Question

Question on Mean Value Theorum.

Answer 1

are you asking someone to answer this for you?

Answer 2

no, i'm asking for help

Answer 3

@hartnn @iPwnBunnies @Luigi0210

Answer 4

I may be wrong about this. I can't believe I can't remember this.
I believe you wanna first find the average rate of change on that interval, using the slope formula, which is provided.
Next, find the derivative of the function. Solve for x when the derivative is equal to the average rate of change in the interval. The x-value should be within that interval.

Answer 5

wait the average rate of change will act as the slope in the formula?

Answer 6

The average rate of change on an interval is the slope of the line created by connecting the two endpoints.

Answer 7

what value do I substitute for a and b

Answer 8

A will be the left most endpoint, when x = 0. B will be the right most endpoint, when x = pi/2. You'll also need the values of the original function.

Answer 9

I don't understand what to do.

Answer 10

\[f'(c) = \frac{f(\pi) - f(0)}{\pi - 0}\]
Plug in 0 and pi into the given function. The difference is the numerator. Then pi will be the denominator. The quotient will be the average rate of change on the interval.

Answer 11

Soz if I'm confusing you. /.\ Explaining calculus in text looks so bad.

Answer 12

@iPwnBunnies

Answer 13

I wonder if they meant sin(2x). o_o
They should've put parentheses.

Answer 14

2*sin(pi) + sin(2pi) = 0
So, they both equal 0, great.
That means, f'(c) = 0/(pi - 0) = 0

Answer 15

okay. do I have to do anything else now?

Answer 16

Yes, we have to find C. C is an x-value between 0 and pi. The slope of the line tangent to the point when x = C will be 0.

Answer 17

What we do now is find the derivative of the function, set it equal to 0, and solve for x.

Answer 18

you mean the derivative of f(x)?

Answer 19

Yes.

Answer 20

I think of the derivative as the equation that gives us the instantaneous slope of a point on the original function's graph

Answer 21

f'(x)=2(cos(x)+cos(2x))

Answer 22

Try the second term again. Use the chain rule.
sin(2x)'s derivative isss: 2*cos(2x)

Answer 23

We're not plugging in 0. We're solving for 0, because that's what the average rate of change is on the original function. We need to find that X value between 0 and pi.

Answer 24

I'm still getting that as the derivative.

Answer 25

OHHH, you factored out the 2. Gotcha :P

Answer 26

Now, set it equal to 0, and solve for x. This one might be a bit messy. >_>

Answer 27

0=2(cos(x)+cos(2x))

Answer 28

x=2pin+pi
x=(1/3)(6pin+pi)
x=(1/3)(6pin-pi)

Answer 29

Hmm, I'm not sure what you did. :o

Answer 30

Ok, now I see. This is a strange question. Apparently, the derivative is 0 at pi. Ugh, I hope I didn't screw up. I will graph the original function and check it real quick.

Answer 31

Ok, I was right. The derivative equals 0 at Pi and its odd multiples. \
This means the slope of the line tangent to the original function at x=pi is 0.

Answer 32

so what am I writing exactly?

Answer 33

It says Apply the Mean Value Theorem. I suppose to write out the steps you did. :3
Then write a conclusion.

Answer 34

yes but what is the answer im confused?

Answer 35

It's an explanation of the Mean Value Theorem. On a closed interval of a function, there exists a value for the average rate of change. Within the interval, there is one point where the instantaneous slope of the tangent line equals the average rate of change.

Answer 36

http://www.sosmath.com/calculus/diff/der11/der11.html