Question

Calculus

Answer 1

Start at x =0. Is f(x) above the x-axis?

Answer 2

some of it is

Answer 3

I think it's decreasing at (-2,0]U[2,4]
is that correct?

Answer 4

There is no "some of it" AL x = 0. It either is or isn't.

Answer 5

f(x) is decreasing on (-2,0).
Does this say anything about F(x)?

In calculating the Riemann Integral, where does f(x) contribute positive area to F(x)?

Answer 6

(-2,2]

Answer 7

and (5,6)

Okay, now consider the definition of F(x). What portion on (0,6) contributes positive area?

Answer 8

(-2,2]U (5,6)

Answer 9

No, you did not consider the definition of F(x). You thought only about f(x). F(x) doesn't exist until x = 0. Try again.

Answer 10

[0,6)

Answer 11

Think a little harder about (3,5). Does f(x) contribute POSITIVE area to F(x) on (3,5)?

Answer 12

no, it has a negative area

Answer 13

so it's positive from (0,2)U(5,6]?

Answer 14

Now, answer the question. Where is F(x) DECREASING?

Answer 15

(3,5)?

Answer 16

Done!

Answer 17

Can you help me on part 2? @tkhunny

Answer 18

Can you find the first derivative of F(x)?

Answer 19

Isn't it just the slope?

Answer 20

The slope of what? We don't have a picture of F(x).

The picture of f(x) IS the first derivative of F(x). You need to see this.

Answer 21

oh ok. The only place I see it is concave down is at (1,3)

Answer 22

NO, you're not quite seeing what you need to see.

You are trying to answer where f(x) is concave down. That is not the question. The question is when is F(x) concave down?

You still haven't answered the question. Do you see the relationship?

F(x) is defined in terms of f(x).
The first derivative of F(x) is F'(x) = f(x).
The second derivative of F(x) is F"(x) = f'(x).

In order to answer questions about the concavity of F(x), you need the second derivative of F(x). You need F"(x) = f'(x).

The second derivative of F(x) is the slope of f(x). Where is the slope of f(x) negative?

Answer 23

(2,4)

Answer 24

I think you have it.

This is a very difficult problem. It is very hard to keep track of what you are talking about. F(x) and f(x) are related, but they should not be confused.

Answer 25

Thank you so much for the help! And sorry if I frustrated you for not understanding this. @tkhunny

Answer 26

No frustration at all. This is very abstract. it is likely that it will take a few tries to get the hang of it. :-)