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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.4, Problem 78E

(a)

To determine

**To sketch:** The given data points and the exponential model.

Expert Solution

The excel calculator is used to draw the given data points and the exponential model below as shown in Figure 1.

From Figure 1, it is observed that

Hence, the fit is good since

(b)

To determine

**To estimate:** The rates of population growth in 1800 and 1850 by averaging slopes of the secant lines.

Expert Solution

The average slope of the secant lines are

**Formula used:**

The slope of a line at two points

**Calculation:**

Obtain the average slope of the secant line.

For 1800, the slope between 1790 to 1800 is computed as follows,

The slope between 1810 to 1800 is computed as follows,

The average slope of the secant line is,

Thus, the average slope of the secant line is,

For 1850, the slope between 1850 to 1840 is computed as follows,

For 1850, the slope between 1860 to 1850 is computed as follows,

The average slope of the secant line is,

Thus, the average slope of the secant line is

Therefore the average slope of the secant lines are

(c)

To determine

**To estimate:** The rate of growth in 1800 and 1850 by using part (a) and compare in part (b).

Expert Solution

The rate of growth in 1800 and 1850 are

The derivative of the fit equation is

Substitute

Substitute

Therefore, the rate of growth in 1800 and 1850 are

(d)

To determine

**To explain:** The discrepancy between value of the exponential model to predict the population in 1870 and the actual population of 38,558,000.

Expert Solution

The value of the exponential model to predict the population in 1870 is given by 45,375,883 but the actual value is 38,558,000.

Thus, value of the exponential model to predict the population in 1870 is much larger than the actual value 38,558,000 because the model used is simplistic.

This model is basically assigns population to dates at 10 year intervals and used to predict a rough answer, but cannot predict with uncertainty