To illustrate the Mean Value Theorem with a specific function, let's consider f(x) = x3 - x, a = 0, b 4. Sincef is a polynomial, it is continuous and differentiable for all x, so it is certainly continuous on [O, 4] and У 120 EXAMPLE 3 100 differentiable on (0, 4). Therefore, by the Mean Value Theorem, there is a numberc in (0, 4) such that f(4) f(0) f(c) (4 0) 80 = 60 and f'(x)3x- 1 Now f(4) 60 f(0) so this equation becomes = 0 40 20 f'(c)(4) 3c2 9c2 60 1 4 4 Video Example 16 which gives c2 that is, c= ± But c must be in (0, 4), so 3 4 The figure illustrates this calculation: The tangent line at this value of c is parallel to the C = 3 secant line.

Question
To illustrate the Mean Value Theorem with a specific function, let's consider f(x) = x3 - x, a = 0,
b 4. Sincef is a polynomial, it is continuous and differentiable for all x, so it is certainly continuous on [O, 4] and
У
120
EXAMPLE 3
100
differentiable on (0, 4). Therefore, by the Mean Value Theorem, there is a numberc in (0, 4) such that
f(4) f(0) f(c) (4 0)
80
=
60
and f'(x)3x- 1
Now f(4) 60
f(0)
so this equation becomes
= 0
40
20
f'(c)(4) 3c2
9c2
60
1
4
4
Video Example
16
which gives c2
that is, c= ±
But c must be in (0, 4), so
3
4
The figure illustrates this calculation: The tangent line at this value of c is parallel to the
C =
3
secant line.
Expand
Transcribed Image Text

To illustrate the Mean Value Theorem with a specific function, let's consider f(x) = x3 - x, a = 0, b 4. Sincef is a polynomial, it is continuous and differentiable for all x, so it is certainly continuous on [O, 4] and У 120 EXAMPLE 3 100 differentiable on (0, 4). Therefore, by the Mean Value Theorem, there is a numberc in (0, 4) such that f(4) f(0) f(c) (4 0) 80 = 60 and f'(x)3x- 1 Now f(4) 60 f(0) so this equation becomes = 0 40 20 f'(c)(4) 3c2 9c2 60 1 4 4 Video Example 16 which gives c2 that is, c= ± But c must be in (0, 4), so 3 4 The figure illustrates this calculation: The tangent line at this value of c is parallel to the C = 3 secant line.

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