   Chapter 2.6, Problem 27E

Chapter
Section
Textbook Problem

Let y = x2 + x. a. Find the average rate of change of y with respect to x in the interval from x = 2 to x = 3, from x = 2 to x = 2.5, and from x = 2 to x = 2.1. b. Find the (instantaneous) rate of change of y at x = 2. 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 [2,3],[2,2.5],[2,2.1] .

Explanation

Given:

The given function is,

y=f(x)=x2+x (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 [2,3] value of x is 2 and value of (x+h) is 3, for the average rate of change of the given function over interval [2,3] first compute f(3) and f(2) .

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

f(3)=32+3=9+3=12

Substitute 2 for x in equation (1) to calculate the value of f(2) ,

f(2)=22+2=4+2=6

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

Averagerateofchange=f(3)f(2)32=f(3)f(2)1

Substitute 12 for f(3) and 6 for f(2) in the above equation,

Averagerateofchange=1261=6

The average rate of change in interval [2,3] is 6.

For given interval [2,2.5] value of x is 2 and value of (x+h) is 2.5, for the average rate of change of the given function over interval [2,2.5] first compute f(2.5) .

Substitute 2.5 for x in equation (1) to calculate the value of f(2.5) ,

f(2.5)=(2.5)2+2.5=6.25+2.5=8

(b)

To determine

To find: The instantaneous rate of change of y at x=2 .

(c)

To determine

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

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