   Chapter 2.6, Problem 28E

Chapter
Section
Textbook Problem

Let y = f(x) = x2 − 4x. a. Find the average rate of change of y with respect to x in the interval from x = 3 to x = 4, from x = 3 to x = 3.5, and from x = 3 to x = 3.1. b. Find the (instantaneous) rate of change of y at x = 3. c. Compare the results obtained in part (a) with the result of part (b).

(a)

To determine

To find: The average rate of change of y with respect to x in the interval [3,4],[3,3.5],[3,3.1] .

Explanation

Given information:

The given function is,

y=f(x)=x24x (1)

Formula used:

The formula to calculate the average rate of change of function with respect to x over the interval [x,x+h] is,

Averagerateofchange=f(x+h)f(x)(x+h)x (2)

Calculation:

For given interval [3,4] value of x is 3 and value of (x+h) is 4, for the average rate of change of the given function over interval [3,4] first compute f(4) and f(3) .

Substitute 4 for x in equation (1) to calculate the value of f(4) ,

f(4)=424(4)=1616=0

Substitute 3 for x in equation (1) to calculate the value of f(3) ,

f(3)=324(3)=912=3

Substitute 4 for (x+h) and 3 for x in equation (2) to find the average rate of change over interval [3,4] ,

Averagerateofcahange=f(4)f(3)43=f(4)f(3)1

Substitute 0 for f(4) and 3 for f(3) in the above equation,

Averagerateofchange=0(3)1=3

The average rate of change in interval [3,4] is 3.

For given interval [3,3.5] value of x is 3 and value of (x+h) is 3.5, for the average rate of change of the given function over interval [3,3.5] first compute f(3.5) .

Substitute 3.5 for x in equation (1) to calculate the value of f(3.5) ,

f(3.5)=(3.5)24(3.5)=12.2514=1.75

Substitute 3.5 for (x+h) and 3 for x in equation (2) to find the average rate of change over interval [3,3

(b)

To determine

To find: The instantaneous rate of change of y at x=3 .

(c)

To determine

To describe: The comparison of the result of part (a) and (b).

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Expand each expression in Exercises 122. (y1y)2

Finite Mathematics and Applied Calculus (MindTap Course List)

Differentiate the function. y = ln(csc x cot x)

Single Variable Calculus: Early Transcendentals

In Exercises 15-22, use the laws of logarithms to solve the equation. logx103=3

Finite Mathematics for the Managerial, Life, and Social Sciences 