Math 101 Section 1.2: Exponential Functions Date:

What is an exponent? It is a shortcut to repeating multiplication.

a1 = a a2 = a · a a3 = a · a · a a4 = a · a · a · a . . an =

Exponent Identities (There are more that you could find, but these should get you pretty far.)

n m n+m an • a a = a • = an−m am

• (an)m = anm • a0 = 1 as long as a 6= 0

n n n • (ab) = a b 1 • = an a−n 1 √ • = a−n • n a = a1/n an Practice with these rules. (The goal is to get used to using them to manipulate the symbols correctly.

6 1/3 1 1. (2 ) 5. (27x2)−1

2. 5−2 6. (x + 1)3

6x3y2 3. (2x3yz)4 7. 2x−1y4

4. (x4)1/2 8. (x + 1)1/4 Today we’re going on functions of the form f(x) = ax

An exponential function f(x) = ax satisfies the following idea: for every unit of x, the function changes by the same ratio.

Example. At time t = 0, you start the culture with 100 bacteria. After 3 hours, the population is twice as big. After another 3 hours, the population is twice as big again.

t a. Why might it seem practical to use an exponential function f(t) = P0a to model this situation? What is tricky or different here?

b. Find P0 and the base a.

c. What is the size of the population after 15 hours?

The following are graphs of functions of the form ax or a−x. Label each with a possible value of a. Which graphs correspond to a negative exponent? Why? What are the domain and range?

Exponential Growth Exponential Decay The number e, a special base for exponential functions that will be important in Calculus: e ≈ 2.171828... is an irrational, transcendental number that will make some formulas easier later. Change of base: ekt = (ek)t = at. Example: Write e0.2t in the form at for some number a.

Practice exponent rules using base e. Simplify the following as much as possible.

1. (ex)2 3. (e3x)1/3

2. e4ex 4. e2x + e2

After class: Go through sections 1.1 and 1.2 to find the definitions of these words. Write a sentence and draw a picture or two to describe them. • A function is decreasing if

• A function is increasing if

• A function is concave up if

• A function is concave down if

Use a graphing website like Wolfram Alpha or Desmos to look at the graph of y = ekx for different k values (positive, negative, large numbers, small fractions). For what values of k is the exponential function y = ekx • increasing? • decreasing? • concave up? • concave down?