Math
Inverse Trigonometric functions
12. [4/7 Points]
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Tutorial Exercise
Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given
interval.
TT
f(x) = cos(x),
[:)
3
Step 1
Recall that the Mean Value Theorem is stated as follows.
If fis continuous on the closed interval [a, b], then there exists a number c in the closed interval [a, b] such
b
that
* S**
f(x) dx = f(c)(b  a).
In terms of area, this means there is a rectangle with side lengths f(c) and (b  a) that has the same area as
found by
f(x) dx.
We are given the function f(x) = cos(x) and the interval
3 3
The function f(x) is is continuous on the given interval. Therefore, we must find the value of c in

[
that makes the following equation true.
3 3
2/3
L
cos(x) dx = cos(c)(
/3
3
cols
Step 2
We will first evaluate the given integral.
#/3
1/3
(cos(x)) dx =
1/3
(sin(x)]
 1/3
NT
= sin
 in()
w
=
Step 3
We have evaluated the given definite integral. Substitute this value on the left side of the equation and then
solve for f(x) = cos(c) in the Mean Value Theorem.
1/3
(cos(x)) dx = cos(c)
#/3
*
= cos(1)()
V3  costil(3)
V3 = cos(c)( V3
27
3
0.5973
3V3
27
= cos(C)
Step 4
Solve for c. Use inverse trigonometric functions to find the value of c guaranteed by the Mean Value Theorem
for the function over the given interval. (Round your answers to four decimal places. Enter your answer as a
commaseparated list.)
313
cos(C)
21
=
arccos(cos(c)) = arccos
cos(325)
C =
89.1729,270.8271 X
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