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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

The number N of locations of a popular coffeehouse chain is given in the table. (The numbers; of locations as of October 1 are given.)

Year 2004 2006 2008 2010 2012
N 8569 12,440 16,680 16,858 18,066

(a) Find the average rate of growth

(i) from 2006 to 2008

(ii) from 2008 to 2010

In each case, include the units. What can you conclude?

(b) Estimate the instantaneous rate of growth in 2010 by taking the average of two average rates of change. What are its units?

(c) Estimate the instantaneous rate of growth in 20!0 by measuring the slope of a tangent.

(a)

To determine

To find: The average rate of growth N(t) from the given intervals.

Explanation

Given:

Let N(t) be the number of locations of a popular coffeehouse chain with respect to time period t, where t is in years.

Formula used:

The average rate of growth N with respect to t on the interval [t1,t2] is,

ΔNΔx=N(t2)N(t1)t2t1 (1)

Calculation:

Section (i)

Obtain the average rate of growth N(t) from t=2006 to t=2008.

Substitute 2006 for t1 and 2008 for t2 in equation (1),

ΔNΔx=N(2008)N(2006)20082006

Consider the given table, the values of N(2008)=16680 and N(2006)=12440.

ΔNΔt=166801244020082006=42402=2120

Thus, the average rate of growth N(t) from t=2006 to t=2008 is 2120 locations per year

(b)

To determine

To estimate: The instantaneous rate of growth of N(t) in t=2010 and find its units.

(c)

To determine

To estimate: The instantaneous rate of growth in 2010 by measuring the slope of the tangent.

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