The text mentioned two ways to define the number e. One

Chapter 5, Problem 62

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The text mentioned two ways to define the number e. One involved the expression (1 1/x) x and the other involved a tangent line. In this exercise youll see that these two approaches are, in fact, related. For convenience, well write the expression (1 1/x) x using the letter n rather than x (so that we can use x for something else in a moment.) Thus, we have as n becomes larger and (1) larger without bound Working from this approximation, well obtain evidence that the slope of the tangent to the curve y ex at x 0 is 1. (A rigorous proof requires calculus.) (a) Define x by the equation x 1/n. As n becomes larger and larger without bound, what happens to the corresponding values of x? Complete the following table. n 102 103 106 109 x (b) Substitute x 1/n in approximation (1) to obtain as x approaches 0 Next, raise both sides to the power x to obtain as x approaches 0 (2) Approximation (2) says that the curve y e x looks more and more like the line y 1 x as x approaches 0. For numerical perspective on this, complete the following tables. These results suggest (but do not prove) that the line y 1 x is tangent to the curve y e x at x 0. Thus, the results suggest that the slope of the tangent to y e x at x 0 is 1. x 0.3 0.2 101 102 103 104 x 1 ex x 0.3 0.2 101 102 103 104 x 1 ex 11 x2 ex 11 x2 1

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