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

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

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

Chapter5: A Survey Of Other Common Functions

Section5.4: Combining And Decomposing Functions

Problem 14E: Decay of Litter Litter such as leaves falls to the forest floor, where the action of insects and...

See similar textbooks

Similar questions

Find the average rates of change of f(x)=x2+2x (a) from x1=3 to x2=2 and (b) from x1=2 to x2=0.
Decay of Litter Litter such as leaves falls to the forest floor, where the action of insects and bacteria initiates the decay process. Let A be the amount of litter present, in grams per square meter, as a function of time t in years. If the litter falls at a constant rate of L grams per square meter per year, and if it decays at a constant proportional rate of k per year, then the limiting value of A is R=L/k. For this exercise and the next, we suppose that at time t=0, the forest floor is clear of litter. a. If D is the difference between the limiting value and A, so that D=RA, then D is an exponential function of time. Find the initial value of D in terms of R. b. The yearly decay factor for D is ek. Find a formula for D in term of R and k. Reminder:(ab)c=abc. c. Explain why A=RRekt.
bThe average rate of change of the linear function f(x)=3x+5 between any two points is ________.
aThe average rate of change of a function f between x=a and x=b is the slope of the ___________ line between (a,f(a)) and (b,f(b)).
5-6 Yes or No? If No, give a reason aIs the average rate of change of a function between x=a and x=b the slope of the secant line through (a,f(a)) and (b,f(b))?