1 Nonlinear Models Chapter 2 Quadratic Functions and Models Exponential Functions and Models Logarithmic Functions and Models Logistic Functions and Models Lectures 2 & 3

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

3 Quadratic Function Every quadratic function has a parabola as its graph. a > 0a < 0

4 Features of a Parabola Vertex: x – intercepts y – intercept symmetry

5 Sketch of a Parabola Vertex: x – intercepts y – intercept Ex.

6 Application Ex. 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

7 Exponential Function An exponential function with base b and exponent x is defined by Ex. where A and b are constants.

8 Laws of Exponents LawExample

9 Graphing Exponential Functions Ex. (0,1) x y

10 Finding the Exponential Curve Through Two Points Ex. Find an equation of the exponential curve that passes through (1,10) and (3,40). Plugging in we get A = 5

11Example Ex. A certain bacteria culture grows according to the following exponential growth model. The bacteria numbered 20 originally, find the number of bacteria present after 6 hours. So about 830 bacteria

12 Compound Interest A = the future value P = Present value r = Annual interest rate m = Number of times/year interest is compounded t = Number of years

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

14 The Number e e is an irrational constant. If $1 is invested for 1 year at 100% interest compounded continuously (m gets very large) then A converges to e:

15 Continuous Compound Interest A = Accumulated amount P = Present value r = Annual interest rate t = Number of years

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

17 Logarithms The base b logarithm of x is the power to which we need to raise b in order to get x. Ex.

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

19 Change-of-Base Formula To compute logarithms other than common and natural logarithms we can use: Ex.

20 Logarithmic Function Graphs Ex. (1,0)

21 Properites of Logarithms

22 Ex. How long will it take a $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

23 Logarithmic Function A logarithmic function has the form Also: Ex.

24Example Suppose that the temperature T, in degrees Fahrenheit, of an object after t minutes can be modeled using the following equation: 1. Find the temperature after 5 minutes. 2. Find the time it takes to reach 190°.

25 Logistic Function where A, N, b are constants. A logistic function is a function that may be expressed in the form:

26 Logistic Function NN b >1 0 < b <1 N is called the limiting value

27 Logistic Function for Small x Thus it grows approximately exponentially with base b. For small values of x we have:

28 Modeling Ex. A small school district has 2400 people. Initially 10 people have heard a particular rumor and the number who have heard it is increasing at 50%/day. It is anticipated that eventually all 2400 people will hear the rumor. Find a logistic model for the number of people who have heard the rumor after t days. Using (0,10):

29 For small value of t: in 1 day 15 people will know so b = A = 239