1. Announcements Topics: -roughly sections 1.4, 2.2, and 2.3 * Read these sections and study solved examples in your textbook! Work On: -Practice problems from the textbook and assignments from the coursepack as assigned on the course web page (under the link “SCHEDULE + HOMEWORK”)

2. Piecewise Functions A piecewise function f(x) is a function whose definition changes depending on the value of x. Example: Absolute Value Function The absolute value of a number x, denoted by |x|, is the distance between x and 0 on the real number line.

3. Piecewise Functions Example: Sketch the graph of f(x).

4. Conversions To be studied independently… however we can take up questions in tutorials look at tables in section 1.2, do conversion, etc.

5. How to read and understand math in journal articles, books about science, and other sources? Math in math textbooks is not exactly the same as math found elsewhere (we will see examples soon). Although the differences are mathematically insignificant (such as different notation), it takes some time to get used to math written in a non-textbook format What do we do? We learn concepts, formulas and algorithms using math textbooks, because it is easier that way Then we use our knowledge to apply to contexts takes from various disciplines

6. Example – Journal Article What is in it?

7. Source, title, author and contact information

8. Abstract

9. Introduction and body of the article

10. Math parts – diagrams (graphs), tables …

11. Math parts – formulas

12. End … conclusion and references

13. Our focus is on math parts Compare to formulas in our textbook - what’s different?

14. Lots of symbols! Some uppercase, some lowercase, Greek alphabet No f(x) notation … we need to read the text to figure out what symbols represent dependent and independent variables, and what symbols represent parameters As well, what is a variable and what is a parameter might change within the article (as in the textbook example on the body mass index)

15. Many parameters are not given numeric values Numeric values are approximate, written in the form 0.68 plus/minus 0.17 Not all text is mathematically relevant (big difference from textbook exercises and examples!)

16. Parentheses around the second term on the right side are not necessary (and in math we would not use them)

17. How do we make ourselves understand this formula?

18. If C is a function and t is independent variable: (1)Which quantities are parameters (2)Replace all parameters by numbers – do you recognize the equation? (3)Keep the parameters (do not give them numeric values); what is the graph of C (you will need to make assumptions, for instance could be positive, negative, or zero) (4)If C is a function of A, what is its graph? (5)If C is a function of r, what is its graph? (6)If C is a function of W, what is its graph? Exercise - Equation Analysis

19. 1.4 Working With Functions Review addition, subtraction, multiplication, division, and composition of functions on your own… Review transformations of graphs and inverse functions (we’ll do a brief review here)

20. Inverse Functions The function is the inverse of if and. Each of and undoes the action of the other. Some simple examples:

21. What Functions Have Inverses? A function has an inverse if and only if it is a one-to-one function. A function f is one-to-one if for every y-value in the range of f, there is exactly one x-value in the domain of f such that y=f(x).

22. Horizontal Line Test If every horizontal line intersects the graph of a function in at most one point, then the graph represents a one-to-one function.

23. Finding the Inverse of a Function Algorithm: 1.Write the equation y=f(x). 2.Solve for x in terms of y. 3.Replace x by (x) and y by x. Note: The domain and range are interchanged Example: Find the inverse of the following. State domain and range.

24. Temperature Conversion The relationship between degrees Celsius (C) and degrees Fahrenheit (F) is linear. We know that corresponds to and corresponds to.

25. Temperature Conversion (a) Find the function that converts to Note: input is and output is Data Points: Function:

26. Temperature Conversion (a) Find the function that converts to Note: input is and output is Data Points: Function:

27. Temperature Conversion (a) Find the function that converts to Note: input is and output is Data Points: Function:

28. Temperature Conversion (a) Find the function that converts to Note: input is and output is Data Points: Function:

29. Temperature Conversion (a) Find the function that converts to Note: input is and output is Data Points: Function:

30. Temperature Conversion (b) Find the function that converts to Note: input is and output is One approach: Find the INVERSE of F(c): Note: we do not interchange variables at the end since F and C have a physical meaning

31. Temperature Conversion (b) Find the function that converts to Note: input is and output is One approach: Find the INVERSE of F(c): Note: we do not interchange variables at the end since F and C have a physical meaning

32. Temperature Conversion (b) Find the function that converts to Note: input is and output is One approach: Find the INVERSE of F(C): Note: we do not interchange variables at the end since F and C have a physical meaning

33. Temperature Conversion (b) Find the function that converts to Note: input is and output is One approach: Find the INVERSE of F(C): Note: we do not interchange variables at the end since F and C have a physical meaning

34. Temperature Conversion (b) Find the function that converts to Note: input is and output is One approach: Find the INVERSE of F(C): Note: we do not interchange variables at the end since F and C have a physical meaning

35. Temperature Conversion (b) Find the function that converts to Note: input is and output is One approach: Find the INVERSE of F(C): Note: we do not interchange variables at the end since F and C have a physical meaning

36. Temperature Conversion (b) Find the function that converts to Note: input is and output is One approach: Find the INVERSE of F(C): Note: we do not interchange variables at the end since F and C have a physical meaning

37. Exponential Functions An exponential function is a function of the form where is a positive real number called the base and is a variable called the exponent. Domain: Range: *Note: Review EXPONENT LAWS on your own!

38. Graphs of Exponential Functions When a>1, the function is increasing. When a<1, the function is decreasing. y=0 is a horizontal asymptote

39. Transformation of an Exponential Function Graph Recall: is a special irrational number between 2 and 3 that is commonly used in calculus Approximation:

40. Logarithmic Functions The inverse of an exponential function is a logarithmic function, i.e. Cancellation equations: In general:For exponentials & logarithms:

41. Graphs of Logarithmic Functions Recall: For inverse functions, the domain and range are interchanged and their graphs are reflections in the line Example: Graph

42. Graphs of Logarithmic Functions

44. Memorize!!!

45. The Natural Logarithm Domain: Range: Graph: The graph increases from negative infinity near x=0 (vertical asymptote) and rises more and more slowly as x becomes larger. Note: and

46. Laws of Logs For x,y>0 and p any real number:

47. Semilog graphs Definition: A semilog graph plots the logarithm of the output against the input. The semilog graph of a function has a reduced range making the key features of the function easier to distinguish.

48. Semilog graphs Example: Compare the graphs and semilog graphs for and

49. Semilog graphs Original GraphsSemilog Graphs

50. Exponential Models When the change in a measurement is proportional to its size, we can describe the measurement as a function of time by the formula where is the value of the measurement at time is the initial value of the measurement, and is a parameter which describes the rate at which the measurement changes

51. Doubling Time. Example: A bacterial culture starts with 100 bacteria and after 3 hours the population is 450 bacteria. Assuming that the rate of growth of the population is proportional to its size, find the time it takes for the population to double.

52. Half-Lives of Drugs

53. Example: Thinking in Half-Lives ** Many drugs are not effective when less than 5% of their original level remains in the body. # of half-livesamount left in body% amount left in body 0M(0)100 1 2 3 4 5

54. Half-Lives of Drugs Example: Thinking in Half-Lives ** Many drugs are not effective when less than 5% of their original level remains in the body. # of half-livesamount left in body% amount left in body 0M(0)100 10.5M(0) 2 3 4 5

55. Half-Lives of Drugs Example: Thinking in Half-Lives ** Many drugs are not effective when less than 5% of their original level remains in the body. # of half-livesamount left in body% amount left in body 0M(0)100 10.5M(0)50 2 3 4 5

56. Half-Lives of Drugs Example: Thinking in Half-Lives ** Many drugs are not effective when less than 5% of their original level remains in the body. # of half-livesamount left in body% amount left in body 0M(0)100 10.5M(0)50 20.5 2 M(0) 3 4 5

57. Half-Lives of Drugs Example: Thinking in Half-Lives ** Many drugs are not effective when less than 5% of their original level remains in the body. # of half-livesamount left in body% amount left in body 0M(0)100 10.5M(0)50 20.5 2 M(0)25 3 4 5

58. Half-Lives of Drugs Example: Thinking in Half-Lives ** Many drugs are not effective when less than 5% of their original level remains in the body. # of half-livesamount left in body% amount left in body 0M(0)100 10.5M(0)50 20.5 2 M(0)25 30.5 3 M(0) 4 5

59. Half-Lives of Drugs Example: Thinking in Half-Lives ** Many drugs are not effective when less than 5% of their original level remains in the body. # of half-livesamount left in body% amount left in body 0M(0)100 10.5M(0)50 20.5 2 M(0)25 30.5 3 M(0)12.5 4 5

60. Half-Lives of Drugs Example: Thinking in Half-Lives ** Many drugs are not effective when less than 5% of their original level remains in the body. # of half-livesamount left in body% amount left in body 0M(0)100 10.5M(0)50 20.5 2 M(0)25 30.5 3 M(0)12.5 40.5 4 M(0) 5

61. Half-Lives of Drugs Example: Thinking in Half-Lives ** Many drugs are not effective when less than 5% of their original level remains in the body. # of half-livesamount left in body% amount left in body 0M(0)100 10.5M(0)50 20.5 2 M(0)25 30.5 3 M(0)12.5 40.5 4 M(0)6.25 5

62. Half-Lives of Drugs Example: Thinking in Half-Lives ** Many drugs are not effective when less than 5% of their original level remains in the body. # of half-livesamount left in body% amount left in body 0M(0)100 10.5M(0)50 20.5 2 M(0)25 30.5 3 M(0)12.5 40.5 4 M(0)6.25 50.5 5 M(0)

63. Half-Lives of Drugs Example: Thinking in Half-Lives ** Many drugs are not effective when less than 5% of their original level remains in the body. # of half-livesamount left in body% amount left in body 0M(0)100 10.5M(0)50 20.5 2 M(0)25 30.5 3 M(0)12.5 40.5 4 M(0)6.25 50.5 5 M(0)3.125

64. Trigonometric Functions Trigonometric functions are used to model quantities that oscillate.

65. Trigonometric Models Example: Seasonal Growth A population of river sharks in New Zealand changes periodically with a period of 12 months. In January, the population reaches a maximum of 14, 000, and in July, it reaches a minimum of 6, 000. Using a trigonometric function, find a formula which describes how the population of river sharks changes with time.

66. Trigonometric Models Example: (A41, #2.)

67. Graphs of Trigonometric Functions Example:

68. Inverse Trigonometric Functions Since the 3 main trigonometric functions are not one-to-one on their natural domains we must first restrict their domains in order to define inverses.

69. Inverse of Sine Restrict the domain of to Now the function is one-to-one on this interval so we can define an inverse.

70. Inverse of Sine The inverse of the restricted sine function is denoted by or Cancellation equations: Calculate: (domain of sin x) (domain of arcsin x)

71. Graphs of Sine and Arcsine y = sin xy = arcsin x domain: range: domain: range:

72. Inverse of Tangent Restrict the domain of to This portion of tangent passes the HLT so tangent is one-to-one here

73. Inverse of Tangent The inverse of the restricted tangent function is denoted by or Cancellation equations: Calculate: (restricted domain of tan x) (domain of arctan x)

74. Graphs of Tangent and Arctangent y = cos xy = arccos x domain: range: domain: range: y = tan x y = arctan x

75. Real-life Use of Arctangent Example: Model for World Population One of the many models used to analyze human population growth is given by where t represents a calendar year and P(t) is the population in billions.