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 Example: alculate the slope of the line containing the points (3,4) and (2,8) Slide 7 / 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 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

4 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 Slide 11 / 180 Equations of Lines Return to Table of ontents 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.

5 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. - 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 of the line, in point-slope form, that has a slope of -5 and contains the point (3,15). Slide 15 / 180

6 - Write the equation of the line, in point-slope form, that contains the points (5,3) and (-3,-6). Slide 16 / Write the equation of the line, in point-slope form, that contains the points (-4,3) and (2,9). Slide 17 / 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).

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

8 - Write the equation of the line, in slope-intercept form, that contains the points (16,14) and (-2,-7). Slide 22 / 180 Slide 23 / 180 Functions Return to Table of ontents 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: 1. 2.

9 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: 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 Slide 27 / 180 Sometimes it is useful to consider relations that are not functions. 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.

10 Failing the Vertical Line Test You can see that both examples do not pass the Vertical Line Test: Slide 28 / Slide 29 / 180 Slide 30 / 180

11 6 Is the following relation a function? Slide 31 / 180 Yes No x y Is the following relation a function? Slide 32 / 180 Yes No nswer Slide 33 / 180

12 Slide 34 / 180 Slide 35 / 180 Slide 36 / 180

13 10 What is the value of f(x+2) given Slide 37 / 180 Slide 38 / 180 Graphing Functions Return to Table of ontents 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.

14 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 11 Which of the following is the graph of? Slide 41 / Which of the following is the graph of? Slide 42 / 180

15 13 Which of the following is the graph of? Slide 43 / Which of the following is a graph of? Slide 44 / Which of the following is a graph of? Slide 45 / 180

16 16 Which of the following is the graph of? Slide 46 / 180 Further hallenge From the previous slide's question, see if you can write the equations for the other graphs: Slide 47 / Which of the following is a graph of? Slide 48 / 180

17 Further hallenge From the previous question, see if you can you write the equations for the three other graphs: Slide 49 / 180 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.

18 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. 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:

19 Graphing Piecewise Functions Slide 55 / 180 Now we can practice graphing the same piecewise function. Evaluating Piecewise Functions Slide 56 / 180 Example: Evaluate the piecewise function at the given values: Graphing Piecewise Functions Slide 57 / 180 Example: Graph the following piecewise function:

20 Slide 58 / 180 Slide 59 / 180 Slide 60 / 180

21 Slide 61 / 180 Slide 62 / Given the following piecewise function, find the value of Slide 63 / 180

22 - Given the following piecewise function, find the value of Slide 64 / Given the following piecewise function, find the value of Slide 65 / 180 Slide 66 / 180

23 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).

24 Evaluating omposite Functions Slide 70 / 180 Example: Given and find Slide 71 / What is the value of given the following functions: Slide 72 /

25 23 What is the value of given the following functions: Slide 73 / What is the value of given the following functions: Slide 74 / Find the value of Slide 75 /

26 Slide 76 / Given and, find h(x) if Slide 77 / 180 Slide 78 / 180

27 28 Given and, find h(x) if Slide 79 / Given and, find h(x) if Slide 80 / 180 Slide 81 / 180 Function Roots Return to Table of ontents

28 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: 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:

29 Slide 85 / 180 Slide 86 / 180 Slide 87 / 180

30 Slide 88 / 180 omain and Range Return to Table of ontents 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

31 omain and Range Slide 91 / 180 Example: Find the omain and Range of the following function: omain and Range Slide 92 / 180 Example: Find the omain and Range of the following function: 33 What is the omain and Range for the following function: Slide 93 / 180

32 34 What is the omain and Range of the following function: Slide 94 / What is the omain and Range for the following function: Slide 95 / 180 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

33 36 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 37 What is the omain (only) for the following function: Slide 98 / 180 Slide 99 / 180 Inverse Functions Return to Table of ontents

34 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. 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.

35 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. Slide 104 / Is the following graph a One-to-One function? Slide 105 / 180 Yes No

36 39 Is the following graph a One-to-One function? Slide 106 / 180 Yes No Slide 107 / 180 Finding the Inverse Slide 108 / 180 Example: Find the inverse of f(x), given:

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

38 40 Which of the following is the correct notation for the Inverse Function of? Slide 112 / 180 E 41 Given the following function, which is its inverse function? Slide 113 / 180 E Not Invertable 42 Given, Find Slide 114 / 180 E Not Invertable

39 43 Given, Find Slide 115 / 180 E Not Invertable 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

40 Trig Functions Slide 118 / 180 These are the six trig functions you are familiar with from Geometry and Precalculus. 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

41 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? -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 Radians Slide 123 / 180 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 Teacher Notes

42 Trigonometry Slide 124 / 180 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 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

43 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. 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

44 Slide 130 / 180 Slide 131 / Evaluate Slide 132 / 180 E

45 Slide 133 / 180 Slide 134 / 180 Slide 135 / 180

46 50 Evaluate Slide 136 / 180 E 51 Evaluate Slide 137 / 180 E 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

47 Slide 139 / Evaluate Slide 140 / 180 Slide 141 / 180

48 Slide 142 / 180 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 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:

49 Evaluating Slide 145 / 180 Example: Evaluate In other words, we must find what angles have sin values of, remembering our range restrictions. Slide 146 / Evaluate Slide 147 / 180 E

50 57 Evaluate Slide 148 / 180 E Slide 149 / 180 Exponents Return to Table of ontents Properties of Exponents Slide 150 / 180

51 Practice Simplify each of the following expressions. Slide 151 / 180 nswer Slide 152 / Simplify. Slide 153 / 180 E None of the above

52 60 Simplify: Slide 154 / 180 E None of the above Slide 155 / 180 Slide 156 / 180

53 63 Simplify: Slide 157 / 180 E None of the above Slide 158 / Simplify: Slide 159 / 180 E None of the above

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

55 Slide 163 / 180 Slide 164 / 180 Slide 165 / 180

56 69 Find Slide 166 / 180 E Log Properties: Slide 167 / 180 hange of ase formula: Logarithms Slide 168 / 180 Example: Find

57 70 Find Slide 169 / 180 E 71 Find Slide 170 / 180 E 72 Find Slide 171 / 180 E Undetermined

58 73 Find Slide 172 / 180 E 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: 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.

59 Exponential and Logarithm Equations Slide 175 / 180 Example: Solve for x: (remember domain requirements for log) Slide 176 / 180 Slide 177 / 180

60 Slide 178 / 180 Slide 179 / Solve for x: Slide 180 / 180 E None of the above