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AP CALCULUS AB CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.3: EXPONENTIAL FUNCTIONS What you’ll learn about… Exponential Growth Exponential Decay Applications The Number e …and why Exponential functions model many growth patterns. Exponential Function Let a be a positive real number other than 1. The function f ( x) = a x is the exponential function with base a. The domain of f(x) = ax is (-, ) and the range is (0,). Compound interest investment and population growth are examples of exponential growth. Exponential Growth If a 1 the graph of f looks like the graph of y= 2 x in Figure 1.22a Exponential Growth If 0 a 1 the graph of f looks like the graph of y = 2- x in Figure 1.22b. Section 1.3 – Exponential Functions Example: Graph the function State its domain and range. f x 3 2x 3 Section 1.3 – Exponential Functions You try: Graph each function State its domain and range. 1. y 3x 6 2. y 2x 4 Rules for Exponents If a > 0 and b > 0, the following hold for all real numbers x and y. x 1. a x ×a y = a x + y 4. a x ×b x = (ab) ax 2. y = a xa æ öx a x a÷ a 5. çç ÷ = x çèbb÷ ø b 3. (a x y x y y x ) = (a ) = a xy Rules for Exponents Rules for Exponents If a > 0 and b > 0, then the following hold for all real numbers x and y. Rule Example 1. a x a y a x y 2. 32 34 36 ax x y a ay x7 3 x x4 3. a a 4. a x b x ab 5. x y y x a x x ax a x b b 96 xy 3 2 4 38 2 x 3x 6 x 3 x3 x3 x 3 3 27 3 Half-life Exponential functions can also model phenomena that produce decrease over time, such as happens with radioactive decay. The half-life of a radioactive substance is the amount of time it takes for half of the substance to change from its original radioactive state to a non-radioactive state by emitting energy in the form of radiation. Note: Carbon-14 half-life is about 5730 years. Half-life The half-life of a radioactive substance is the amount of time it takes for half of the substance to change from its original radioactive state to a non-radioactive state by emitting energy in the form of radiation. In the following equation, n represents the half-life. 1 y k 2 x n Section 1.3 – Exponential Functions Example: Suppose the half-life of a certain radioactive substance is 12 days and that there are 8 grams present initially. When will there be only 1.5 grams of the substance remaining? (Hint: Solve graphically) Section 1.3 – Exponential Functions You try: The half-life of a radioactive substance is 20 days. The number of grams present initially is 10 grams. Determine when 4 grams of the substance will remain. Exponential Growth and Exponential Decay The function y = k ×a x , k > 0, is a model for exponential growth if a > 1, and a model for exponential decay if 0 < a < 1. Zeros of Exponential Functions To find the zeros of an exponential function using a graphing calculator (TI-83 or 84): 1. Enter the equation in y1. 2. Graph in the appropriate window. 3. Use the following keystrokes: CALC (2nd TRACE) ZERO When it says “Left Bound?”, go just left of the x-intercept and hit ENTER. When it says “Right Bound?”, go just right of the x-intercept and hit ENTER. When it says “Guess?”, go to approximately the x-intercept and hit ENTER. It will print out ZERO x = ________ y = ________ The zero is the x-value. Example Exponential Functions Use a grapher to find the zero's of f (x)= 4x - 3. f (x)= 4x - 3 [-5, 5], [-10,10] Section 1.3 – Exponential Functions Example: Find the zeros of graphically. f x 7 1.25x Section 1.3 – Exponential Functions You try: Find the zeros of each function graphically. 1. f x 2 1.20x 2. f x 4 1.25x The Number e Many natural, physical and economic phenomena are best modeled by an exponential function whose base is the famous number e, which is 2.718281828 to nine decimal places. x æ 1 ö÷ We can define e to be the number that the function f (x)= çç1 + ÷ çè x ÷ ø approaches as x approaches infinity. The Number e The exponential functions y = e x and y = e- x are frequently used as models of exponential growth or decay. Interest compounded continuously uses the model y = P ×e r t , where P is the initial investment, r is the interest rate as a decimal and t is the time in years. Example The Number e [0,100] by [0,120] in 10’s Remember Compounding Formulas: 1. Simple Interest: At A0 1 r t 2. 3. Compounded n times per year: r At A0 1 n nt Compounded continuously: Pt P0e rt , where r is the decimal form on the percent.