1. Welcome Back!

2. Who is Mrs. Meyer? WO graduate, Class of ’98
GVSU graduate, Class of ‘02
Cornerstone University , masters 2008
MATH NERD!
Running in the Hot Chocolate 15K in November
Love to travel (but not rustic camping!!)
I love to read!
Who is Mrs. Meyer?

3. WO Powder Puff, Class of 1998

4. Married to Mr. Meyer… yup… the orchestra guy!

5. Prerequisites for Calculus  WE NEED TEXTBOOKS 
Chapter 1
Prerequisites for Calculus 
WE NEED TEXTBOOKS 

6. Section 1.1: Lines Learning Targets:
I can write an equation and sketch a graph of a line given specific information.
I can identify the relationships between parallel/perpendicular lines and slopes.

7. Example 1
If a particle moves from the point (a, b) to the point (c, d), the slope would be:

8. Slopes
With your partner, come up with as many ways to name slope as you can. (4 min)

9. Parallel and Perpendicular Lines
equivalent
The slopes of parallel lines are ______________
The slopes of perpendicular lines are _____________________
(or the product of the two slopes is _______)
Opposite reciprocals -1

10. y = mx + b Ax + By = C y – y1 = m(x – x1) Equations of Lines
Slope-intercept form:
Standard form (General Linear Equation):
Point-Slope form:
y = mx + b
Ax + By = C
y – y1 = m(x – x1)

11. Example 1:
Example 1:
Write an equation for the line through the point (-1, 2) that is (a) parallel, and (b) perpendicular to the line y = 3x – 4. (Leave your answers in point slope!!)
Parallel: Perpendicular:

12. Section 1.2 Notes: Functions and Graphs
Learning Targets:
I can identify the domain and range of a function using its graph or equation.
I can recognize even and odd functions using equations and graphs.
I can interpret and find formulas for piecewise defined functions.
I can write and evaluate compositions of two functions.

13. What is a function? Brainstorm w/ partners
Dependent variables:
Independent variables:
Domain:
Range: y x
input
{x: }
[ ]
output
{y: }
[ ]

14. Viewing and Interpreting Graphs
Identify the domain and range, and then sketch a graph of the function. No Calculator

15. Graph Viewing Skills Recognize that the graph is reasonable.
See all important characteristics of the graph.
Interpret those characteristics.
Recognize grapher/calculator failure.

16. Example 3:
Use calculator to identify the domain and range, and then draw a graph of the function.

17. Even Functions and Odd Functions
Even functions: (Symmetric about the y-axis)
Odd functions: (Rotation symmetric about the origin)
f(-x) = f(x)
f(-x) = -f(x)

18. Example 4: Odd or even

19. Example 5: Piecewise

20. Example 6: Absolute Values
Draw the graph of Then find the domain and range.

21. Example 7: Composites Find a formula for f(g(2)), and g(f(2))
f(g(2)) g(f(2))

22. Warm Up
Solve the equations:
1. x3 = 17
2. x10 =
Simplify: 3.

23. Section 1.3 Notes: Exponential Functions
Learning Targets:
I can determine the domain, range, and graph of an exponential function.
I can solve problems involving exponential growth and decay.
I can use exponential regression to solve problems.

24. Exponential Growth
Definition: Let a be a positive real number other than 1. The function f(x) = ax is the exponential function with base a.

25. Example 1:
Graph the function y = 3(2x) – 4. State the domain and range.

26. Example 2:
Find the zeros (solutions) of f(x) = (1/3)x - 4 graphically. (Sketch a picture of the solution).

27. Rules for Exponents
If a > 0 and b > 0, the following hold for all real numbers x and y.

28. Exponential Decay
Definition: The half-life of a radioactive substance is the amount of time it takes for half of the substance to change from its original radioactive state to a nonradioactive state by emitting energy in the form of radiation.

29. Example 3:
Suppose the half-life of a certain radioactive substance is 20 days and that there are 5 grams present initially. When will there be only 1 gram of the substance remaining?

30. Growth and Decay
Definition: The function f(x) = kax , k > 0 is a model for exponential growth if a > 1, and a model for exponential decay if 0 < a < 1.

31. The Number e e
The number e is used in problems where interest (for example) is being compounded continuously with the formula A(t) = Pert.
What is the formula that we use if we are not compounding continuously?

32. Example 4
Graph y =

33. Example 5
Graph y =

34. The number e is used in problems where interest (for example) is being compounded continuously with the formula A(t) = Pert.
What is the formula that we use if we are not compounding continuously?

35. Example 6:
Chenelle opened a bank account at a 1.25% interest rate compounded quarterly. She put $500 in the account 10 years ago and has not touched the account since then. How much should be in her account today?

36. Example 7:
How long would it take Chenelle’s investment to double if the account was compounded continuously?

37. Study note!
To help you study for your quiz over , you may want to practice the quiz using the “Quick Quiz” on page 29 of your textbook. These serve as a good review, but also great AP testing practice!

38. Parametric Equations
Please complete the activity on pages 10 – 12 in your packets. Work with your table partner. Your calculator should be in radian mode and parametric mode.

39. Section 1.4 Notes: Parametric Functions
Objectives: Relations, circles, ellipses, lines and other curves. Parametric equations can be used to obtain graphs of relations and functions.

40. Relations
Definition: A relation is a set of ordered pairs (x, y) of real numbers.
The graph of a relation is the set of points in the plane that correspond to the ordered pairs of the relation.
If x and y are functions of a third variable t, called a parameter, then we use the parametric mode of our calculator.

41. Example 1:
Describe the graph of the relation determined by when . Indicate the direction in which the curve is being traced. Find a Cartesian equation for a curve that contains the parametrized curve.

42. Definitions
Definition: If x and y are given as functions over an interval of t-values, then the set of points defined by these equations is a parametric curve.
NOTE: If we are graphing a parametric curve on a closed interval [a, b], we consider the point (f(a), g(a)) the initial point and (f(b), g(b)) the terminal point.

43. Circles
Open your book to page 31. With your partner, complete the Exploration 1: Parametrizing Circles. Record your answers/responses below so we can discuss as a group: 1. 2. 3. 4.

44. Example 2:
Describe the graph of the relation determined by x = 2 cos t, y = 2 sin t, when Find the initial and terminal points, if any, and indicate the direction in which the curve is traced.
Find a Cartesian equation for a curve that contains the parametrized curve.

45. Ellipses Example 3:
Graph the parametrized curve x = 3 cos t, y = 4 sin t, Find the Cartesian equation for a curve that contains the parametric curve. Find the initial and terminal points, if any, and indicate the direction in which the curve is traced.

46. Lines and Other Curves Example 4:
Draw and identify the graph of the parametric curve determined by x = 3t, y = 2 – 2t,

47. Example 5:
Find a parametrization for the line segment with endpoints (-2, 1) and (3, 5).

48. Section 1.5 Notes: Functions and Log
Learning Targets:
I can identify one to one functions.
I can determine the algebraic representation and the graphical representation of a function and its inverse.
I can use parametric equations to graph inverse functions.
I can apply the properties of logarithms.
I can use logarithmic regression equations to solve problems.

49. One-to-one Functions
Definition: A function f(x) is one-to-one on a domain D if f(a) f(b) whenever a b.
NOTE: A one-to-one function passes the vertical line test AND the horizontal line test!

50. Example 1:
Determine if the following functions are one- to-one:
f(x) = |x|
g(x) =
No, although this is a function it does not pass horizontal line test
Yes, passes both!
Also, note: we know this does not include the bottom half of the square root. If it included a + / - in front it would!

51. Inverses
Definition: The function defined by reversing a one-to-one function f is the inverse of f.
NOTE: If , then f and g are inverses.

52. Finding Inverses Writing f-1 as a function of x:
Solve the equation y = f(x) for x in terms of y.
Interchange x and y. The resulting formula will be y = f -1(x).

53. Example 2:
Show that the function y = f(x) = -2x + 4 is one-to-one and find its inverse function.

54. Graph the one-to-one function
f(x) = x2, and its inverse and the line y = x,
Express the inverse of
f as a function of x.

55. Logarithmic Functions
The base a logarithm function is the inverse of the base a exponential function ( ).

56. Example 4: x =e 3t + 5 Solve for x: (A) ln x = 3t + 5 (B) e2x = 10
Base of the log is the base of the exponent
x =e 3t + 5

57. Properties of Logarithms:
For any real numbers x > 0 and y > 0,
Product Rule:
Quotient Rule:
Power Rule:

58. Example 5: Solve a) ln(2x – 1) = ln 16 b) ln 56 – ln x = 4
c) ln (x + 4) + ln x = ln 12

59. Definition: Change of Base Formula:

60. Example 5:
Graph

61. Example 6:
Sarah invests $1000 in an account that earns 5.25% interest compounded annually. How long will it take the account to reach $2500? (Solve algebraically and confirm graphically!)

62. Use the logarithm rules to simplify:
Warm Up (Do this warmup instead of the “Friday” warm-up. You can put it in the Thursday spot
Use the logarithm rules to simplify:
Addition prop
Subtract prop
Addition prop

63. Section 1.6 Notes: Trigonometric Functions
Learning Targets:
I can convert between radians and degrees, and find arc length.
I can identify the periodicity and even-odd properties of the trigonometric functions.
I can find values of trigonometric functions.
I can generate the graphs of the trigonometric functions and explore different transformations on these graphs.
I can use the inverse trigonometric functions to solve problems.

64. Example 1:
Find all the trigonometric values of x if sin x = -3/5 and tan x < 0.

65. Transformations of Trigonometric Graphs

66. Example 2: Using the fxn, determine the (a) period, (b) domain,
(c) range,
(d) draw the graph of the function

67. Inverse Trig Functions:
Domain
Range

68. Example 3:
Find the measure of cos-1 (-0.5) in degrees and radians (NO CALCULATOR!) How many solutions should you expect?

69. Example 4:
Solve: (A) sin x = 0.7, (B) tan x = -2,
How many solutions should you How many solutions should you expect to have? expect to have?

70. You are done with Chapter 1!!! Quiz on Monday. Test on Wednesday!
HOORAY!
You are done with Chapter 1!!!
Quiz on Monday.
Test on Wednesday!