This exercise introduces the approximation provided that x

Chapter 5, Problem 26

(choose chapter or problem)

This exercise introduces the approximation provided that x is close to zero This approximation has an important consequence when we study population growth in Section 5.7. (The details are in Exercise 61 in Section 5.7.) (a) Use the information in Figure 3 to explain why the equation of the line tangent to the curve y e x at the point (0, 1) is y x 1. (b) Verify visually that the line y x 1 is tangent to the curve y e x at the point (0, 1) by using a graphing utility to display the graphs of both functions in the same standard viewing rectangle. Note that the curve and the line are virtually indistinguishable in the immediate vicinity of the point (0, 1). (c) Zoom in on the point (0, 1); use a viewing rectangle in which x extends from 0.05 to 0.05 and y extends from 0.95 to 1.05. Again, note that the line and the curve are virtually indistinguishable in the immediate vicinity of the point (0, 1). For a numerical look at this, complete the following table. In the columns for e x and e x (x 1), round each entry to four decimal places. When you are finished, observe that the closer x is to 0, the closer the agreement between the two quantities x 1 and e x . x x 1 ex ex (x 1) 0.04 0.03 0.02 0.01 0.00 0.01 0.02 0.03 0.04 G G ex