Explanation The total change in the dependent variable over the total change in the independent variable is called the average rate of change. The average rate of change between two points of a function is given by: f ( x 2 ) − f ( x 1 ) x 2 − x 1 , Where, ( x 1 , f ( x 1 ) ) is the starting point of the interval and ( x 2 , f ( x 2 ) ) is the endpoint of the interval. The average rate of change of a function can be negative, positive, or even zero just like the slope of the secant line. To know how fast the particle moves at a particular instant of time, we define the instantaneous rate of change. Given: f ( x ) = x ( 1 − x ) over [ 0 , 1 ] Formula Used: The average rate of change of f with respect to x is given by: Δ f Δ x = f ( x 2 ) − f ( x 1 ) x 2 − x 1 Calculation: Sketch the graph of the function f ( x ) = x ( 1 − x ) over [ 0 , 1 ]

4th Edition

ISBN: 9781319055844

Author: FRANZOSA

Publisher: VST

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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Question

Chapter 2.1, Problem 24E

To determine

a. The average rate of change over [0,1]

b. The instantaneous rate of change at x=12.

c. The values of x at which the rate of change is positive.

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Chapter 2.1, Problem 23E

Chapter 2.1, Problem 25E

Chapter 2 Solutions

EBK CALCULUS

Ch. 2.1 - Prob. 1PQ Ch. 2.1 - Prob. 2PQ Ch. 2.1 - Prob. 3PQ Ch. 2.1 - Prob. 4PQ Ch. 2.1 - Prob. 1E Ch. 2.1 - Prob. 2E Ch. 2.1 - Prob. 3E Ch. 2.1 - Prob. 4E Ch. 2.1 - Prob. 5E Ch. 2.1 - Prob. 6E

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a. The average rate of change over [ 0 , 1 ] b. The instantaneous rate of change at x = 1 2 . c. The values of x at which the rate of change is positive.

Solution Summary: The author explains that the average rate of change of a function can be negative, positive, or even zero just like the slope of the secant line.

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a. The average rate of change over [0,1]

b. The instantaneous rate of change at x=12.

c. The values of x at which the rate of change is positive.

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Chapter 2 Solutions

a. The average rate of change over [ 0 , 1 ] b. The instantaneous rate of change at x = 1 2 . c. The values of x at which the rate of change is positive.

Solution Summary: The author explains that the average rate of change of a function can be negative, positive, or even zero just like the slope of the secant line.

Concept explainers

Videos

a. The average rate of change over [0,1] b. The instantaneous rate of change at x=12. c. The values of x at which the rate of change is positive.

Want to see the full answer?

Chapter 2 Solutions

a. The average rate of change over [0,1]

b. The instantaneous rate of change at x=12.

c. The values of x at which the rate of change is positive.