To illustrate the Mean Value Theorem with a specific function, let's consider f(x) = x3 - x, a = 0, b = 8. Since f is a polynomial, it is continuous and differentiable for all x, so it is certainly continuous on [0, 8] and differentiable on (0, 8). Therefore, by the Mean Value Theorem, there is a number c in (0, 8) such that f(8) f(0) = f'(c)(8-0). Now f(8) 504 f(0) = 0 , and f'(x) = 3n² - 1 , so this equation becomes 7 = f'(c)(8) = 0 (8) - 504 which gives c2 21.33 , that is, c = ± 4:6188 . But c must be in (0, 8), so c = 4.62 The following figure illustrates the calculation that the tangent line at this value of c is parallel to the secant line. 700 600 500 400 300 200 100 8 Ⓡ
To illustrate the Mean Value Theorem with a specific function, let's consider f(x) = x3 - x, a = 0, b = 8. Since f is a polynomial, it is continuous and differentiable for all x, so it is certainly continuous on [0, 8] and differentiable on (0, 8). Therefore, by the Mean Value Theorem, there is a number c in (0, 8) such that f(8) f(0) = f'(c)(8-0). Now f(8) 504 f(0) = 0 , and f'(x) = 3n² - 1 , so this equation becomes 7 = f'(c)(8) = 0 (8) - 504 which gives c2 21.33 , that is, c = ± 4:6188 . But c must be in (0, 8), so c = 4.62 The following figure illustrates the calculation that the tangent line at this value of c is parallel to the secant line. 700 600 500 400 300 200 100 8 Ⓡ
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![To illustrate the Mean Value Theorem with a specific function, let's consider f(x) = x³ - x, a = 0, b = 8. Since f is a polynomial, it is continuous and differentiable for all x, so it is certainly continuous on [0, 8] and
differentiable on (0, 8). Therefore, by the Mean Value Theorem, there is a number c in (0, 8) such that
f(8) f(0) = f'(c)(8-0).
Now f(8) 504 ,f(0) = 0
and f'(x) = 3n² - 1
, so this equation becomes
7
= f'(c)(8) = 0
(8) - 504
which gives c2. 21.33
, that is, c = ± 4:6188
. But c must be in (0, 8), so c = 4.62
The following figure illustrates the calculation that the tangent line at this value of c is parallel to the secant line.
y
8
Ⓡ
700
600
500
400
300
200
100](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4b90110e-d69f-4502-ad47-61daee248c92%2F0a26b3d2-5af1-4205-b617-3e556b0fb963%2Fn1hahtb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:To illustrate the Mean Value Theorem with a specific function, let's consider f(x) = x³ - x, a = 0, b = 8. Since f is a polynomial, it is continuous and differentiable for all x, so it is certainly continuous on [0, 8] and
differentiable on (0, 8). Therefore, by the Mean Value Theorem, there is a number c in (0, 8) such that
f(8) f(0) = f'(c)(8-0).
Now f(8) 504 ,f(0) = 0
and f'(x) = 3n² - 1
, so this equation becomes
7
= f'(c)(8) = 0
(8) - 504
which gives c2. 21.33
, that is, c = ± 4:6188
. But c must be in (0, 8), so c = 4.62
The following figure illustrates the calculation that the tangent line at this value of c is parallel to the secant line.
y
8
Ⓡ
700
600
500
400
300
200
100
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