EXAMPLE 4 If an object moves in a straight line with position function s f(t), then the average velocity between t a and t = b is f(b)-f(a) b-a and the velocity at t = c is f'(c). Thus the Mean Value Theorem tells us that at some time t = c between a and b the instantaneous velocity f'(c) is equal to the average velocity. For instance, if a car traveled 500 km in 5 hours, X km/h at least once. then the speedometer must have read In general, the Mean Value Theorem can be interpreted as saying that there is a number at which the instantaneous rate of change is equal to the average rate of change over an interval EXAMPLE 5 Suppose that f(0) = -2 and f'(x) s 4 for all values of x. How large can f(3) possibly be? SOLUTION We are given that fis differentiable (and therefore continuous) everywhere. In particular, we can apply the Mean Value Theorem on the interval 0, 31. There exists a number c such that f'(c) (3) - го) So f'(c) f(3) f(0) f'(c) = -2 We are given that f'(x) s 4 for all x, so in particular we know that f'(c) s Multiplying both sides of this inequality by 3, we have 3f'(c) So f(3) 2 f'(c) -2 The largest possible value for f(3) is

Question
EXAMPLE 4
If an object moves in a straight line with position function s
f(t), then the average velocity between
t a and t = b is
f(b)-f(a)
b-a
and the velocity at t = c is f'(c). Thus the Mean Value Theorem tells us that at some time t = c between a and b
the instantaneous velocity f'(c) is equal to the average velocity. For instance, if a car traveled 500 km in 5 hours,
X km/h at least once.
then the speedometer must have read
In general, the Mean Value Theorem can be interpreted as saying that there is a number at which the
instantaneous rate of change is equal to the average rate of change over an interval

Image Transcription

EXAMPLE 4 If an object moves in a straight line with position function s f(t), then the average velocity between t a and t = b is f(b)-f(a) b-a and the velocity at t = c is f'(c). Thus the Mean Value Theorem tells us that at some time t = c between a and b the instantaneous velocity f'(c) is equal to the average velocity. For instance, if a car traveled 500 km in 5 hours, X km/h at least once. then the speedometer must have read In general, the Mean Value Theorem can be interpreted as saying that there is a number at which the instantaneous rate of change is equal to the average rate of change over an interval

EXAMPLE 5
Suppose that f(0) = -2 and f'(x) s 4 for all values of x. How large can f(3) possibly be?
SOLUTION
We are given that fis differentiable (and therefore continuous) everywhere. In particular, we can apply
the Mean Value Theorem on the interval 0, 31. There exists a number c such that
f'(c)
(3) - го)
So
f'(c)
f(3) f(0)
f'(c)
= -2
We are given that f'(x) s 4 for all x, so in particular we know that f'(c) s
Multiplying both sides of this
inequality by 3, we have 3f'(c)
So
f(3) 2
f'(c) -2
The largest possible value for f(3) is

Image Transcription

EXAMPLE 5 Suppose that f(0) = -2 and f'(x) s 4 for all values of x. How large can f(3) possibly be? SOLUTION We are given that fis differentiable (and therefore continuous) everywhere. In particular, we can apply the Mean Value Theorem on the interval 0, 31. There exists a number c such that f'(c) (3) - го) So f'(c) f(3) f(0) f'(c) = -2 We are given that f'(x) s 4 for all x, so in particular we know that f'(c) s Multiplying both sides of this inequality by 3, we have 3f'(c) So f(3) 2 f'(c) -2 The largest possible value for f(3) is

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