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Applied Calculus for the Manageria...

10th Edition
Soo T. Tan
ISBN: 9781285464640
Textbook Problem

Let y = f(x) = x2 − 4x.

  1. 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.
  2. b. Find the (instantaneous) rate of change of y at x = 3.
  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).

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