AP Calculus AB. Table of Contents. Slide 1 / 180. Slide 2 / 180. Slide 3 / 180. Review Unit

Transcription

1 Slide 1 / 180 Slide 2 / 180 P alculus Review Unit Table of ontents lick on the topic to go to that section Slide 3 / 180 Slopes Equations of Lines Functions Graphing Functions Piecewise Functions Function omposition Function Roots omain and Range Inverse Functions Trigonometry Exponents Logs and Exponential Functions

2 Slide 4 / 180 Slopes Return to Table of ontents Slope Slide 5 / 180 Recall from lgebra, The SLOPE of a line is the ratio of the vertical movement to the horizontal movement. In other words, it describes both the steepness and direction of a line. alculating Slope Slide 6 / 180 One way to determine the slope is calculate it from two points. onsider two points, (x 1,y 1) and (x 2,y 2) The slope, m, is: *Note: a slope is not defined for a vertical line (where x 1=x 2)

3 alculating Slope Slide 7 / 180 Example: alculate the slope of the line containing the points (3,4) and (2,8) alculating Slope Slide 7 (nswer) / 180 Example: alculate the slope of the line containing the points nswer (3,4) and (2,8) slope m= -4 1 What is the slope of the line containing the points: (15,-7) and (3,5)? Slide 8 / 180 m= 1 m= -1 m= -1/11 m= 2/8

4 1 What is the slope of the line containing the points: (15,-7) and (3,5)? m= 1 nswer m= -1 Slide 8 (nswer) / 180 m= -1 m= -1/11 m= 2/8 2 What is the slope of the line containing the points: (2,2) and (8,3)? Slide 9 / 180 m= 6 m= 5 m= 1/2 m= 1/6 2 What is the slope of the line containing the points: (2,2) and (8,3)? m= 6 nswer m= 1/6 Slide 9 (nswer) / 180 m= 5 m= 1/2 m= 1/6

5 3 What is the slope of the line containing the points: (17,23) and (-6,-18)? Slide 10 / 180 m= 41/23 m= 23/41 m= -2 m= -23/41 3 What is the slope of the line containing the points: (17,23) and (-6,-18)? Slide 10 (nswer) / 180 m= 41/23 m= 23/41 m= -2 m= -23/41 nswer m= 41/23 Slide 11 / 180 Equations of Lines Return to Table of ontents

6 Point-Slope Form Slide 12 / 180 Once you have the slope of a line, it is important to be able to write the equation for the line. If you have the slope of the line, m, and any one point, (x 1, y 1), you can write the equation of the line. Let be a point, then This form is called Point-Slope Form of an equation. Point-Slope Form is extremely useful in alculus and it is important that you are comfortable using it. Point-Slope Form Slide 13 / 180 Example: Find the equation of the line that has a slope of 4 and passes through the point (-2, 5). Write the answer in Point-Slope form. Point-Slope Form Slide 13 (nswer) / 180 Example: Find the equation of the line that has a slope of 4 and passes through the point (-2, 5). Write the answer in Point-Slope form. nswer

7 - Write the equation for the line in point-slope form, that has a slope of 4 and contains the point (5,-8). Slide 14 / Write the equation for the line in point-slope form, that has a slope of 4 and contains the point (5,-8). Slide 14 (nswer) / 180 nswer - Write the equation of the line, in point-slope form, that has a slope of -5 and contains the point (3,15). Slide 15 / 180

8 - Write the equation of the line, in point-slope form, that has a slope of -5 and contains the point (3,15). Slide 15 (nswer) / 180 nswer - Write the equation of the line, in point-slope form, that contains the points (5,3) and (-3,-6). Slide 16 / 180 First, find slope: - Write the equation of the line, in point-slope form, that contains the points (5,3) and (-3,-6). nswer Next, choose point and plug into Point-Slope form: Slide 16 (nswer) / 180 or

9 - Write the equation of the line, in point-slope form, that contains the points (-4,3) and (2,9). Slide 17 / 180 Slide 17 (nswer) / 180 Slope-Intercept Form Slide 18 / 180 Recall from lgebra, another common way to express the equation of a line is called slope-intercept form. This is written as: Where m is the slope, and the y-intercept is at (0,b).

10 Slope-Intercept Form Slide 19 / 180 Example: Find the equation of the line with a slope of 3, containing the point (4,5). Express your answer in slope-intercept form. Slope-Intercept Form Example: Find the equation We of the know line from with the a slope Point-Slope of 3, containing the point (4,5). Express your form answer that: in slope-intercept form. Teacher Notes onverting to slope-intercept: Slide 19 (nswer) / Write the equation of the line, in slope-intercept form, that has a slope of 5 and contains the point (23,15). Slide 20 / 180

11 - Write the equation of the line, in slope-intercept form, that has a slope of 5 and contains the point (23,15). nswer Slide 20 (nswer) / Write the equation of the line, in slope-intercept form, that has a slope of -3 and contains the point (6,8). Slide 21 / Write the equation of the line, in slope-intercept form, that has a slope of -3 and contains the point (6,8). Slide 21 (nswer) / 180 nswer

12 - Write the equation of the line, in slope-intercept form, that contains the points (16,14) and (-2,-7). Slide 22 / Write the equation of the line, in slope-intercept form, that contains the points (16,14) and (-2,-7). Slide 22 (nswer) / 180 nswer Slide 23 / 180 Functions Return to Table of ontents

13 What is a Function? Slide 24 / 180 function is a relationship between x and y such that for any value x, there will be one and only one value of y. For example: What is a Function? We can show that these are function is a functions, relationship evaluate between the x and equations. y such that for any value x, there Put will be in any one value and only for one x, there value will of y. be only one value for y. Teacher Notes For example: 1. if x is 7, y is 30. if x is 1, y is 24 etc if x is 3, y is 24. if x is 2, y is 19 etc. Slide 24 (nswer) / 180 Vertical Line Test For the function definition given on the previous slide to be true, the function will also pass what is called the Vertical Line Test. This states that a graph is of a function if and only if there is no vertical line that crosses the graph more than once. Slide 25 / 180 For the same examples, let's look at their graphs: 1. 2.

14 Vertical Line Test For the function definition given on the previous slide to be true, the function will also pass what is called rag the Vertical purple vertical Line Test. line This states that a graph is of a function and if notice and only it will if there not cross is no the vertical line that crosses the graph more graph than once. more than once, for both examples. For the same examples, let's look at their graphs: Teacher Notes Slide 25 (nswer) / [This object is a pull tab] Functions as a Table Slide 26 / 180 third way to demonstrate functions is in tabular form. Sometimes functions can be represented as a set of ordered pairs, or a relation. This is used often when the equation itself is unknown. Here is an example of how that would be expressed: x y There is no given equation for this relation, but it is a function since there is only one y value for each x value. Equations Which are Not Functions Sometimes it is useful to consider relations that are not functions. Slide 27 / 180 If for any input there is more than one output, it is not a function. Here are examples of equations that are not functions: 1. 2.

15 Equations Which are Not Functions Sometimes it is useful to consider relations that are not functions. Show students that there is more If for any input there is more than than one one possible output, y-value it is not for a each function. Here are examples of equations x, in both that examples. are not functions: Teacher Notes Example 1: if we choose x=19, then y=4 or y=-4 Example 2: if we choose x=3, then y=5 or y=-5 Slide 27 (nswer) / 180 Failing the Vertical Line Test You can see that both examples do not pass the Vertical Line Test: Slide 28 / Failing the Vertical Line Test You can see that both examples do not pass the Vertical Line Test: Teacher Notes Note: The purple vertical 1. line crosses 2. both graphs more than once. Slide 28 (nswer) / 180

16 Slide 29 / 180 Slide 29 (nswer) / 180 Slide 30 / 180

17 Slide 30 (nswer) / Is the following relation a function? Slide 31 / 180 Yes No x y Is the following relation a function? Slide 31 (nswer) / 180 Yes No x nswer y No, x=-1-2 has 2 values for y-1 0

18 7 Is the following relation a function? Slide 32 / 180 Yes No nswer Slide 33 / 180 Slide 33 (nswer) / 180

19 Slide 34 / 180 Slide 34 (nswer) / 180 Slide 35 / 180

20 Slide 35 (nswer) / 180 Slide 36 / 180 Slide 36 (nswer) / 180

21 10 What is the value of f(x+2) given Slide 37 / 180 Slide 37 (nswer) / 180 Slide 38 / 180 Graphing Functions Return to Table of ontents

22 Graphing Functions Slide 39 / 180 It is important to be able to graph functions. t this point, you should be familiar with methods for doing so. You should also be able to understand parent graphs, and identify shapes and orientations of different, common functions. Transforming Functions Slide 40 / 180 Functions, like equations, are transformed in a predictable manner. Each letter below has a separate effect on a given function. Identify how each letter transforms a function. y = a f( bx c) ± d Transforming Functions Slide 40 (nswer) / 180 Functions, like equations, are transformed in a predictable manner. Each letter below has a separate effect on a given function. Identify how each letter transforms a function. y = a f( bx c) ± d n

23 11 Which of the following is the graph of? Slide 41 / Which of the following is the graph of? Slide 41 (nswer) / 180 nswer 12 Which of the following is the graph of? Slide 42 / 180

24 12 Which of the following is the graph of? Slide 42 (nswer) / 180 nswer 13 Which of the following is the graph of? Slide 43 / Which of the following is the graph of? Slide 43 (nswer) / 180 nswer

25 14 Which of the following is a graph of? Slide 44 / Which of the following is a graph of? Slide 44 (nswer) / 180 nswer 15 Which of the following is a graph of? Slide 45 / 180

26 15 Which of the following is a graph of? nswer Slide 45 (nswer) / Which of the following is the graph of? Slide 46 / Which of the following is the graph of? Slide 46 (nswer) / 180 nswer

27 Further hallenge From the previous slide's question, see if you can write the equations for the other graphs: Slide 47 / 180 Further hallenge From the previous slide's question, see if you can write the equations for the other graphs: : : Teacher Notes : Slide 47 (nswer) / Which of the following is a graph of? Slide 48 / 180

28 17 Which of the following is a graph of? nswer Slide 48 (nswer) / 180 Further hallenge From the previous question, see if you can you write the equations for the three other graphs: Slide 49 / 180 Further hallenge From the previous question, see if you can you write the equations for the three other graphs: : : Teacher Notes : Slide 49 (nswer) / 180

29 Slide 50 / 180 Piecewise Functions Return to Table of ontents Piecewise Functions Slide 51 / 180 Piecewise functions can be thought of as several functions at once, each defined on a specific interval, or each used in a different region. To graph a piecewise function you do not plot the entire graph of each individual section - graph only the parts defined by x. Piecewise Functions simple example of a piecewise function is the absolute value function. Slide 52 / 180 The graph of this function looks like this: Note, that at the point x=0, the two function pieces meet. This is not always the case.

30 iscontinuity Notation Some piecewise functions can be discontinuous. When you have a piecewise function in which the different sections do not meet, there is special notation for the end points. Slide 53 / 180 included endpoint/ solid circle discluded endpoint/ open circle Evaluating Piecewise Functions Slide 54 / 180 Evaluating a piecewise function is the same as a continuous function, however we must pay close attention to the endpoint definitions. Example: Evaluate the following piecewise function at the given points: Evaluating Piecewise Functions Slide 54 (nswer) / 180 Evaluating a piecewise function is the same as a continuous function, however we must pay close attention to the endpoint definitions. Example: Evaluate the following piecewise function at the given points: nswer

31 Graphing Piecewise Functions Slide 55 / 180 Now we can practice graphing the same piecewise function. Graphing Piecewise Functions Slide 55 (nswer) / 180 Now we can practice graphing the same piecewise function. nswer Evaluating Piecewise Functions Slide 56 / 180 Example: Evaluate the piecewise function at the given values:

32 Evaluating Piecewise Functions Slide 56 (nswer) / 180 Example: Evaluate the piecewise function at the given values: nswer Graphing Piecewise Functions Slide 57 / 180 Example: Graph the following piecewise function: Graphing Piecewise Functions Slide 57 (nswer) / 180 Example: Graph the following piecewise function: nswer

33 Slide 58 / 180 Slide 58 (nswer) / 180 Slide 59 / 180

34 Slide 59 (nswer) / 180 Slide 60 / 180 Slide 60 (nswer) / 180

35 Slide 61 / 180 Slide 61 (nswer) / 180 Slide 62 / 180

36 Slide 62 (nswer) / Given the following piecewise function, find the value of Slide 63 / Given the following piecewise function, find the value of Slide 63 (nswer) / 180 nswer

37 - Given the following piecewise function, find the value of Slide 64 / Given the following piecewise function, find the value of Slide 64 (nswer) / 180 nswer - Given the following piecewise function, find the value of Slide 65 / 180

38 - Given the following piecewise function, find the value of Slide 65 (nswer) / 180 nswer Slide 66 / 180 Slide 66 (nswer) / 180

39 Slide 67 / 180 Function omposition Return to Table of ontents Slide 68 / 180 Evaluating omposite Functions Slide 69 / 180 To evaluate composite functions, you must start from the innermost "layer" and work your way out. For example, if and To evaluate, first x passes through the function g(x), and that output is then plugged into f(x).

40 Evaluating omposite Functions Slide 69 (nswer) / 180 To evaluate composite functions, you must start from the innermost "layer" and work your way out. For example, if nswer and To evaluate, first x passes Students through often the get function confused when g(x), and they have to substitute an entire that output is then plugged into f(x). expression into another function, so work slowly and allow time for [This discussion. object a pull tab] Evaluating omposite Functions Slide 70 / 180 Example: Given and find Slide 70 (nswer) / 180

41 Slide 71 / 180 Slide 71 (nswer) / What is the value of given the following functions: Slide 72 /

42 22 What is the value of given the following functions: Slide 72 (nswer) / nswer 23 What is the value of given the following functions: Slide 73 / What is the value of given the following functions: Slide 73 (nswer) / 180 nswer

43 24 What is the value of given the following functions: Slide 74 / What is the value of given the following functions: Slide 74 (nswer) / nswer Find the value of Slide 75 /

44 25 Find the value of Slide 75 (nswer) / nswer Slide 76 / 180 Slide 76 (nswer) / 180

45 26 Given and, find h(x) if Slide 77 / Given and, find h(x) if Slide 77 (nswer) / 180 nswer Slide 78 / 180

46 Slide 78 (nswer) / Given and, find h(x) if Slide 79 / Given and, find h(x) if nswer Slide 79 (nswer) / 180

47 29 Given and, find h(x) if Slide 80 / Given and, find h(x) if nswer Slide 80 (nswer) / 180 Slide 81 / 180 Function Roots Return to Table of ontents

48 Roots of a Function nother important idea to understand regarding functions is the roots of the function. root, sometimes called a zero solution of f(x), is the value of x such that f(x)=0. It can also be called the x-intercept. Slide 82 / 180 roots/ zeroes/ x-intercepts alculating Roots One method for finding roots is to factor and use the zero product property. For quadratics that are unfactorable, the quadratic formula can be used. Slide 83 / 180 Example: Find the roots of the following: Slide 83 (nswer) / 180

49 Quadratic Formula Slide 84 / 180 Sometimes the equations are not as easily factorable, and the quadratic formula is required. Recall: ; Example: Find the roots of the following equation: Quadratic Formula Slide 84 (nswer) / 180 Sometimes the equations are not as easily factorable, and the quadratic formula is required. Recall: ; nswer Example: Find the roots of the following equation: so and Slide 85 / 180

50 Slide 85 (nswer) / 180 Slide 86 / 180 Slide 86 (nswer) / 180

51 Slide 87 / 180 Slide 87 (nswer) / 180 Slide 88 / 180 omain and Range Return to Table of ontents

52 omain and Range Slide 89 / 180 Recall from lgebra II, the omain of a function is the set of all possible inputs for a function, typically the x-values. Similarly, the Range of a function is the set of all possible outputs for a function, typically the y-values. omain and Range ertain conditions must be avoided in order for the omains and Ranges of functions to be real. Slide 90 / 180 Watch for values which may cause: zero in the denominator square roots of negative numbers logs of zero logs of negative numbers omain and Range Slide 91 / 180 Example: Find the omain and Range of the following function:

53 omain and Range Slide 91 (nswer) / 180 Example: Find the omain and Since Range this function of the is a following general polynomial, function: any value can be plugged in yielding a real number, so the omain is all real numbers: Teacher Notes This is a line with a slope of 3. It will pass through all y-values, so the range is all real numbers. omain and Range Slide 92 / 180 Example: Find the omain and Range of the following function: Slide 92 (nswer) / 180

54 33 What is the omain and Range for the following function: Slide 93 / What is the omain and Range for the following function: Slide 93 (nswer) / 180 nswer 34 What is the omain and Range of the following function: Slide 94 / 180

55 34 What is the omain and Range of the following function: Slide 94 (nswer) / 180 nswer 35 What is the omain and Range for the following function: Slide 95 / What is the omain and Range for the following function: Slide 95 (nswer) / 180 nswer

56 More hallenging Example Sometimes more complicated functions are presented. In this case, finding ranges might be very difficult, and finding domains are more important. Example: find the omain for the following function: Slide 96 / 180 More hallenging Example We must make sure the denominator is not zero. Sometimes more complicated Since we cannot functions factor are the denominator presented. to cancel In this case, finding ranges might out with be the very numerator, difficult, we must and finding the zeros domains of the denominator. are more important. Teacher Notes Example: find the omain for the following function: Therefore, the domain is all real numbers except x=-1 and x=3 written as: Slide 96 (nswer) / What is the omain (only) for the following function: Slide 97 / 180 omain: ll real numbers omain: ll real numbers except x=-3, x=2 and x=-5 omain: ll real numbers except x=-3 and x=-5 omain: ll real numbers except x=3 and x=5

57 36 What is the omain (only) for the following function: Slide 97 (nswer) / 180 omain: ll real numbers nswer omain: ll real numbers except x=-3, x=2 [This object and is a pull x=-5 tab] omain: ll real numbers except x=-3 and x=-5 omain: ll real numbers except x=3 and x=5 37 What is the omain (only) for the following function: Slide 98 / What is the omain (only) for the following function: nswer Note, this answer shows the omain has to avoid division by zero, and square root of negative numbers. Slide 98 (nswer) / 180

58 Slide 99 / 180 Inverse Functions Return to Table of ontents One-to-One Functions Slide 100 / 180 In order to study inverse functions, it is first necessary to specify which kind of functions are appropriate. We know that for a relation to be a function, every value in the domain must have exactly one value in the range. For a function to have an inverse, we further require that every value in the range must have exactly one value in the domain. In other words, no two values of x yield the same y. This relationship is called a One-to-One Function. One-to-One Functions Slide 100 (nswer) / 180 In order to study inverse functions, it is first necessary to specify which kind of functions are appropriate. We know that for a relation to be a function, every value in the domain must have exactly Note: one value Even in the if the range. function For a is function to have an inverse, we further not require one-to-one, that every it can value still in the range must have exactly one value in the domain. have an inverse, but the In other words, no two values inverse of x yield is not the a same function. y. This relationship is called a One-to-One Function. Teacher Notes [This object is a teacher notes pull tab]

59 Horizontal Line Test You must determine if a function is One-to-One, in order for you to then find it's inverse. If given ordered pairs, simply look to see if there are no repeated y- values. If given an equation that is easy to plot, you can use the Horizontal Line Test. This states that if it is possible to draw a Horizontal line anywhere on the graph, and it crosses the graph more than once, it fails the Oneto-One requirement. Slide 101 / 180 Failing the Horizontal Line Test Slide 102 / 180 Example: Notice: The line crosses the graph twice and fails the Horizontal Line Test. Therefore, it is not a One-to- One function. Passing the Horizontal Line Test Slide 103 / 180 Example: Notice: The line does not cross the graph more than once and Passes the Horizontal Line Test. Therefore it is a One-to-One function.

60 Slide 104 / Is the following graph a One-to-One function? Slide 105 / 180 Yes No 38 Is the following graph a One-to-One function? Slide 105 (nswer) / 180 Yes No nswer No

61 39 Is the following graph a One-to-One function? Slide 106 / 180 Yes No 39 Is the following graph a One-to-One function? Slide 106 (nswer) / 180 Yes No nswer Ye s Slide 107 / 180

62 Finding the Inverse Slide 108 / 180 Example: Find the inverse of f(x), given: Finding the Inverse Slide 108 (nswer) / 180 Example: Find the inverse of f(x), given: nswer Inverse efinition Slide 109 / 180 Step 5 involves the previously discussed Function omposition. (click for link) Inverse Function can be defined as: Given two One-to-One Functions if: and and then and are Inverses of each other.

63 Inverses Slide 110 / 180 Example: Given: re these two functions inverses of each other? heck to make sure it follows the definition. Inverses Slide 110 (nswer) / 180 Example: Given: re these two functions inverses of each other? heck to make sure it follows the definition. and nswer inverses Slide 111 / 180 Terminology The Inverse of and the Inverse of is is

64 Slide 111 (nswer) / 180 Terminology The inverse notation can cause confusion, the "-1" superscript is NOT an The Inverse exponent: of is Teacher Notes and the Inverse of is [This object is a teacher notes pull tab] 40 Which of the following is the correct notation for the Inverse Function of? Slide 112 / 180 E 40 Which of the following is the correct notation for the Inverse Function of? Slide 112 (nswer) / 180 E nswer

65 41 Given the following function, which is its inverse function? Slide 113 / 180 E Not Invertable 41 Given the following function, which is its inverse function? nswer Slide 113 (nswer) / 180 E Not Invertable 42 Given, Find Slide 114 / 180 E Not Invertable

66 42 Given, Find nswer Slide 114 (nswer) / 180 E Not Invertable 43 Given, Find Slide 115 / 180 E Not Invertable 43 Given, Find Slide 115 (nswer) / 180 E Not Invertable nswer

67 Graphs of Inverses Slide 116 / 180 nother special relationship that you may recall about functions and their inverses is that their graphs are a reflection across the line y=x. Slide 117 / 180 Trigonometry Return to Table of ontents Trig Functions Slide 118 / 180 These are the six trig functions you are familiar with from Geometry and Precalculus.

68 Trig Functions Slide 118 (nswer) / 180 These are the six trig functions Review you are these familiar relationships with from with students: Geometry and Precalculus. Teacher Notes [This object is a teacher notes pull tab] Trig - Right Triangles Slide 119 / 180 ll these trig functions are defined in terms of a right triangle: Opposite Hypotenuse djacent The graphs of these functions should be easily recognizable: Slide 120 / 180

69 Range of Trig Functions Slide 121 / 180 The ranges for these functions can also be determined. What is: The range of sin and cos? The range of csc and sec? The range of tan and cot? Slide 121 (nswer) / 180 -S-T- nother important matter is the sign of the trig functions in each quadrant. The letters -S-T- represent the positive values. ll other trig functions will be negative in those quadrants. : ll trig functions are positive in the 1st quadrant. S: Sin values are positive in the 2nd quadrant. T: Tan values are positive in the 3rd quadrant. : os values are positive in the 4th quadrant. Slide 122 / 180

70 -S-T- nother important matter is the sign of the trig functions in each quadrant. The letters -S-T- represent the positive values. ll other trig functions will be negative in those quadrants. Most students try to remember the ST order : ll trig functions are positive with in a the catchy 1st quadrant. phrase like: S: Sin values are positive in ll the Students 2nd quadrant. Take alculus Teacher Notes T: Tan values are positive in the 3rd quadrant. : os values are positive in the 4th quadrant. Slide 122 (nswer) / 180 [This object is a teacher notes pull tab] In alculus class almost all problems are in radians, not in degrees. This table shows the "special" angles, in both, that you should be familiar with. egrees Radians 0 Radians Teacher Notes Slide 123 / 180 Trigonometry In Geometry and Pre-calculus you learned quite a bit about trigonometry. To be successful in calculus, it is very important that you know how to evaluate trig functions at various angles. Many real life situations behave in a trigonometric pattern (i.e. traffic flow), therefore you will see that trig functions are very prevalent in the course and on the P Exam. Teacher Notes Slide 124 / 180

71 1. THE UNIT IRLE Slide 125 / 180 This method requires you to memorize values for each ordered pair. Recall that the x value of each ordered pair is the cosine value, while the y value of the ordered pair is the sine value. The Unit ircle The Unit ircle is divided into 4 quadrants. They are listed below. Slide 126 / 180 II I III IV Slide 127 / 180 Special ngles in the II, III, and IV Quadrants The x and y coordinates for special angles in the other quadrants can be determined by knowing the similar 1st quadrant angle's value. The x and y values will be the same, but the signs will (or can) be different.

72 2. THE TRIG TLE Slide 128 / 180 This method requires you memorize values from the table and remember: 3. SPEIL RIGHT TRINGLES Slide 129 / 180 This method requires you to draw any of the above triangles on a set of axes depending on given angle, and remember: Teacher Notes Slide 130 / 180

73 Slide 130 (nswer) / 180 Slide 131 / 180 Slide 131 (nswer) / 180

74 46 Evaluate Slide 132 / 180 E 46 Evaluate nswer E Slide 132 (nswer) / 180 [This object is a pull E tab] Slide 133 / 180

75 Slide 133 (nswer) / 180 Slide 134 / 180 Slide 134 (nswer) / 180

76 Slide 135 / 180 Slide 135 (nswer) / Evaluate Slide 136 / 180 E

77 50 Evaluate Slide 136 (nswer) / 180 nswer E 51 Evaluate Slide 137 / 180 E 51 Evaluate Slide 137 (nswer) / 180 nswer E E

78 The following Trig Identities are some of the more common ones, you may recall from Pre-calculus. Pythagorean Identity Trig Identities Sum Identities Slide 138 / 180 ouble ngle Formulas Half ngle Formulas Slide 139 / 180 Slide 139 (nswer) / 180

79 52 Evaluate Slide 140 / Evaluate nswer Slide 140 (nswer) / 180 Slide 141 / 180

80 Slide 141 (nswer) / 180 Slide 142 / 180 Slide 142 (nswer) / 180

81 Inverse Trig Functions Slide 143 / 180 Inverse Trig Functions follow the same rules as other Inverse Functions we learned earlier. (lick here) They "undo" what the trig function does. For example if the function is then the inverse trig function is. You may also see the following terminology. Inverse Trig Functions Slide 143 (nswer) / 180 Inverse Trig Functions follow the same rules as other Inverse Functions we learned earlier. (lick here) Remind students that the "-1" They "undo" what the trig function does. For example if the function is then the inverse trig is function not an is exponent.. Teacher Notes You may also see the following terminology. [This object is a teacher notes pull tab] Inverse Functions Remember that Inverse Functions must be One-to-One. Recalling our basic trig graphs, (lick here) we can see that none of them are One-to-One. Therefore, we must restrict the range. Slide 144 / 180 For sinx: For cosx: For tanx:

82 Evaluating Slide 145 / 180 Example: Evaluate In other words, we must find what angles have sin values of, remembering our range restrictions. Evaluating Slide 145 (nswer) / 180 Example: Evaluate This value can be easily found using In other words, we must the find Unit what ircle angles or trig have table. sin values of, remembering our range restrictions. nswer: nswer Slide 146 / 180

83 Slide 146 (nswer) / Evaluate Slide 147 / 180 E 56 Evaluate E nswer Slide 147 (nswer) / 180

84 57 Evaluate Slide 148 / 180 E 57 Evaluate E nswer Slide 148 (nswer) / 180 Slide 149 / 180 Exponents Return to Table of ontents

85 Properties of Exponents Slide 150 / 180 Practice Simplify each of the following expressions. Slide 151 / 180 nswer Slide 152 / 180

86 Slide 152 (nswer) / Simplify. Slide 153 / 180 E None of the above 59 Simplify. nswer Slide 153 (nswer) / 180 E None of the above

87 60 Simplify: Slide 154 / 180 E None of the above 60 Simplify: nswer E correct answer = 1 Slide 154 (nswer) / 180 E None of the above Slide 155 / 180

88 Slide 155 (nswer) / 180 Slide 156 / 180 Slide 156 (nswer) / 180

89 63 Simplify: Slide 157 / 180 E None of the above 63 Simplify: E None of the above nswer Slide 157 (nswer) / 180 Slide 158 / 180

90 Slide 158 (nswer) / Simplify: Slide 159 / 180 E None of the above 65 Simplify: E None of the above nswer Slide 159 (nswer) / 180

91 Slide 160 / 180 Logs and Exponential Functions Return to Table of ontents Slide 161 / 180 Slide 162 / 180

92 Slide 162 (nswer) / 180 Slide 163 / 180 Slide 163 (nswer) / 180

93 Slide 164 / 180 Slide 164 (nswer) / 180 Slide 165 / 180

94 Slide 165 (nswer) / Find Slide 166 / 180 E 69 Find E nswer Slide 166 (nswer) / 180

95 Log Properties: Slide 167 / 180 hange of ase formula: Teacher Notes Log Properties: The following are log properties that are calculated, but might help students if they knew them without having to stop and obtain them: hange of ase formula: Slide 167 (nswer) / 180 [This object is a teacher notes pull tab] Logarithms Slide 168 / 180 Example: Find

96 Slide 168 (nswer) / Find Slide 169 / 180 E 70 Find E nswer Slide 169 (nswer) / 180

97 71 Find Slide 170 / 180 E 71 Find E nswer Slide 170 (nswer) / Find E Undetermined Slide 171 / 180

98 72 Find E Undetermined nswer Slide 171 (nswer) / Find E Slide 172 / Find E nswer Hint: Use hange of ase formula with a new base of 6. nswer: E Slide 172 (nswer) / 180

99 Special ase of Log Slide 173 / 180 This is called the natural log, and it has a base of. follows the same rules and has the same properties as. Note that: Special ase of Log Slide 173 (nswer) / 180 Students should be familiar This is called the natural with log, and, it it has a value base of of:. follows the same rules and has the same properties as. Teacher Notes Note that: [This object is a teacher notes pull tab] Exponential and Logarithm Equations Slide 174 / 180 Using what we learned about the relationships between logs and exponents, we can now solve equations containing them.

100 Exponential and Logarithm Equations Slide 175 / 180 Example: Solve for x: (remember domain requirements for log) Exponential and Logarithm Equations Slide 175 (nswer) / 180 Example: Solve for x: (remember domain requirements for log) nswer potential answers are: each answer must be checked with the original problem, to make sure that we are not taking the log of a negative number... nswer: Slide 176 / 180

101 Slide 176 (nswer) / 180 Slide 177 / 180 Slide 177 (nswer) / 180

102 Slide 178 / 180 Slide 178 (nswer) / 180 Slide 179 / 180

103 Slide 179 (nswer) / Solve for x: Slide 180 / 180 E None of the above 78 Solve for x: nswer Hint: Isolate the exponential term, then take of both sides. nswer: Slide 180 (nswer) / 180 E None of the above