Thomas Calculus 13th [Solutions]

Section 4.2 The Mean Value Theorem 233
4.2 THE MEAN VALUE THEOREM
1. When
2
f ( x) x 2x 1 for 0 x 1, then
f (1) f (0)
f ( c) 3 2c 2 c 1 .
1 0 2
2. When
2/3
f ( x)
x for 0 x 1, then
f (1) f (0)
1/3
f ( c) 1 2 c c 8 .
1 0
3 27
3. When f ( x)
x 1 for 1 x 2
x 2, then
f (2) f (1/2)
2 1/2
f ( c) 0 1 1 c 1.
2
c
4. When f ( x) x 1 for 1 x 3, then
f (3) f (1)
2
f 1 3
3 1
( c ) c
2 2 1 2
.
c
5. When
1
f ( x) sin ( x ) for 1 x 1, then
2
c 1 4 c 1 4 0.771
2 2
( ) f (1) f ( 1) 1 2 2
2 2 2
f c 4
1 ( 1) 2 2
1 c 1 c
2
1 c
6. When f(x) = ln(x 1) for 2 x 4, then
f (4) f (2)
f ( c) 1 ln 3 ln1 c 1 2 c 1 2 2.820
4 2 c 1 2 ln 3 ln 3
3 2
f (2) f ( 1)
2 1 7
7. When f ( x)
x x for 1 x 2, then
f ( c)
2 3c 2 c c .
2 ( 1)
3
1 7 1 7
1.22 and 0.549 are both in the interval 1 x 2.
3 3
3
x 2 x 0 g(2) g( 2)
8. When g( x) , then
g ( c) 3 g ( c ). If 2 x 0, then g ( x)
2
2 ( 2)
x 0 x 2
2
3c 3 c 1. Only c 1 is in the interval. If 0 2,
9. Does not; f ( x ) is not differentiable at x 0 in ( 1, 8).
2
3x
3 g ( c)
x then g ( x) 2x 3 g ( c) 2c 3 c 3 .
2
10. Does; f ( x ) is continuous for every point of [0, 1] and differentiable for every point in (0, 1).
11. Does; f ( x ) is continuous for every point of [0, 1] and differentiable for every point in (0, 1).
12. Does not; f ( x ) is not continuous at x 0 because
13. Does not; f is not differentiable at x 1 in ( 2, 0).
lim f ( x) 1 0 f (0).
x 0
14. Does; f ( x ) is continuous for every point of [0, 3] and differentiable for every point in (0, 3).
15. Since f ( x ) is not continuous on 0 x 1, Rolles Theorem does not apply:
lim f ( x) lim x 1 0 f (1).
x 1 x 1
16. Since f ( x ) must be continuous at x 0 and x 1 we have lim f ( x) a f (0) a 3 and
x 0
lim f ( x) lim f ( x) 1 3 a m b 5 m b . Since f ( x ) must also be differentiable at x 1
x 1 x 1
we have lim f ( x) lim f ( x)
2x 3| x 1 m| x 1 1 m . Therefore, a 3, m 1and b 4.
x 1 x 1
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