Question

Verify that the functions satisfy the hypotheses of the Mean Value Theorem on the given interval. The find all numbers that satisfy the conclusion of the Mean Value Theorem.
a) f(x)=(0.5)x+sin(x) [0,2pi]

b) f(x)=2x+sin(2x) [0,pi]

Answer 1

1. find the values at the endpoints
2. find the slope of the straight line connecting the end point:
\[m = \frac{f(x_1)-f(x_2)}{x_1-x_2} \]

3. then you have to show that there is at least one point over this interval where the derivative is equal to that slope, so you set:

\[f'(x)=m\]

and solve

Answer 2

What do you mean find the values at the endpoints?

Answer 3

your given an interval to work on, [0,2pi], so the endpoints would be 0 and 2pi

Answer 4

ohh and do I plug each one as x and that would give me x1 and x2?

Answer 5

then I find the slope using that?

Answer 6

yeah.
The mean Value Theorem is basically this:
-If your given a certain interval of a function, that interval will have an average slope (to get from one endpoint to the other)
-We can then say that the derivative function will have to equal that average at some point.

this picture illustrates this fact:
http://upload.wikimedia.org/wikipedia/commons/thumb/6/61/Lagrange_mean_value_theorem.svg/438px-Lagrange_mean_value_theorem.svg.png

Answer 7

thank you!

Answer 8

yeah, find the slope using:

\[m=slope=\frac{rise}{run}=\frac{\Delta y}{\Delta x}\]

Answer 9

you have to find all the values of x, at which the derivative is equal to the average slope

Answer 10

Okay, I'll try it. Thank you!

Answer 11

best of luck

Answer 12

@Mathmuse Okay, I have a question!

Answer 13

@monroe17 shoot