Question

Hey, so I'm having trouble with this AP Calculus problem and I was wondering if anyone would explain why the answer is what it is? I'll attach the document- I don't understand Question 5, part b.

Answer 1

this looks like a differential equation to solve

Answer 2

oh silly me the solutions are given! i didn't scroll down that far

Answer 3

I don't understand why the second derivative of W being greater than zero proves part (a) to be an underestimate.

Answer 4

And honestly, I don't really understand part (c) at all either...

Answer 5

part b i can explain

Answer 6

the second derivative being positive means the function is concave up
the real definition of concave up is that any tangent line lies below the graph
or at least the real definition when you write down what exactly the above means
in any case a picture will show that if the function is concave up the line lies below the graph

Answer 7

Oh, wow, I never knew that. I'm glad to know that now. Thank you very much!

Answer 8

actually i lied, the real definition of concave up is that any secant line lies above the curve

Answer 9

but both things are true

Answer 10

yw

Answer 11

Awesome :) Yeah, I didn't know that so when I saw the problem's explanation I just felt even more confused.

Answer 12

as for C solving the differential equation, the steps are there in the solution
if you haven't seen them before it could be confusing, but they are all there i think

Answer 13

basically any time you have
\[f'(x)=f(x)\] or some expression involving \(f\) (in this case it is \(f'(x)=\frac{1}{25}(f(x)-300)\) then the solution will be the exponential in some form

Answer 14

I think I need to review separation of variables and differentiable equations. I'll keep that in mind- I have trouble trying to pick my first step so that'll definitely help. Thank you :)