If x is a rational number, x = p/q, where p and q are integers and q > 0, then

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1 6.2 Exponential Functions and Their Derivatives An is a function of the form where a is a positive constant. Recall: If x = n, a positive integer, then If x = 0, then If x = n, where n is a positive integer, then If x is a rational number, x = p/q, where p and q are integers and q > 0, then Question: What is the meaning of a x if x is an irrational number? Let s consider 2 3 : 1

2 In general, if a is any positive number, we define This definition makes sense because any irrational number can be approximated as closely as we like by a rational number. Graphs of Some Members of the Family of Exponential Functions: Notes: All graphs of the function y = a x pass through the point (0, 1) because a 0 = 1 for a 0. As the base a gets larger, the exponential function grows more rapidly (for x > 0). There are basically three kinds of exponential functions y = a x : If 0 < a < 1,. If a = 1,. If a > 1, 2

3 Caution: Do not confuse the exponential function f(x) = a x with the power function f(x) = x a. The following figure shows how the exponential function y = 2 x compares with the power function f(x) = x 2. Ultimately, 2 x grows far more rapidly than x 2. Example 1. Make a rough sketch of the graph of the function f(x) = ( ) x 1 2 by hand. 2 Example 2. Find the exponential function f(x) = Ca x whose graph is given. 3

4 Laws of Exponents: If a > 0 and a 1, then f(x) = a x is a continuous function with domain all real numbers and range (0, ). In particular, for all x. If 0 < a < 1, f(x) = a x is a decreasing function; if a > 1, f is an increasing function. If a, b > 0 and x, y R, then (1) a x+y = (2) a x y = (3) (a x ) y = (4) (ab) x = (5) ( a b ) x = Example 3. Find the domain of g(t) = 1 2 t. If a > 1, then and. If 0 < a < 1, then and. In particular, if a 1, then the x-axis is a horizontal asymptote of the graph of the exponential function y = a x. Example 4. Find the limit: lim x (1.001)x Applications of Exponential Functions Exponential functions occur in a variety of natural and social science settings, the most commonly studied being population models. 4

5 Derivatives of Exponential Functions Example 5. Try to find the derivative of the exponential function y = a x using the definition of the derivative. Of all possible choices for the base a in f (x) = f (0)a x, the simplest differentiation formula would occur when f (0) = 1. It is traditional to denote the value of this base by the letter e. Notice: It must be true that 2 < e < 3. e is the number such that Geometrically, this means that of all the possible exponential functions y = a x, the function f(x) = e x is the one whose tangent line at (0, 1) has a slope f (0) that is exactly 1. We call the function f(x) = e x the. 5

6 If we put a = e and, therefore, f (0) = 1, we get the following derivative of the natural exponential function: Thus, the exponential function has the property that it is its own derivative. Thus, the slope of a tangent line to the curve y = e x at any point is equal to the y-coordinate of the point. Example 6. Differentiate the function. a) f(x) = e x + x e b) k(r) = e 2r c) y = e 2t cos 4t 6

7 d) f(t) = sin(e t ) + e sin t e) y = e k tan x, k is a constant f) f(t) = 1 + te 2t Note: We are able to find equations of tangent and normal lines, analyze the behavior of the function, differentiate implicitly, etc for exponential functions, just as we did for polynomials, rational functions, algebraic functions, and trigonometric functions. Exponential Graphs The exponential function y = e x is one of the most frequently occurring functions in calculus and its applications, so it is important to be familiar with its graph and properties. 7

8 Example 7. Starting with the graph of y = e x, find the equation of the graph that results from shifting two units right then reflecting across the x-axis. Sketch the graph. Properties of the Natural Exponential Function: The exponential function f(x) = e x has domain and range. Thus e x > 0 for all x. Also So the x-axis is a horizontal asymptote of f(x) = e x. Example 8. Find the limit: lim x 2 e3/(2 x) Integration Because the exponential function y = e x has a simple derivative, its integral is also simple. Example 9. Evaluate the integral. a) 5 5 e x dx 8

9 (1 + e x ) 2 b) dx e x c) 1 + e x dx e x 9