§3.3 Exponential Functions.

1. §3.3 Exponential Functions. The student will learn about exponential functions, compound interest, present value, the number e, continuous compounding, and exponential growth.

2. - 2 ≤ x ≤ 4 - 1 ≤ y ≤ 30 The Exponential Function Definition: The equation f (x) = b ax for b > 0, b ≠ 1, defines an exponential function. The number b is the base. The domain of f is the set of all real numbers and the range of f is the set of all positive numbers. Graph y = 3 x

3. Properties of the graph of f (x) = b ax. • For all b the y intercept is 1. That is (0,1) is a point on all graphs (unless shifted). • All graphs are continuous curves. • The x axis is a horizontal asymptote (unless shifted). • If a > 0, then b ax is an increasing function. • If a < 0, then b ax is an decreasing function.

4. Examples Graph the following y = 3x -4  x  4 0  y  30. y = 3-x -4  x  4 0  y  30. y = -(3 - x ) -4  x  4 -30  y  0.

5. Exponential Function Properties • Exponential laws: • a x a y = a x + y • Exponential laws: • a x a y = a x + y a x / a y = a x – y • Exponential laws: • a x a y = a x + y a x / a y = a x – y (a x) y = a xy • Exponential laws: • a x a y = a x + y a x / a y = a x – y (a x) y = a xy • Exponential facts: • (ab) x = a x b x (a/b) x = a x / b x 3. a x = a y if and only if x = y 4. For x ≠ 0. a x = b x if and only if a = b.

6. Compound Interest Let P = principal, r = annual interest rate, t = time in years, n = number of compoundings per year, and A = amount realized at the end of the time period. Compound interest

7. Example: Generous Grandma Your Grandma gifts you with $1,000 in a bank at 5% daily. Calculate the amount after 5 years. Compound interest (daily) $1,284.00

8. Present Value Present ValueCompound Interest in Reverse Let P = principal, r = annual interest rate, t = time in years, n = number of compoundings per year, and A = amount realized at the end of the time period. In the compound interest formula the A term may be thought of as the future value. That is, grandma’s $1,000 had a future value of $1,284 in five years. We may reverse that order and ask what do we need to start with to have $1,284 at 5% in 5 years? That question is the question of present value.

9. Example: Present Value How much would you need to put in the credit union at 5% to have $1,284 in five years? Present Value

10. Depreciation by a Fixed Percentage Depreciation by a fixed percentage means that equipment loses some percentage of its value each year. It is the same as compound interest except that the percentage is negative and it is usually compounded only one time per year so that n = 1. Depreciation by percentage

11. The Number “e” There are several special numbers in mathematics. You may already know of the number π. The number e is a natural constant that occurs often in nature and also in economics. Many of our applications of exponential growth will have base e. The number e  2.71828182846. Show how to get this on a calculator.

12. Continuous Compound Interest Let P = principal, r = annual interest rate, t = time in years, n = number of compoundings per year, and A = amount realized at the end of the time period. Continuous compounding A = P e rt.

13. Example: Generous Grandma Your Grandma puts $1,000 in a bank at 5% for you. Calculate the amount after 5 years. Continuous compounding A = P e rt. A = 1000 e (.05 x 5) = $1,284.03

14. Present ValueContinuous Compound Interest in Reverse Let P = principal, r = annual interest rate, t = time in years, n = number of compoundings per year, and A = amount realized at the end of the time period. Future Value A = P e rt. Present Value with Continuous Compounding 14

15. Example – IRA After graduating from York College, Sam Spartan landed a great job with Springettsbury Manufacturing, Inc. His first year he bought a $5,000 Roth IRA and invested it in a stock sensitive mutual fund that grows at 10% continuously a year. He plans to retire in 40 years. a. What will be its value in 40 years? A = P e rt = 5000 e (.10)(40) = $272,991 Tax free. b. The second year he repeats the purchase of a Roth IRA. What will be its value in 39 years? $247,012 Tax free. Show how to become a millionaire!! = $2,212,963

16. Example – IRA After graduating from York College, Sam Spartan landed a great job with Springettsbury Manufacturing, Inc. His first year he bought a $4,500 Roth IRA and invested it in a stock sensitive mutual fund that grows at 10% continuously a year. He plans to retire in 40 years. = $2,212,963 Invested in a 5% bond fund yields $110,648 per year, tax free. The principle remains for your heirs. Don’t count on your Social Security to live on. My father contributed to social security from Dec 1, 1936 until he retired 62 years later at age 80. He receives $8640 per year!!!

17. The Function y = e x As we have seen many of our applications of exponential growth will have base e. The number e  2.71828182846 These equations almost always follow the form A t = A0 e kt A = P e rt. Graph y = ex on a calculator. -4  x  4 0  y  30.

18. Summary. • We learned about basic exponential functions : f (x) = b x • We learned financial applications of exponentiation including compound interest, present value and depreciation. • We learned about the natural number e and its application to continuous interest.

19. ASSIGNMENT §3.3 on my website. 6, 7, 8, 9, 10 , 11, 12, 13, 14.