doesn't sound too hard, does it? Let's look at some examples. First, a physics problem.
I know this is a loaded problem, so we'll take it one step at a time! First, notice the format. You don't need to know anything about physics to work this problem, but you do need to understand derivatives. We are given an equation that represents the particle's position to the time. We know this because the variable x is given in meters, which is a position unit.
In the first question, we are asked for the velocity at time t. This means we need to find a formula for the rate of change of position with respect to time (see your textbook for the definition of velocity). We simply need to find the derivative of the position function with respect to time.
Part (c) might be a challenge if you've never been faced with something like it before. The term "at rest" means just what it sounds like--"not moving." We are asked to find when the particle is not moving. If a particle is at rest, then its position is not changing. Therefore, the change in s is 0, but time is still passing, so the change in t is a value other than 0. So then, it's velocity is 0 divided by some number--which is 0. Thus, when a particle is at rest, its velocity is zero. All we need to do here, then, is find when the velocity of this particle is zero.
Part (d) might also be a challenge for you. Again, you don't need to know physics to solve this problem, you just need to use your brain! Let's think of an example situation. Envision a particle on top of a ruler at the 0 meters mark. That particle moves forward 1 meter in 1 second. The particle's average velocity is the (second position minus the first position) divided by the total time,which comes out to (1-0)/1, or 1 m/s, and the value is positive. Now the particle moves backward 1 meter in 1 second. The average velocity now is (0-1)/1, or -1 m/s. It is negative because the particle moved from the 1 meter mark to the 0 meter mark, instead of the other way.
I hope you can see now that when a particle moves forward, it has a positive velocity, and when it moves backward, it has a negative velocity. In this way, velocity is different than speed.
For this question then, we need to find at what time interval the particle's velocity is positive. To do so, we can look at the graph of the velocity function, or we can find the roots of the velocity function and then determine where the velocity is positive. Notice that we already found the roots of the velocity function in part (c).
That wasn't too hard, right? Now let's look at a problem from economics.
Now onto this problem. We are first asked to find the marginal cost function, which is simply the derivative of the cost function given to us.
If you have a word problem type not covered here that you need help with, leave a comment below. If you have any other questions or comments or suggestions for future topics, let me know!