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Chapter 3

Chapter 3 . Exponential Functions. SAT problem of the day. Introduction.

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Chapter 3

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  1. Chapter 3 Exponential Functions

  2. SAT problem of the day

  3. Introduction • Exponential functions look somewhat similar to functions you have seen before, in that they involve exponents, but there is a big difference, in that the variable is now the power, rather than the base. Previously, you have dealt with such functions as f(x) = x2, where the variable x was the base and the number 2 was the power. In the case of exponentials, however, you will be dealing with functions such as g(x) = 2x, where the base is the fixed number, and the power is the variable.

  4. Let's look more closely at the function g(x) = 2x. To evaluate this function, we operate as usual, picking values of x, plugging them in, and simplifying for the answers. But to evaluate 2x, we need to remember how exponents work. In particular, we need to remember that negative exponents mean "put the base on the other side of the fraction line".

  5. Evaluate functions • The first thing you will probably do with exponential functions is evaluate them. • Evaluate 3x at x = –2, –1, 0, 1, and 2. • To find the answer, I need to plug in the given values for x, and simplify:

  6. Example#1 • Given f(x) = 3–x, evaluate f(–2), f(–1), f(0), f(1), and f(2). • To find the answer, I need to plug in the given values for x, and simplify:

  7. Example#2 • Given g(x) = ( 1/3 )x, evaluate for x = –2, –1, 0, 1, and 2. Plug in the given values for x, and simplify:

  8. Student guided practice • Do problems 1-4 from the worksheet exponential function

  9. Graphing Exponential functions

  10. Student guided practice • Do problems 5 to 7 in your book 189

  11. Natural base • There is one very important number that arises in the development of exponential functions, and that is the "natural" exponential • e= 2.7182818281…….. • The number "e" is the "natural" exponential, because it arises naturally in math and the physical sciences (that is, in "real life" situations), just as pi arises naturally in geometry. This number was discovered by a guy named Euler (pronounced "OY-ler"; I think he was Swiss), who described the number and named the number "e", and then swore that this stood for "exponential", and not for his own name. • Your calculator can do computations with e; it is probably a "second function" on your calculator, right above the "ln" or "LN" key on your calculator.

  12. Example#1 • Given f(x) = ex, evaluate f(3), rounding to two decimal places.

  13. Example#2 • Given f(x) = 3ex, evaluate f(2), rounding to two decimal places.

  14. Student guided practice • Do problems 29-32 in your book page 189

  15. One very important exponential equation is the compound-interest formula: • ...where "A" is the ending amount, "P" is the beginning amount (or "principal"), "r" is the interest rate (expressed as a decimal), "n" is the number of compounding a year, and "t" is the total number of years.

  16. Example • Suppose that you plan to need $10,000 in thirty-six months' time when your child starts attending university. You want to invest in an instrument yielding 3.5% interest, compounded monthly. How much should you invest?

  17. Example • Suppose Karen has $1000 that she invests in an account that pays 3.5% interest compounded quarterly. How much money does Karen have at the end of 5 years?

  18. Example • William wants to have a total of $4000 in two years so that he can put a hot tub on his deck. He finds an account that pays 5% interest compounded monthly. How much should William put into this account so that he’ll have $4000 at the end of two years?

  19. Homework!!! • Do the compound interest worksheet

  20. Closure • Today we learned about exponential functions and the compound formula. • Next class we are going to continue with exponential functions

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