1. CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.3: EXPONENTIAL FUNCTIONS AP CALCULUS AB

2. Exponential Growth Exponential Decay Applications The Number e …and why Exponential functions model many growth patterns. What you’ll learn about…

3. Exponential Function The domain of f(x) = a x is (-, ) and the range is (0,). Compound interest investment and population growth are examples of exponential growth.

4. Exponential Growth

6. Section 1.3 – Exponential Functions Example: Graph the function State its domain and range.

7. Section 1.3 – Exponential Functions You try: Graph each function State its domain and range.

8. Rules for Exponents x

9. If a > 0 and b > 0, then the following hold for all real numbers x and y. RuleExample 1. 2. 3. 4. 5.

10. 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.

11. 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.

12. 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)

13. 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.

14. Exponential Growth and Exponential Decay

15. 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 y 1. 2.Graph in the appropriate window. 3.Use the following keystrokes: CALC (2 nd 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.

16. Example Exponential Functions [-5, 5], [-10,10]

17. Section 1.3 – Exponential Functions Example: Find the zeros of graphically.

18. Section 1.3 – Exponential Functions You try: Find the zeros of each function graphically. 1. 2.

19. The Number e

21. Example The Number e [0,100] by [0,120] in 10’s

22. Remember Compounding Formulas: 1. Simple Interest: 2. Compounded n times per year: 3. Compounded continuously: