Explanation 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. Given: y = 4 x − 9 Formula Used: f ( x 2 ) − f ( x 1 ) x 2 − x 1 Calculation: a. Δ y Δ x for the interval [ 2 , 5 ] where y = 4 x − 9 = [ 4 ( 5 ) − 9 ] − [ 4 ( 2 ) − 9 ] 5 − 2 = 11 + 1 3 = 12 3 = 4 Thus, Δ y Δ x for the interval [ 2 , 5 ] is 4 b. To compute the instantaneous rate of change of y with respect to x at x = 2 , we need to compute the rate of change for many interval lying to the left hand side and right hand side of x = 2 ▪ Compute Δ y Δ x for Left-hand side of § x = 2 • Δ y Δ x for the interval · [ 1.5 , 2 ] = [ 4 ( 2 ) − 9 ] − [ 4 ( 1.5 ) − 9 ] 2 − 1.5 = − 1 + 3 0.5 = 2 0.5 = 4 Thus, the average rate of change for the interval [ 1.8 , 2 ] is 4 • Δ y Δ x for the interval [ 1.8 , 2 ] = [ 4 ( 2 ) − 9 ] − [ 4 ( 1

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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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 4E

To determine

a. ΔyΔx for the interval [2,5], where y=4x−9

b. The instantaneous rate of change of y with respect to x at x=2

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Chapter 2.1, Problem 3E

Chapter 2.1, Problem 5E

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Chapter 2 Solutions

Calculus - Standalone book

Ch. 2.1 - Prob. 1PQ Ch. 2.1 - Prob. 2PQ Ch. 2.1 - Prob. 3PQ Ch. 2.1 - Prob. 4PQ Ch. 2.1 - Prob. 5PQ 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 Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E Ch. 2.1 - Prob. 11E Ch. 2.1 - Prob. 12E Ch. 2.1 - Prob. 13E Ch. 2.1 - Prob. 14E Ch. 2.1 - Prob. 15E Ch. 2.1 - Prob. 16E Ch. 2.1 - Prob. 17E Ch. 2.1 - Prob. 18E Ch. 2.1 - Prob. 19E Ch. 2.1 - Prob. 20E Ch. 2.1 - Prob. 21E Ch. 2.1 - Prob. 22E Ch. 2.1 - Prob. 23E Ch. 2.1 - Prob. 24E Ch. 2.1 - Prob. 25E Ch. 2.1 - Prob. 26E Ch. 2.1 - Prob. 27E Ch. 2.1 - Prob. 28E Ch. 2.1 - Prob. 29E Ch. 2.1 - Prob. 30E Ch. 2.1 - Prob. 31E Ch. 2.1 - Prob. 32E Ch. 2.1 - Prob. 33E Ch. 2.1 - Prob. 34E Ch. 2.1 - Prob. 35E Ch. 2.1 - Prob. 36E Ch. 2.2 - Prob. 1PQ Ch. 2.2 - Prob. 2PQ Ch. 2.2 - Prob. 3PQ Ch. 2.2 - Prob. 4PQ Ch. 2.2 - Prob. 5PQ Ch. 2.2 - Prob. 6PQ Ch. 2.2 - Prob. 7PQ Ch. 2.2 - Prob. 1E Ch. 2.2 - Prob. 2E Ch. 2.2 - Prob. 3E Ch. 2.2 - Prob. 4E Ch. 2.2 - Prob. 5E Ch. 2.2 - Prob. 6E Ch. 2.2 - Prob. 7E Ch. 2.2 - Prob. 8E Ch. 2.2 - Prob. 9E Ch. 2.2 - Prob. 10E Ch. 2.2 - Prob. 11E Ch. 2.2 - Prob. 12E Ch. 2.2 - Prob. 13E Ch. 2.2 - 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Prob. 31E Ch. 2.7 - Prob. 32E Ch. 2.7 - Prob. 33E Ch. 2.7 - Prob. 34E Ch. 2.7 - Prob. 35E Ch. 2.7 - Prob. 36E Ch. 2.7 - Prob. 37E Ch. 2.7 - Prob. 38E Ch. 2.7 - Prob. 39E Ch. 2.7 - Prob. 40E Ch. 2.7 - Prob. 41E Ch. 2.7 - Prob. 42E Ch. 2.7 - Prob. 43E Ch. 2.7 - Prob. 44E Ch. 2.7 - Prob. 45E Ch. 2.8 - Prob. 1PQ Ch. 2.8 - Prob. 2PQ Ch. 2.8 - Prob. 3PQ Ch. 2.8 - Prob. 4PQ Ch. 2.8 - Prob. 5PQ Ch. 2.8 - Prob. 1E Ch. 2.8 - Prob. 2E Ch. 2.8 - Prob. 3E Ch. 2.8 - Prob. 4E Ch. 2.8 - Prob. 5E Ch. 2.8 - Prob. 6E Ch. 2.8 - Prob. 7E Ch. 2.8 - Prob. 8E Ch. 2.8 - Prob. 9E Ch. 2.8 - Prob. 10E Ch. 2.8 - Prob. 11E Ch. 2.8 - Prob. 12E Ch. 2.8 - Prob. 13E Ch. 2.8 - Prob. 14E Ch. 2.8 - Prob. 15E Ch. 2.8 - Prob. 16E Ch. 2.8 - Prob. 17E Ch. 2.8 - Prob. 18E Ch. 2.8 - Prob. 19E Ch. 2.8 - Prob. 20E Ch. 2.8 - Prob. 21E Ch. 2.8 - Prob. 22E Ch. 2.8 - Prob. 23E Ch. 2.8 - Prob. 24E Ch. 2.8 - Prob. 25E Ch. 2.8 - Prob. 26E Ch. 2.8 - Prob. 27E Ch. 2.8 - Prob. 28E Ch. 2.8 - Prob. 29E Ch. 2.8 - Prob. 30E Ch. 2.8 - Prob. 31E Ch. 2.8 - 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a. Δ y Δ x for the interval [ 2 , 5 ] , where y = 4 x − 9 b. The instantaneous rate of change of y with respect to x at x = 2

Solution Summary: The author calculates the instantaneous rate of change of y with respect to x at left[2,5right] .

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a. ΔyΔx for the interval [2,5], where y=4x−9

b. The instantaneous rate of change of y with respect to x at x=2

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Chapter 2 Solutions

a. Δ y Δ x for the interval [ 2 , 5 ] , where y = 4 x − 9 b. The instantaneous rate of change of y with respect to x at x = 2

Solution Summary: The author calculates the instantaneous rate of change of y with respect to x at left[2,5right] .

Concept explainers

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a. ΔyΔx for the interval [2,5], where y=4x−9 b. The instantaneous rate of change of y with respect to x at x=2

Want to see the full answer?

Chapter 2 Solutions

a. ΔyΔx for the interval [2,5], where y=4x−9

b. The instantaneous rate of change of y with respect to x at x=2