Chapter 3 – Mean Value Theorem WKST
Name:
A Graphing Calculator is allowed and NEEDED for these problems.
For problems #1 to #6:
(a) Determine if the function satisfies the two hypotheses of the Mean Value Theorem on the indicated interval.
(b) If the function satisfies the hypotheses of the Mean Value Theorem, find the value of f ‘ (c) guaranteed by
the theorem on the indicated interval.
1.
f ( x) 
1 on [ 1 , 3 ]
x
2.
(a)
(b)
5.
f ( x)  x 3 on [ 0 , 1 ]
(a)
(b)
f ( x)  x  1 on [ 1 , 3 ]
(a)
(b)
x on [ 1 , 3 ]
x2
(a)
(b)
2
3.
f ( x) 
4.
6.
f ( x)  x  2 on [ 1 , 3 ]
(a)
(b)
f ( x) 
(a)
(b)
( x  1) 2
x 1
on [ 1 , 3 ]
7. Which of the following satisfy the hypotheses of Rolle’s Theorem on the interval [ 0 , 2 ] ?
I.
f ( x) 
II.
f ( x)  x  1
1
x 1
III. f ( x)  x 2  2 x
(A) I only
(B) II only
(C) III only
(D) II and III
(E) I, II, and III
8. Which of the following satisfy the hypotheses of Mean Value Theorem on the interval [ 0 , 2 ] ?
I.
f ( x)  sin  x  cos 2 x
II.
f ( x)  3 x  8
III. f ( x)  x 2  2 x
(A) I only
(B) II only
(C) III only
(D) I and II
(E) I, II, and III
9. How many values of c satisfy the conclusion of the Mean Value Theorem for f ( x)  x 3  1 on the
interval [ -1 , 1 ] ?
(A) 0
(B) 1
(C) 2
(D) 3
10. How many values of c are guaranteed by Rolle’s Theorem for f ( x) 
(A) 4
(B) 5
(C) 6
(E) 4
sin x
on the interval [ -10 , 10 ] ?
x
(D) 7
(E) the theorem
does not apply
11. How many values of c are guaranteed by Rolle’s Theorem for f (x) below on the interval [ -10 , 10 ] ?
 sin x

f ( x)   x

 1
(A) 4
(B) 5
, if x  0
, if x  0
(C) 6
(D) 7
(E) the theorem
does not apply
ANSWERS:
1a)yes
b) 3
2a)no
b)none
3a)yes
b)8/27
4a)yes
b)1
5a)yes
b)all real
7)C
8)E
9)C
10)E
11)B
6a)no
b)none