Lecture 7: Sections 2.3 and 2.4 Rational and Exponential Functions. Recall that a power function has the form f(x) = x r where r is a real number.

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1 L7-1 Lecture 7: Sections 2.3 and 2.4 Rational and Exponential Functions Recall that a power function has the form f(x) = x r where r is a real number. f(x) = x 1/2 f(x) = x 1/3 ex. Sketch the graph of f(x) = 1 x.

2 L7-2 Def. A rational function has the form f(x) = p(x) q(x) where p(x) and q(x) are polynomial functions, q(x) 0. Asymptotes Def. The line x = a is a vertical asymptote of the graph of function f(x) if the function values f(x) increase or decrease without bound as x gets closer and closer to a (from one or both sides). To find the vertical asymptotes of a rational function: ex. Find the domain and each vertical asymptote of the given function: 1. f(x) = x2 1 x 2 + 4

3 L f(x) = 2x + 3 x 2 x 2 3. f(x) = x2 5x + 4 x 4 Def. The line y = L is a horizontal asymptote of the graph of function f if function values f(x) approach L as x increases or decreases without bound. Later we will define asymptotes more formally in terms of limits.

4 L7-4 ex. Sketch the graph of f(x) = 2x + 1 x 2.

5 L7-5 ex. A company is planning to produce a new smartphone. The fixed costs of production are $100,000 and each phone will cost $350 to produce. Find the cost function C(x) and the average cost function C. Find the horizontal asymptote of C. What does it mean? Exponential Functions (Sec. 2.4) Be sure to review sections R6 and R7. Def. An exponential function with base a is the function defined by where x is any real number, a > 0 and a 1.

6 L7-6 How to evaluate a x for all x? Consider the specific exponential function f(x) = 4 x and let n be a positive integer. Evaluate: 1. f(n) 2. f(0) 3. f( n) 4. f(3/2) 5. f( 2) Properties of Exponents Let x, y be real numbers, with a > 0 and b > 0. 1) a x a y = 2) ax a y = 3) (a x ) y = 4) (ab) xy = 5) ( a b ) x =

7 L7-7 ex. Simplify the expression ( 83n ) n To solve equations with exponentials, we use the following property: 6) Let x and y be real numbers with a > 0, a 1. If a x = a y, then NOTE: What happens if a = 1? ex. Solve for x: ( ) 2x 5 1 = 81 x 27

8 The Graph of an Exponential Function Graph on the axes below: 1) f(x) = 2 x L7-8 2) f(x) = ( ) x 1 2 3) f(x) = 2 x

9 Characteristics of Exponential Functions 1. Domain Range L Intercept(s) 3. Asymptote(s) ex. Sketch the graph of f(x) = 3 (x+2) 1.

10 L7-10 The base e Consider the following table of values (p. 82): m ( ) m m , , ,000, Note that as m increases without bound, approaches a single number, called e. We say that as m, ( m) m e. ( ) m m NOTE: e

11 L7-11 ex. Sketch the graph of f(x) = e x. Now sketch the graph of f(x) = e x. An Application: Compound Interest Simple Interest Simple Interest is computed on the original principal (initial amount borrowed/invested) only: ex. Suppose $2000 is invested at 4% simple interest. How much is in the account after one year? After t years?

12 L7-12 Simple Interest Formula If principal P is invested at an annual rate of interest r (percentage rate expressed as a decimal), the amount A in the account after t years is given by Compound Interest More commonly, interest earned is added to the principal at regular intervals and earns interest at the same rate. ex. Suppose our $2000 is invested at 4% annual interest compounded annually. How much is in the account 1. after one year 2. after two years 3. after t years

13 L7-13 Now suppose interest is calculated quarterly (four times a year or every three months). How much is in the account after one quarter? After one year? Compound Interest Formula Suppose that principal P is invested at an annual interest rate r (as a decimal), compounded m times per year. The compound amount A after a term of t years is given by

14 L7-14 ex. Mr. Adams invested $1000 in an account earning 4 1 % interest per year. How much will be in the 2 account after 3 years if interest is compounded 1. quarterly 2. monthly

15 L7-15 Continuous Compounding Now suppose the number of compoundings m increases without bound. We have continous compounding. Continuous Compounding Formula If principal P is invested at an annual interest rate r (as a decimal) compounded continuously, the compound amount A after t years is given by ex. If $1000 is invested at 4 1 % per year compounded continuously, how much will be in the ac- 2 count after 3 years? Proof(page 83 in text):

16 L7-16 Now You Try It! 1. Simplify the expression ( x 3n+6 y 9n (x 3 y n 1 ) 6n ) 1/3. 2. Solve for x: 5 x x2 = 1 25 x 3. Sketch the graph of a) g(x) = 1 2 x+3 and b) f(x) = e 2 x. 4. Given f(x) = Be kx for real numbers B and k. If f(0) = 80 and f(1) = 140, find the following: a) The value of B b) Rewrite f(x) = Be kx as f(x) = Ba x. Hint: remember that e kx = (e k ) x. 5. The population of a new development is doubling every 4 years. If the initial population was 500 people, find a formula for P (t), the population t years after it was opened. Hint: P (t) = Ab kt. Find the population for years 0, 4, 8 and 12 and try to find a pattern. Then use your formula to estimate the population 10 years after it was opened. 6. If you have $5000 to invest for two years, which would give the greater return: 5% compounded monthly or 4.85% compounded continuously? 7. Suppose that the amount of information retained by an individual student in one of his college courses can be modeled by the function f(x) = 76e 0.5x + 24 where f(x) is the percentage of information remembered after x weeks, starting from the last week of the class. How much information did he retain initially? After 4 weeks? Find the percentage that should be retained long term, by letting x increase without bound. Hint: consider the graph of y = e x.