Chapter 2 Nonlinear Models Sections 2.1, 2.2, and 2.3

Nonlinear Models  Quadratic Functions and Models  Exponential Functions and Models  Logarithmic Functions and Models

Quadratic Function A quadratic function of the variable x is a function that can be written in the form Example: where a, b, and c are fixed numbers

The graph of a quadratic function is a parabola. a > 0a < 0 Quadratic Function

Vertex coordinates are: x – intercepts are solutions of y – intercept is: symmetry Vertex, Intercepts, Symmetry

Vertex: x – intercepts y – intercept Graph of a Quadratic Function Example 1: Sketch the graph of

Vertex: x – intercepts y – intercept Graph of a Quadratic Function Example 2: Sketch the graph of

Vertex: x – intercepts y – intercept Graph of a Quadratic Function Example 3: Sketch the graph of no solutions

Example: For the demand equation below, express the total revenue R as a function of the price p per item and determine the price that maximizes total revenue. Maximum is at the vertex, p = $100 Applications

Example: As the operator of Workout Fever health Club, you calculate your demand equation to be q  0.06p + 84 where q is the number of members in the club and p is the annual membership fee you charge. 1. Your annual operating costs are a fixed cost of $20,000 per year plus a variable cost of $20 per member. Find the annual revenue and profit as functions of the membership price p. 2. At what price should you set the membership fee to obtain the maximum revenue? What is the maximum possible revenue? 3. At what price should you set the membership fee to obtain the maximum profit? What is the maximum possible profit? What is the corresponding revenue? Applications

The annual revenue is given by Solution The annual cost as function of q is given by The annual cost as function of p is given by

Thus the annual profit function is given by Solution

The graph of the revenue function is

The profit function is

Nonlinear Models  Quadratic Functions and Models  Exponential Functions and Models  Logarithmic Functions and Models

An exponential function with (constant) base b and exponent x is defined by Notice that the exponent x can be any real number but the output y = b x is always a positive number. That is, Exponential Functions

Example: where A is an arbitrary but constant real number. We will consider the more general exponential function defined by

Graph of Exponential Functions when b > 1

Graph of Exponential Functions when 0 < b < 1

xy -41/16 -31/8 -21/4 1/ xy -41/16 -31/8 -21/4 1/ Graph of Exponential Functions when b > 1

Graphing Exponential Functions xy /2 2 1/4 3 1/8 4 1/16 xy /2 2 1/4 3 1/8 4 1/16

Graphing Exponential Functions

Laws of Exponents LawExample

Finding the Exponential Curve Through Two Points Example: Find an exponential curve y  Ab x that passes through (1,10) and (3,40). Plugging in b  2 we get A  5

A certain bacteria culture grows according to the following exponential growth model. If the bacteria numbered 20 originally, find the number of bacteria present after 6 hours. Thus, after 6 hours there are about 830 bacteria Exponential Functions-Examples When t  6

Compound Interest A = the future value P = Present value r = Annual interest rate (in decimal form) m = Number of times/year interest is compounded t = Number of years

Find the accumulated amount of money after 5 years if $4300 is invested at 6% per year and interest is reinvested each month = $ Compound Interest

where e is an irrational constant whose value is The exponential function with base e is called “The Natural Exponential Function” The Number e

The Natural Exponential Function

A way of seeing where the number e comes from, consider the following example: If $1 is invested in an account for 1 year at 100% interest compounded continuously (meaning that m gets very large) then A converges to e: The Number e

Continuous Compound Interest A = Future value or Accumulated amount P = Present value r = Annual interest rate (in decimal form) t = Number of years

Example: Find the accumulated amount of money after 25 years if $7500 is invested at 12% per year compounded continuously. Continuous Compound Interest

Example: Human population The table shows data for the population of the world in the 20th century. The figure shows the corresponding scatter plot. Exponential Regression

The pattern of the data points suggests exponential growth. Therefore we try to find an exponential regression model of the form P(t)  Ab t Exponential Regression

We use a graphing calculator with exponential regression capability to apply the method of least squares and obtain the exponential model Exponential Regression

Nonlinear Models  Quadratic Functions and Models  Exponential Functions and Models  Logarithmic Functions and Models

How long will it take a $800 investment to be worth $1000 if it is continuously compounded at 7% per year? A New Function Input Output

Basically, we take the exponential function with base b and exponent x, and interchange the role of the variables to define a new equation This new equation defines a new function. A New Function

xy 1/16 1/8 1/4 1/ xy 1/16-4 1/8-3 1/4-2 1/ Graphing The New Function Example: graph the function x  2 y

Logarithms The logarithm of x to the base b is the power to which we need to raise b in order to get x. Example:Answer:

xy 1/16 1/8 1/4 1/ xy 1/16-4 1/8-3 1/4-2 1/ Graphing y  log 2 x Recall that y  log 2 x is equivalent to x  2 y

Common Logarithm Natural Logarithm Abbreviations Base 10 Base e Logarithms on a Calculator

To compute logarithms other than common and natural logarithms we can use: Example: Change of Base Formula

Graphs of Logarithmic Function

Properties of Logarithms

Example: How long will it take an $800 investment to be worth $1000 if it is continuously compounded at 7% per year? Apply ln to both sides Application About 3.2 years

Logarithmic Functions A more general logarithmic function has the form or, alternatively, Example: