2-2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Chapter 3 Derivatives
83
Thus
Thus, tanx+ cotx-
Example 9: Suppose that f(0) --3 and f(x)55 for all values of x. How large can f(2) possibly be?
Solution: Suppose that
f(x) 55
This implies f is differentiable (and therefore is continuous) everywhere.
f(0) --3
and
for all values of x.
In particular, we choose f is defined on [0, 2]. Then by Mean Value Theorem on (0, 2], there
exists a number c in (0, 2) such that
f(2) – f(0) _ f2)*
f'(c) =
2-0
- f(2) = 2f'(c) – 3< 2(5) - 3 = 10-3 =7.
This shows the largest possible value for f(2) is 7.
Exercise 3.6
Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given
interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem.
1.
a. f(x) = 5 – 12x + 3x², in [1, 3]
b. f(x) = x³ – x² – 6x + 2, in [0, 3]
c. f(x) = cos 2x, in [n/8, 7n/8]
Let f(x) = 1 - x2/3, Show that f(-1) = f(1) but there is no number c in (-1, 1) such that f'(c) = 0.
Why does this not contradict Rolle's Theorem?
2.
3.
Let f(x) = tanx. Show that f(0) = f(T) but there is no number c in (0, t) such that f'(c) = 0. Why
does this not contradict Rolle's Theorem?
Verify that the function satisfies the three hypotheses the Mean Value Theorem on the given
interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem.
4.
a. f(x) = 2x² – 3x + 1, in [0, 2].
b. f(x) = x³ + x -1 [0, 2]
C f(x) =e-2, in [0, 3]
d. f(x) =+2 in [1, 4).
Let f(x) = (x - 3)-2. Show that there is no value of c in (1, 4) such that f(4) – f(1) = f'(c) (4 – 1).
Why does this not contradict the Mean Value Theorem?
6.
Show that the equation x3 - 15x + c = 0 has at most one root in the interval [-2, 2].
7. If f(1) = 10 and f'(x) 2 for 1<x<4, how small can f(4) possibly be?
8.
Use the Mean Value Theorem to prove the inequality sina - sinb| < la -b|for all a and b.
9.
Use the method of Example 3 to prove the identity 2 sin-lx = cos-'(1 – 2x²) for x 2 0.
43)
5ha)d
ne theoremn G Soie
Heror de
(0.3) Surh hat
(3) - )
%3D
3-
-2
Heorem t ver1d
Given funetlon s $m)= Cocen
polynomi al funeton Is
= -2 sine
Teal n and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F128b56e9-7d96-404e-a92a-7b54b4ed5ac6%2Fd71d1dc1-0ebc-435d-b26d-3bcb7f5b139f%2Fj705by_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Chapter 3 Derivatives
83
Thus
Thus, tanx+ cotx-
Example 9: Suppose that f(0) --3 and f(x)55 for all values of x. How large can f(2) possibly be?
Solution: Suppose that
f(x) 55
This implies f is differentiable (and therefore is continuous) everywhere.
f(0) --3
and
for all values of x.
In particular, we choose f is defined on [0, 2]. Then by Mean Value Theorem on (0, 2], there
exists a number c in (0, 2) such that
f(2) – f(0) _ f2)*
f'(c) =
2-0
- f(2) = 2f'(c) – 3< 2(5) - 3 = 10-3 =7.
This shows the largest possible value for f(2) is 7.
Exercise 3.6
Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given
interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem.
1.
a. f(x) = 5 – 12x + 3x², in [1, 3]
b. f(x) = x³ – x² – 6x + 2, in [0, 3]
c. f(x) = cos 2x, in [n/8, 7n/8]
Let f(x) = 1 - x2/3, Show that f(-1) = f(1) but there is no number c in (-1, 1) such that f'(c) = 0.
Why does this not contradict Rolle's Theorem?
2.
3.
Let f(x) = tanx. Show that f(0) = f(T) but there is no number c in (0, t) such that f'(c) = 0. Why
does this not contradict Rolle's Theorem?
Verify that the function satisfies the three hypotheses the Mean Value Theorem on the given
interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem.
4.
a. f(x) = 2x² – 3x + 1, in [0, 2].
b. f(x) = x³ + x -1 [0, 2]
C f(x) =e-2, in [0, 3]
d. f(x) =+2 in [1, 4).
Let f(x) = (x - 3)-2. Show that there is no value of c in (1, 4) such that f(4) – f(1) = f'(c) (4 – 1).
Why does this not contradict the Mean Value Theorem?
6.
Show that the equation x3 - 15x + c = 0 has at most one root in the interval [-2, 2].
7. If f(1) = 10 and f'(x) 2 for 1<x<4, how small can f(4) possibly be?
8.
Use the Mean Value Theorem to prove the inequality sina - sinb| < la -b|for all a and b.
9.
Use the method of Example 3 to prove the identity 2 sin-lx = cos-'(1 – 2x²) for x 2 0.
43)
5ha)d
ne theoremn G Soie
Heror de
(0.3) Surh hat
(3) - )
%3D
3-
-2
Heorem t ver1d
Given funetlon s $m)= Cocen
polynomi al funeton Is
= -2 sine
Teal n and
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