1. 4_1 Day 2 The natural base: e.

2. WARM-UP: Graph (5/3)^x (5/3)^-x -(5/3)^x (5/3) ^x (5/3)^(x-2)

3. e-the Natural Base In the definition on the first page we use an unspecified base a to introduce exponential functions. It happens that in many applications the convenient choice for a base is the irrational number e, where e = 2.71828…., called the NATURAL BASE. The function f(x) = e x is called the natural exponential function, where e is the constant and x is the variable.

4. e on your calculator y = e^x Graph on Window x=[-5, 5], y=[-2,10] Fill in the table: (notice the table and graph are very similar to the ex1 in warmup xy 0 2 3 4

5. e on your calculator Type in your “y=“ Graph on Window x=[0, 100], y=[-2,4] Use y= to fill in the table below xy 1 10 50 100 1000 Conclusion:

6. Compound Interest Pretend that you choose allowance option #1 yesterday, but you invested your first allowance check ($10) into a bank account. (Forget the other years payments for now). Next year your balance is $11, the following year it is $12.10, the following year is $13.31, then $14.64 (see table). What type of function is this? (linear? quadric? Exponential?) Do you see a pattern? $ in bank $10 $11 $12.10 $13.31 $14.64 $16.11 $17.72 yr 0 1 2 3 4 5 6

7. Compound Interest $ in bank $10 $11 $12.10 $13.31 $14.64 $16.11 $17.72 yr 0 1 2 3 4 5 6 Check this 10.00 = 10(1.1) 0 = 10(1+.1) 0 11.00 = 10(1.1) 1 = 10(1+.1) 1 12.10 = 10(1.1) 2 = 10(1+.1) 2 13.31 = 10(1.1) 3 = 10(1+.1) 3 14.64 = 10(1.1) 4 = 10(1+.1) 4 16.11 = 10(1.1) 5 = 10(1+.1) 5 17.72 = 10(1.1) 6 = 10(1+.1) 6 *We call.1 the interest rate = r *P for Principle meaning the starting value so this case $10 *t is the time in years Find how much the balance is after ten years (so when you are 20 how much will your original $10 be worth)? P P(1+r) 1 P(1+r) 2 P(1+r) 3 P(1+r) 4 P(1+r) 5 P(1+r) 6

8. The formulas: #1. For n compounding per year :  In our previous example, notice n= 1 thus we say its compounded annually or once per year.  If the bank compounded your interest quarterly, what would your balance been at the end of the 10 years?  What if they compounded it daily so (365 days a year? #2 Compounded Continuously:

9. Ex.#6 A total of $12,000 is invested at an annual percentage rate of 9%. Find the balance after five years if it is compounded: a. quarterly b. Semiannually c. continuously

10. Half-Life Ex.#7 Let y represent the mass of a particular radioactive element whose half-life is 25 years. After t years, the mass in grams is given by. a. What is the initial mass (when t = 0)? b. How much of the element is present after 80 years?

11. Exponential Growth Ex.#8 The number of fruit flies in an experimental population after t hours is given by Q(t) = 20e 0.03 t, t > 0. a.) Find the initial number of fruit flies in the population b.) How large is the population after 72 hours? c.) Sketch a graph of Q(t) d.) After how many hours would you expect to see about 300 fruit flies? y x

12. Homework Pg. 290 # 35, 36, 48,49, 50, 52, 53 E.C. Euler worksheet #2 & #3 only on the backside