Exponential Functions

Transcription

1 Exponential Functions MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011

2 Objectives In this lesson we will learn to: recognize and evaluate exponential functions with base a, graph exponential functions and use the One-to-One Property, recognize and evaluate exponential functions with base e, use exponential functions to model and solve real-world problems.

3 Exponential Functions Definition The exponential function f with base a is denoted by f (x) = a x where a > 0, a 1, and x is any real number.

4 Exponential Functions Definition The exponential function f with base a is denoted by f (x) = a x where a > 0, a 1, and x is any real number. Example Use a calculator if necessary to evaluate each of the following expressions = = = = =

5 Exponential Functions Definition The exponential function f with base a is denoted by f (x) = a x where a > 0, a 1, and x is any real number. Example Use a calculator if necessary to evaluate each of the following expressions = = = = =

6 Exponential Functions Definition The exponential function f with base a is denoted by f (x) = a x where a > 0, a 1, and x is any real number. Example Use a calculator if necessary to evaluate each of the following expressions = = = = =

7 Graphs, a > 1 When a > 1 the graph of an exponential function resembles the following. x 2 x x

8 Effect of the Base If we increase the value of the base a, the graph becomes steeper x

9 Graphs, 0 < a < 1 When 0 < a < 1 the graph of an exponential function resembles the following. x ( ) x x

10 Asymptote Regardless of the base a, the function f (x) = a x approaches, but never touches the line y = 0 (the x-axis). In situations like this we may say, y = 0 is an asymptote of the graph of f (x) = a x, or y = 0 is a horizontal asymptote of the graph of f (x) = a x, or the graph of f (x) = a x approaches the line y = 0 asymptotically.

11 General Concepts of Exponential Functions For a > 1: For 0 < a < 1: a x > 0 a x increases on (, ) and is called an exponential growth function. a 0 = 1, so the point (0, 1) is on the graph. a x approaches the x-axis for negative values of x (The x-axis is a horizontal asymptote). a x > 0 a x decreases on (, ) and is called an exponential decay function. a 0 = 1, so the point (0, 1) is on the graph. a x approaches the x-axis for positive values of x (The x-axis is a horizontal asymptote).

12 One-to-One Property Since exponential functions are either always increasing or always decreasing, they all pass the Horizontal Line Test and are therefore one-to-one functions. One-to-One Property For a > 0 and a 1, a x = a y if and only if x = y.

13 One-to-One Property Since exponential functions are either always increasing or always decreasing, they all pass the Horizontal Line Test and are therefore one-to-one functions. One-to-One Property For a > 0 and a 1, a x = a y if and only if x = y. Example Use the One-to-One Property to solve the following equation. 2 x 3 = 16

14 One-to-One Property Since exponential functions are either always increasing or always decreasing, they all pass the Horizontal Line Test and are therefore one-to-one functions. One-to-One Property For a > 0 and a 1, a x = a y if and only if x = y. Example Use the One-to-One Property to solve the following equation. 2 x 3 = 16 2 x 3 = 2 4

15 One-to-One Property Since exponential functions are either always increasing or always decreasing, they all pass the Horizontal Line Test and are therefore one-to-one functions. One-to-One Property For a > 0 and a 1, a x = a y if and only if x = y. Example Use the One-to-One Property to solve the following equation. 2 x 3 = 16 2 x 3 = 2 4 x 3 = 4 x = 7

16 Natural Base e In many applications (particularly to the physical and social sciences) we will prefer to use the base e This is called the natural base and the function f (x) = e x is called the natural exponential function.

17 Natural Base e In many applications (particularly to the physical and social sciences) we will prefer to use the base e This is called the natural base and the function f (x) = e x is called the natural exponential function. Example Use a calculator if necessary to evaluate each of the following expressions. e 1.2 = e 3 = e 2.7 = e 0.2 = e π =

18 Natural Base e In many applications (particularly to the physical and social sciences) we will prefer to use the base e This is called the natural base and the function f (x) = e x is called the natural exponential function. Example Use a calculator if necessary to evaluate each of the following expressions. e 1.2 = e 3 = e 2.7 = e 0.2 = e π =

19 Natural Base e In many applications (particularly to the physical and social sciences) we will prefer to use the base e This is called the natural base and the function f (x) = e x is called the natural exponential function. Example Use a calculator if necessary to evaluate each of the following expressions. e 1.2 = e 3 = e 2.7 = e 0.2 = e π =

20 Application: Compound Interest Definition After t years, the balance A in an account with principal P and annual interest rate r (expressed as a decimal) is given by the following formulas. ( 1 For n compounding periods per year: A = P 1 + r ) n t n 2 For continuous compounding: A = Pe r t

21 Example Determine the balance A at the end of 20 years if $1500 is invested at 6.5% interest and the interest is compounded quarterly monthly weekly continuously

22 Example Determine the balance A at the end of 20 years if $1500 is invested at 6.5% interest and the interest is compounded quarterly ( A = ) (4)(20) monthly weekly continuously

23 Example Determine the balance A at the end of 20 years if $1500 is invested at 6.5% interest and the interest is compounded quarterly ( A = ) (4)(20) monthly ( A = ) (12)(20) weekly continuously

24 Example Determine the balance A at the end of 20 years if $1500 is invested at 6.5% interest and the interest is compounded quarterly ( A = ) (4)(20) monthly weekly ( A = ) (12)(20) ( A = ) (52)(20) continuously

25 Example Determine the balance A at the end of 20 years if $1500 is invested at 6.5% interest and the interest is compounded quarterly ( A = ) (4)(20) monthly weekly ( A = ) (12)(20) ( A = ) (52)(20) continuously A = 1500e (0.065)(20)

26 Application: Exponential Growth The number of fruit flies in an experimental population after t hours is given by Q(t) = 20e 0.03t for t 0. 1 Find the initial number of fruit flies in the population. 2 How large is the population of fruit flies after 72 hours?

27 Application: Exponential Growth The number of fruit flies in an experimental population after t hours is given by Q(t) = 20e 0.03t for t 0. 1 Find the initial number of fruit flies in the population. Q(0) = 20e (0.03)(0) = 20 2 How large is the population of fruit flies after 72 hours?

28 Application: Exponential Growth The number of fruit flies in an experimental population after t hours is given by Q(t) = 20e 0.03t for t 0. 1 Find the initial number of fruit flies in the population. Q(0) = 20e (0.03)(0) = 20 2 How large is the population of fruit flies after 72 hours? Q(72) = 20e (0.03)(72) 173

29 Application: Radioactive Decay Let Q represent the mass of carbon-14 ( 14 C) in grams. The quantity of carbon-14 present after t years is given by Q(t) = 10 ( ) 1 t/ Find the initial quantity of carbon-14 present. 2 Determine the quantity present after 2500 years.

30 Application: Radioactive Decay Let Q represent the mass of carbon-14 ( 14 C) in grams. The quantity of carbon-14 present after t years is given by Q(t) = 10 ( ) 1 t/ Find the initial quantity of carbon-14 present. Q(0) = 10 ( ) 1 0/5715 = 10 grams 2 2 Determine the quantity present after 2500 years.

31 Application: Radioactive Decay Let Q represent the mass of carbon-14 ( 14 C) in grams. The quantity of carbon-14 present after t years is given by Q(t) = 10 ( ) 1 t/ Find the initial quantity of carbon-14 present. Q(0) = 10 ( ) 1 0/5715 = 10 grams 2 2 Determine the quantity present after 2500 years. Q(2500) = 10 ( ) / grams 2

32 Half-Life Q(t) = 10 ( ) 1 t/5715 2

33 Homework Read Section 3.1. Exercises: 1, 5, 9, 13,..., 69, 73