Unsure how to do these calculus word problems

[Lunethis42]

Hey, I'm having problems figuring out how to do these word problems. I'm not very good at integrals to begin with and I'm not sure what the questions are asking for, much less how to solve them. Any help at all would be appreciated, and thanks for your time.

 

[HallsofIvy]

These appear to be problems involving differentiating exponentials- \(\displaystyle f(x)= a^x\). Do you know how to do that?

One idea is to take the logarithm of both sides: \(\displaystyle log(f(x))= log(a^x)= xlog(a)\). Differentiating both sides \(\displaystyle \frac{1}{f(x)}\frac{df}{dx}= log(a)\). So \(\displaystyle \frac{df}{dx}= f(x)log(a)= log(a)a^x\). (The logarithm here is the natural logarithm.)

 

[Dr.Peterson]

Hey, I'm having problems figuring out how to do these word problems. I'm not very good at integrals to begin with and I'm not sure what the questions are asking for, much less how to solve them. Any help at all would be appreciated, and thanks for your time.

In order to help, we need to know what kind of problems you are having. Apparently it is not necessarily in the integration or differentiation, but interpreting the question. It will help a lot if you tell us what you think they are asking and show how you might start, just giving it your best try, so we can see how close you are, and where you need help.

In particular, don't just assume it's about integration and look for a place to do that; let the problem itself suggest what to do. Often I start by paraphrasing it in my mind, or visualizing what is happening, maybe with a sketch. Anything that makes the story more concrete will probably help you think about it.