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Chapter 6 | Applications of Integration
To put it another way, the area under the curve y =1/ t between t =1 and t = e is 1 ( Figure 6.77 ). The proof that such a number exists and is unique is left to you. ( Hint : Use the Intermediate Value Theorem to prove existence and the fact that ln x is increasing to prove uniqueness.)
Figure 6.77 The area under the curve from 1 to e is equal to one.
The number e can be shown to be irrational, although we won’t do so here (see the Student Project in Taylor and Maclaurin Series (http://cnx.org/content/m53817/latest/) ). Its approximate value is given by e ≈ 2.71828182846. The Exponential Function We now turn our attention to the function e x . Note that the natural logarithm is one-to-one and therefore has an inverse function. For now, we denote this inverse function by exp x . Then, exp(ln x ) = x for x > 0 and ln(exp x ) = x for all x . The following figure shows the graphs of exp x and ln x .
Figure 6.78 The graphs of ln x and exp x .
We hypothesize that exp x = e x . For rational values of x , this is easy to show. If x is rational, then we have ln( e x ) = x ln e = x . Thus, when x is rational, e x =exp x . For irrational values of x , we simply define e x as the inverse function of ln x .
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