Question

I need help on part C and D, how do I go about in solving?

Answer 1

@myeyeshurt

Answer 2

Write a closed form expression for the area for -5 to c. It should be the sum of a few definite integrals

Answer 3

From there divide by the interval c-(-5) to get the average value. You can set this equal to 1/2 as the problem requests

Answer 4

Let me know if you need more detail

Answer 5

So like this?

\( \frac{1}{c+5}\left(\left(\int_{-5}^{-3}f\left(x\right)dx\right)+\left(\int_{-3}^{-2}f\left(x\right)dx\right)+\left(\int_{-2}^{0}f\left(x\right)dx\right)+\left(\int_{0}^{c}f\left(x\right)dx\right)\right)=\frac{1}{2} \)

Answer 6

And then I evaluate the first few using geometry?

\(\frac{1}{c+5}\left(\pi+1+2+\int_{0}^{c}f\left(x\right)dx\right)=\frac{1}{2}\)

Answer 7

How do I solve for C now?

Answer 8

Use FTC corrollary

Answer 9

The integrand is F(c)-F(0)

Answer 10

When evaluated

Answer 11

But you can also just use 1/2 b*h since it's a triangle

Answer 12

Here b=(c-0)=c and h is the y intercept

Answer 13

Ohh I see, let me give that a try

Answer 14

I tried using FTC but I was confused bc I didn't know what F(x) was

Answer 15

So the area of that is just c because length is c and width is 2

Answer 16

\(\frac{1}{c+5}\left(\pi+3+c\right)=\frac{1}{2}\)

I got it now, thanks man. Do you have a suggestion for part D as well? Thank you very much once again

Answer 17

Hold on, unless I did something wrong in the last step, I got a negative value for c.

Answer 18

Nevermind, silly me, the area for the bottom bit is negative of course. Realized my mistake. So back to D, how do I do that?

Answer 19

So h(6)=f(3) and you want h'(6) or f'(3)

Answer 20

Since c is greater than 3

Answer 21

It's just the slope of f

Answer 22

Oh okay I got it, thanks. I find the slope with (c,0) and (0,2).

Answer 23

I appreciate it brother thank you. God bless

Answer 24

Yup!

Answer 25

Wait you have to multiply by one half

Answer 26

Cause of the chain rule

Answer 27

Oh from the x/2? why would that matter?

Answer 28

d/dx f(x/2)=1/2 f'(x/2)

Answer 29

Oh okay I see. Thank you once again.

Answer 30

One more thing, for part b is the answer -3 and 0, because the inflection point is the change in signs for g'(x) or f(t)?

Answer 31

Idr any second derivative stuff tbh :(

Answer 32

it's okay bro :) thank you