To express: The average rate of change of y with respect to x over the interval [ x 1 , x 2 ]

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 2, Problem 11RCC

(a)

To determine

To express: The average rate of change of y with respect to x over the interval [x1,x2]

Expert Solution

The average rate of change of y with respect to x is ΔyΔx=f(x2)f(x1)x2x1.

Explanation of Solution

Result used:

The average rate of change of y with respect to x is

ΔyΔx=f(x0+Δx)f(x0)Δx

Calculation:

The average rate of change of y with respect to x is

ΔyΔx=f(x0+Δx)f(x0)Δx

Where [x0,x]=[x1,x2] and Δx=x2x1.

The average rate of change of y with respect to x over the interval [x1,x2] is

ΔyΔx=f(x1+(x2x1))f(x1)x2x1    =f(x1+x2x1)f(x1)x2x1=f(x2)f(x1)x2x1

Thus the average rate of change of y with respect to x is ΔyΔx=f(x2)f(x1)x2x1.

(b)

To determine

To express: The instantaneous rate of change of y with respect to x at x=x1

Expert Solution

The instantaneous rate of change of y with respect to x at x=x1 is

limxx1ΔyΔx=limxx1f(x)f(x1)xx1.

Explanation of Solution

Result used:

The instantaneous rate of change of the function y=f(x) at the point x0 in its domain is limxx0ΔyΔx=limxx0f(x0)f(x)x0x

Now the instantaneous rate of change of the function y=f(x) at x=x1 is

limxx1ΔyΔx=limxx1f(x1)f(x)x1x          =limxx1(f(x)f(x1))(xx1)=limxx1f(x)f(x1)xx1

Thus the instantaneous rate of change of y with respect to x at x=x1 is

limxx1ΔyΔx=limxx1f(x)f(x1)xx1.

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