Calculus I - Relative Growth Rate help

[Orcrin12]

Here is a picture of the problem:

View attachment 8528

I am firmly stuck on part b). I know that relative growth rate is defined at f'(x)/f(x), but I cannot seem to come up with the right answer. I've tried everything I could conceivably think of. I do not understand why my answer is incorrect, unless I'm completely missing something.

The answers you see in the box are my values for P'(11)/P(11) and P'(21)/P(21). They are wrong.

I have P'(x) = (18.75e^-0.025x)/(5+10e^-0.025x)^2

Please help!

 

[jvicens]

Hello there,

I understand your struggle with part b) of this problem. Let me try to help you out.

First of all, you are correct in your understanding of the relative growth rate formula: f'(x)/f(x). However, in this case, the function P(x) is given to you as P(x) = 5 + 10e^-0.025x. This means that P'(x) = -0.25e^-0.025x.

Now, to find the relative growth rate at x = 11, we need to plug in the value of x = 11 into both P'(x) and P(x). This gives us P'(11) = -0.25e^-0.275 and P(11) = 5 + 10e^-0.275.

Therefore, the relative growth rate at x = 11 is (-0.25e^-0.275)/(5 + 10e^-0.275). Using a calculator, this comes out to be approximately -0.0027.

Similarly, to find the relative growth rate at x = 21, we plug in x = 21 into P'(x) and P(x). This gives us P'(21) = -0.25e^-0.525 and P(21) = 5 + 10e^-0.525.

Therefore, the relative growth rate at x = 21 is (-0.25e^-0.525)/(5 + 10e^-0.525). Using a calculator, this comes out to be approximately -0.0013.

I hope this helps you understand where you went wrong. Always remember to plug in the specific values of x into the functions before calculating the relative growth rate.

Best of luck with your studies!
Scientist