Exponential Functions OBJECTIVE Graph exponential functions. Differentiate exponential functions.

3.1 Exponential Functions DEFINITION: An exponential function f is given by where x is any real number, a > 0, and a ≠ 1. The number a is called the base. Examples:

3.1 Exponential Functions f (x) = a0 · ax, a > 1 positive, increasing, continuous function as x gets smaller, a0 · ax approaches 0 concave up x-axis is the horizontal asymptote

3.1 Exponential Functions f (x) = a0 · ax, 0 < a < 1 positive, decreasing, continuous function as x gets larger, a0 · ax approaches 0 concave up x-axis is the horizontal asymptote

3.1 Exponential Functions Example 1: Graph First, we find some function values.

3.1 Exponential Functions Quick Check 1 For , complete this table of function values. Graph .

3.1 Exponential Functions DEFINITION: We call e the natural base.

3.1 Exponential Functions THEOREM 1 The derivative of the function f given by f (x) = ex is itself:

3.1 Exponential Functions Example 2: Find dy/dx:

3.1 Exponential Functions Example 2 (concluded):

3.1 Exponential Functions Quick Check 2 Differentiate: a.) , b.) , c.) ,

3.1 Exponential Functions THEOREM 2 The derivative of e to some power is the product of e to that power and the derivative of the power. or

3.1 Exponential Functions Example 3: Differentiate each of the following with respect to x:

3.1 Exponential Functions Example 3 (concluded):

3.1 Exponential Functions Quick Check 3 Differentiate: a.) , b.) , c.) ,

3.1 Exponential Functions Example 4: Graph with x ≥ 0. Analyze the graph using calculus. First, we find some values, plot the points, and sketch the graph.

3.1 Exponential Functions Example 4 (continued):

3.1 Exponential Functions Example 4 (continued): a) Derivatives. Since b) Critical values. Since the derivative for all real numbers x. The derivative exists for all real numbers, and the equation has no solution. There are no critical values.

3.1 Exponential Functions Example 4 (continued): c) Increasing. Since the derivative for all real numbers x, we know that h is increasing over the entire real number line. d) Inflection Points. Since we know that the equation h(x) = 0 has no solution. Thus there are no points of inflection and the graph is concave down over the entire real line.

3.1 Exponential Functions Example 4 (concluded): e) Concavity. Since for all real numbers x, we know that h is decreasing and the graph is concave down over the entire real number line.

3.1 Exponential Functions Section Summary An exponential function is given by f (x) = a0 · ax, where x is any real umber, a > 0 and a ≠ 1. The number a is the base, and the y-intercept is (0, a0). If a > 1, then f (x) = a0 · ax is increasing, concave up, increasing without bound as x increases, and has a horizontal asymptote y = 0 as x approaches –∞. If 0 < a < 1, then f (x) = a0 · ax is decreasing, concave up, increasing without bound as x approaches –∞, and has a horizontal asymptote y = 0 as x approaches infinity.

3.1 Exponential Functions Section Summary The exponential function , where e ≈ 2.71828 has the derivative . That is, the slope of a tangent line to the graph of y = ex is the same as the function value at x. In general, The graph of is an increasing function with no critical values, no maximum or minimum values, and no points of inflection. The graph is concave up, with and