f(x) = a x, h(5) = ( 1) 5 1 = 2 2 1


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1 Exponential Functions an their Derivatives Exponential functions are functions of the form f(x) = a x, where a is a positive constant referre to as the base. The functions f(x) = x, g(x) = e x, an h(x) = ( 1) x are all exponential functions. Of course, if we choose x to be a nonnegative integer, then a x has a special interpretation. Consier h(x) above; if we evaluate h at x = 5, we have h(5) = ( 1) 5 1 = In a sense, x just counts how many copies of the 1/ appear. It is extremely important to istinguish between exponential functions an polynomial functions. For example, let s compare the exponential function f(x) = x to the polynomial function p(x) = x. These two functions are completely ifferent: while f(x) raises the constant to the variable power x, p(x) raises the variable x to the constant power. In other wors, exponential functions have constant bases but variable powers, while polynomial functions have variable bases but constant powers. It will be extremely helpful to unerstan the general shape of the graph of an exponential function. The following graph illustrates several exponential functions with bases a > 1: 1
2 The graph below illustrates some exponential functions with bases 0 < a < 1: Properties of Exponential Functions Exponential functions have many of the same properties that we are use to seeing when working with polynomials. The following list etails the omain, range, an rules for combining exponential functions. Let a > 0, b > 0, a a x is continuous. a x has omain (, ) 3. a x has range (0, ) 4. If a > 1, then a x is an increasing function. 5. If 0 < a < 1, then a x is an ecreasing function. 6. a x+y = a x a y 7. a x y = ax a y 8. (a x ) y = a xy 9. (ab) x = a x b x
3 Calculus Properties of Exponential Functions Of course, we woul like to know what calculus has to say about exponential functions. In particular, we woul like to unerstan: 1. limits,. erivatives, an 3. integrals of exponential functions. Limits of Exponential Functions Throughout the rest of the section, assume that a > 0. Since exponential functions are continuous, finite limits agree with function values: lim x c ax = a c, for any real number c. We woul like to unerstan limits at infinity as well. following fact base on the graphs above: 1. If a > 1, then (a) lim x ax = (b) lim x ax = 0. If 0 < a < 1, then (a) lim x ax = 0 (b) lim x ax = You may have alreay guesse the thus Since 1/ < 1, we know that lim 3 ( 1) x. x lim x (1) x = 0; lim 3 ( 1) x = lim x x 3 lim x = ( lim x 3) 0 = 3, (1 ) x since the limit of a constant is the constant. Thus the line y = 3 is a horizontal asymptote of 3 (1/) x. 3
4 Derivative of the Natural Exponential Function We will actually put off learning about the erivatives of most exponential functions until section 3.4; however, in this section, we will learn the erivative of one special exponential function, e x, which we call the natural exponential function. Derivative of e x. The number e is the number so that x ex = e x. In other wors, e x is a function that escribes its own rate of change. x ex/. Since the form of our function is not the same as the form of the rule we ve just learne, we must think of it as a prouct, quotient, or composition. Clearly e x/ is a composition function; so we ll ifferentiate using the chain rule. Recall that we must fin the insie function g(x) an the outsie function f(x); then the chain rule says that x f(g(x)) = g (x)f (g(x)). Let s set up the chart: f(x) = e x f (x) = e x f (g(x)) = e x/ g(x) = x g (x) = 1 So x ex/ = 1 ex/. x ex sin x. Again, we realize that this function is a composition, so that we must use the chain rule to fin its erivative. Thinking of g(x) as the insie function an f(x) as the outsie function, we have f(x) = e x f (x) = e x f (g(x)) = e x sin x g(x) = x sin x g (x) =? Of course, to fin the erivative of x sin x, we ll nee to use the prouct rule: Finishing off the chart, we have x sin x = sin x + x cos x. x f(x) = e x f (x) = e x f (g(x)) = e x sin x g(x) = x sin x g (x) = sin x + x cos x. 4
5 Using the chain rule we see that x f(g(x)) = g (x)f (g(x)), x ex sin x = (sin x + x cos x)e x sin x. x. x eee We have yet another composition function to ifferentiate. Let s set up the chart for the chain rule: f(x) = e x f (x) = e x f (g(x)) = e eex g(x) = e ex g (x) =? Unfortunately, we on t know how to ifferentiate g(x) immeiately; we ll have to apply the chain rule again: out(x) = e x out (x) = e x out (in(x)) = e ex in(x) = e x in (x) = e x Thus returning to the original chart, we have g (x) = e x e ex ; f(x) = e x f (x) = e x f (g(x)) = e eex g(x) = e ex g (x) = e x e ex, so that x x eee = e x e ex e eex. Integral of the Natural Exponential Function The rule for integrating e x shoul be clear from the the rule for ifferentiating it: e x x = e x + C. e sin x x. Clearly, we will nee to use usubstitution to evaluate the integral. The most natural substitution to make seems to be u = so that u = csc x x. 5
6 Consiering the original integral, however, it appears that we have an issue with this substitution: e sin x x. How o we replace the factor of 1/ sin x? Perhaps rewriting the original function will help. We know that 1 = csc x, sin x so e sin x x = csc xe x. Now we can replace all of the terms containing xs with terms containing us! We have e sin x x = csc xe x = e u u = e u + C = e + C. Thus e sin x x = e + C. 6
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