Question

For f(x) = cos(x), find all the values of c on the interval (0, pi) that satisfy the Mean Value Theorem.

Answer 1

@electrokid can u help?

Answer 2

1. find f'(x)

Answer 3

@onegirl write down the first derivative

Answer 4

okay

Answer 5

hold on

Answer 6

its -sin(x)

Answer 7

good.
now, by MVT,
\[f'(x)=\frac{f(b)-f(a)}{b-a}\]
where [a,b] = [0,2pi] your domain

Answer 8

okay so 0 is a and pi b

Answer 9

so,
\[-sin(c)=\frac{\cos(2\pi)-cos(0)}{2\pi-0}
\]

Answer 10

yes

Answer 11

so, what is sin(c) ?

Answer 12

what is it or what equals to it?

Answer 13

yes. what does it equal to?

Answer 14

0

Answer 15

perfect.
\[\sin(c)=0\]
for what values of "c" in the range of [0,2pi] is sin(c)=0?

Answer 16

huh? lol sorry i didn't get that last part

Answer 17

\[\sin(c)=0\]
this can happen only for \(c=0,\pi,2\pi,3\pi,...n\pi\)
but, we are looking only in the range of [0,2pi] (both EXCLUSIVE)

Answer 18

ohh ok so 0, pi, and 2pi?

Answer 19

the interval notations:
\[
[a,b]\qquad\text{closed interval includes both a & b}\\
(a,b)\qquad\text{open interval does not include a & b}
\]

Answer 20

what does your question give you? open interval or closed?

Answer 21

closed

Answer 22

so, there you go

Answer 23

you got your three possible values for "c"

Answer 24

okay but look at these multiply choices for this question.

A. c = 0.6901 and 2.4515
B. c = 0
C. c = -2/pi
D. c = -0.6901 and -2.4515

Answer 25

so it will be A?

Answer 26

we have
\(c=\{0,3.14,6.14\}\)
from the options, only B has one of our values

Answer 27

ok

Answer 28

so maybe your interval is something like this:
\(0\le c<\pi\)
this means, that "0" is included but not "pi"

Answer 29

okay

Answer 30

no wait..

Answer 31

just saw in the question...
you domain is \([0,pi)\)
not 2pi like we did

Answer 32

\[\sin(c)=\frac{\cos(\pi)-\cos(0)}{\pi-0}\]

Answer 33

ohh yes it is!

Answer 34

so sin(c) is 0

Answer 35

no
\(\cos(\pi)=-1\qquad\cos(0)=1\)
\[-\sin(c)=\frac{-1-1}{\pi}
\]
solve for "c"

Answer 36

careful with the signs

Answer 37

okay

Answer 38

hold on

Answer 39

I got c = 0.6901 and 2.4515 which will be A

Answer 40

perfect

Answer 41

okay thanks