Chapter 2 Nonlinear Models Sections 2.1, 2.2, and 2.3

1. Chapter 2Nonlinear ModelsSections 2.1, 2.2, and 2.3

2. Nonlinear Models • Quadratic Functions and Models • Exponential Functions and Models • Logarithmic Functions and Models

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

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

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

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

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

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

9. Applications 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

10. 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?

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

12. Solution Thus the annual profit function is given by

13. The graph of the revenue function is

14. The graph of the revenue function is

15. The profit function is

16. The profit function is

17. Nonlinear Models • Quadratic Functions and Models • Exponential Functions and Models • Logarithmic Functions and Models

18. Exponential Functions 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 outputy=bxis always a positive number. That is,

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

20. Graph of Exponential Functionswhen b > 1

21. Graph of Exponential Functionswhen 0 < b < 1

22. Graph of Exponential Functionswhen b > 1

23. Graphing Exponential Functions

24. Graphing Exponential Functions

25. Laws of Exponents Law Example

26. Finding the Exponential Curve Through Two Points Example: Find an exponential curve yAbx that passes through (1,10) and (3,40). Plugging inb2we getA5

27. When t6 Exponential Functions-Examples 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

28. 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

29. 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 = $5800.06

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

31. The Natural Exponential Function

32. The Number e 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:

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

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

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

36. 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) Abt

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

38. Nonlinear Models • Quadratic Functions and Models • Exponential Functions and Models • Logarithmic Functions and Models

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

40. A New Function 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.

41. Graphing The New Function Example:graph the function x2y

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

43. Graphing ylog2 x Recall thaty  log2x is equivalent to x2y

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

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

46. Graphs of Logarithmic Function

47. Properties of Logarithms

48. Application 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 About 3.2 years

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