Pre Calculus. Intro to Integrals.

Size: px
Start display at page:

Download "Pre Calculus. Intro to Integrals."

Transcription

1 1

2 Pre Calculus Intro to Integrals

3 Riemann Sums Trapezoid Rule Table of Contents click on the topic to go to that section Accumulation Function Antiderivatives & Definite Integrals Fundamental Theorem of Calculus Substitution Method Area Between Curves Volume: Disk Method Volume: Washer Method Volume: Shell Method 3

4 Riemann Sums Return to Table of Contents 4

5 Reimann Sums Consider the following velocity graph: 30 mph How far did the person drive? 5 hrs The area under the velocity graph is the total distance traveled. Integration is used to find the area. 5

6 Reimann Sums But we seldom travel at a constant speed. 50 mph 5 hrs The area under this graph is still the distance traveled but we need more than multiplication to find it. 6

7 Reimann Sums George Riemann (Re mon) studied making these curves into a series of rectangles. So the area under the curve would be the sum of areas of the rectangles, this is called Riemann Sums. 7

8 Reimann Sums Riemann Sums, or Rectangular Approximation Method (RAM), is calculated by drawing rectangles from the x axis up to the curve. The question is: What part of the "top" of the rectangle should be used to determine the height of the rectangle? The right hand corner. (RRAM) The left hand corner. (LRAM) The middle. (MRAM) 8

9 Reimann Sums Example: Find the area between y = x 2, the x axis, and [0,1] using Riemann Sums and 4 partitions. LRAM 0 1/4 1/2 3/4 1 Found the width of the rectangle: (b a)/n= (1 0)/4 = 1/4 Is this approximation an overestimate or an underestimate? 9

10 Reimann Sums Example: Find the area between y = x 2, the x axis, and [0,1] using Riemann Sums and 4 partitions. RRAM 0 1/4 1/2 3/4 1 Is this approximation an overestimate or an underestimate? 10

11 Reimann Sums Example: Find the area between y = x 2, the x axis, and [0,1] using Riemann Sums and 4 partitions. MRAM 0 1/4 1/2 3/4 1 This value falls between LRAM and RRAM. 11

12 Reimann Sums *NOTE: MRAM LRAM + RRAM 2 12

13 Reimann Sums Q: What units should be used? A: Since the area is found by multiplying base times height, the units of the area are the units of the x axis times the units of the y axis. In our example at the beginning of the unit we had a velocity (mph) vs. time (hours) the units would then be 13

14 Reimann Sums 1 When finding the area between and the x axis [1,3] using four partitions, how wide should each interval be? 14

15 Reimann Sums 2 Find the area between and the x axis [1,3] using four partitions and LRAM. 15

16 Reimann Sums 3 Find the area between and the x axis [1,3] using four partitions and RRAM. 16

17 Reimann Sums 4 When finding the area between and the x axis [1,3] using four partitions and MRAM, when in the third rectangle, what x should be used to find the height? 17

18 Reimann Sums 5 Find the area between and the x axis [1,3] using four partitions and MRAM. 18

19 Reimann Sums We can write the four areas using where a k is the area of the k th rectangle. It is just another way of writing what we just did. Σ is the Greek letter sigma and stands for the summation of all the terms evaluated at starting with the bottom number and going through to the top. 19

20 Reimann Sums Selected Rules for Sigma 20

21 Reimann Sums Equivalent Formulas 1 st n integers: 1 st n squares: 1 st n cubes: 21

22 Reimann Sums 6 22

23 Reimann Sums 7 23

24 Reimann Sums 8 24

25 Reimann Sums 9 25

26 Trapezoid Rule Return to Table of Contents 26

27 Reimann Sums Example: Find the area y = x 2 and the x axis [0,1] using 4 partitions. 0 1/4 1/2 3/4 1 Why were areas found using RAM only estimates? How could we draw lines to improve our estimates? What shape do you get? 27

28 Reimann Sums Example: Find the area y = x 2 and the x axis [0,1] using 4 partitions and the trapezoids. Trapezoids 0 1/4 1/2 3/4 1 28

29 Reimann Sums *NOTE: Trapezoid Approximation = LRAM + RRAM 2 We could make our approximation even closer if we used parabolas instead of lines as the tops of our intervals. This is called Simpson's Rule but this is not on the AP Calc AB exam. 29

30 Reimann Sums 10 The area between and the x axis [1,3] is approximated with 4 partitions and trapezoids. What is the height of each trapezoid? 30

31 Reimann Sums 11 The area between and the x axis [1,3] is approximated with 4 partitions and trapezoids. What is the area of the 4 th trapezoid? 31

32 Reimann Sums 12 The area between y = and the x axis [1,3] is approximated with 4 partitions and trapezoids. What is the approximate area? 32

33 Reimann Sums 13 What is the approximate area using the trapezoids that are drawn? 4 y x 33

34 Reimann Sums 14 What is the approximate fuel consumed using the trapezoids rule for this hour flight? Time (minutes) Rate of Consumption (gal/min)

35 Reimann Sums In the last 2 responder questions, the partitioned intervals weren't uniform. The AP will use both. So don't assume. 35

36 Reimann Sums So far we have been summing areas using Σ. Gottfried Leibniz, a German mathematician, came up with a symbol you're going to see a lot of:. It is actually the German S instead of the Greek. It still means summation. As a point of interest, we use the German notation in calculus because Leibniz was the first to publish. Sir Isaac Newton is now given the credit for unifying calculus because his notes predate Leibniz's. 36

37 Accumulation Function Return to Table of Contents 37

38 Accumulation Function V (m/s) y 2 Another way we can calculate area under a function is to use geometry. What's happening during t=0 to t=3? What is the area of t=0 to t=3? What does the area mean? 1 t (sec) x What is the acceleration at t=3? What is happening during t=3 to t=6? What is the area of t=3 to t=6? What does this area mean? 2 Where is the object at t=6 in relation to where it was at t=0? 38

39 Accumulation Function The symbol notation for the area from zero to 3: "The area from t=0 to t=3 is the integral from 0 to 3 of the velocity function with respect to t." In general: 39

40 Accumulation Function y What is the area from x= 4 to x=0? 1 x Note: 4 5 Def: where 40

41 Accumulation Function When solving an accumulation function: (direction)(relation to x axis)(area) 5 y x

42 Accumulation Function 15 4 y 3 2 semicircle 1 x

43 Accumulation Function 16 (round to two decimal places) 4 y 3 2 semicircle 1 x

44 Accumulation Function 17 (round to two decimal places) 4 y 3 2 semicircle 1 x

45 Accumulation Function 18 4 y 3 2 semicircle 1 x

46 Accumulation Function 19 4 y 3 2 semicircle 1 x

47 Antiderivatives & Definite Integrals Return to Table of Contents 47

48 Antiderivatives Area under the curve of f(x) from a to b is We have been using geometry to find A. The antiderivative of f(x) can also be used. 48

49 Antiderivatives Properties of Definite Integrals 49

50 Antiderivatives 20 50

51 Antiderivatives 21 51

52 Antiderivatives 22 52

53 Antiderivatives 23 53

54 Antiderivatives Antiderivative Rules Why + C? 54

55 Antiderivatives Where F(x) is the anti derivative of f(x). 55

56 Antiderivatives Examples: Notice how C always disappears? We don't need C when we do definite integrals. 56

57 Antiderivatives 24 57

58 Antiderivatives 25 58

59 Antiderivatives 26 59

60 Antiderivatives 27 60

61 Antiderivatives 28 61

62 Antiderivatives 29 62

63 Antiderivatives 30 63

64 Antiderivatives You can do definite integrals on your graphing calculator. For the TI 84: use the MATH key 9: fnint( For example: Depending on which version of the operating system you have: 64

65 Antiderivatives The graphing calculator also has a built in integration function. MATH > 9:fnInt( depending on the version of the operating system: fnint( or For example, integrate x 2 3 from 1 to 4 with respect to x. fnint(x 2 3,x,1,4) 65

66 Fundamental Theorem of Calculus Return to Table of Contents 66

67 Fundamental Theorem of Calculus There are 2 parts to the Fundamental Theorem of Calculus, depending on the book, the order will change. Fundamental Theorem of Calculus (F.T.C.) Part 1 If f(x) is continuous at every point of [a,b] and F(x) is the antiderivative of f(x) then (which is what you've been doing) 67

68 Fundamental Theorem of Calculus Fundamental Theorem of Calculus (F.T.C.) Part 2 If F(x) is continuous at every point of [a,b] then has a derivative at every point on [a,b],and 68

69 Fundamental Theorem of Calculus Example: It looks easy, but be aware. When the derivative of the bounds in anything other that 1, need to multiply f(x) by the derivative. 69

70 Fundamental Theorem of Calculus 31 A B C D 70

71 Fundamental Theorem of Calculus 32 A B C D 71

72 Fundamental Theorem of Calculus 33 A B C D HINT 72

73 Fundamental Theorem of Calculus 34 A B C D HINT 73

74 Substitution Method Return to Table of Contents 74

75 Substitution Method Just like with differentiation, there are many integrals that are more complicated to evaluate. In situations like these, we use the Substitution Method to turn a difficult integral into a much simpler one. 75

76 Substitution Method The Substitution Method When we are given the initial function substitution where u is a function of x,, we make a, and 76

77 Substitution Method Ex: Let 77

78 Substitution Method Ex: Let 78

79 Substitution Method 35 What is the value of u? A B C D 79

80 Substitution Method 36 What will the integral be after the substitution is made? A B C D 80

81 Substitution Method 37 Evaluate the integral A B C D 81

82 Substitution Method The Substitution Method If you have a definite integral, you have two options for plugging in the bounds to get the final answer. 1. Plug a and b into the integrated function AFTER you have re substituted the x's back into the function 2. Plug a and b into to create two new bounds, and plug these into the integrated function 82

83 Substitution Method Ex: Let 83

84 Substitution Method 38 What is the value of u? A B C D 84

85 Substitution Method 39 What is the new upper bound? 85

86 Substitution Method 40 What is the new lower bound? 86

87 Substitution Method 41 What will the integral be after the substitution is made? A B C D 87

88 Substitution Method 42 What is the value of the integral? 88

89 Substitution Method 43 What is the value of u? A B C D 89

90 Substitution Method 44 What is the new upper bound? 90

91 Substitution Method 45 What is the new lower bound? 91

92 Substitution Method 46 What will the integral be after the substitution is made? A B C D 92

93 Substitution Method 47 What is the value of the integral? 93

94 Area Between Curves Return to Table of Contents 94

95 Area Between Curves The area between a curve and the x axis is But what about the area between two curves? Such as the area between and from the intersection to x=1. x= (don't round till the end) 95

96 Area Between Curves Area Between Curves Where a and b are the left and right bounds of the region. f(x) is upper curve on a graph and g(x) the lower. So for our example: 96

97 Area Between Curves 48 When finding the area between and, what is the left bounds of x? 97

98 Area Between Curves 49 When finding the area between and, what is the right bounds of x? 98

99 Area Between Curves 50 When finding the area between and, what is integral used? A B C D 99

100 Area Between Curves 51 What is the area between and? 100

101 Area Between Curves Consider: Integrating in terms of y would be easier. f(x) g(x) is right function minus left and the bounds are now the least value of y to the greatest. 101

102 Area Between Curves 52 When finding the area between and the y axis, what is the lower bound of y? 102

103 Area Between Curves 53 When finding the area between and the y axis, what is the upper bound of y? 103

104 Area Between Curves 54 What is the area between and the y axis? 104

105 Example: Find the area between the curves in the first quadrant. Notice that the lower function isn't the same for the entire region. We could find the area in terms of y or divide the region into 2 separate integrals and add their areas. 105

106 Area Between Curves 55 When finding the area between f(x) = x, g(x) = 1/2x, and h(x) =1/2x 1, what would be the area to the left of the y axis? A B C D 106

107 Area Between Curves 56 When finding the area between f(x) = x, g(x) = 1/2x, and h(x) =1/2x 1, what is the left bounds of h(x) f(x)? 107

108 Area Between Curves 57 When finding the area between f(x) = x, g(x) = 1/2x, and h(x) =1/2x 1, what is the right bound of h(x) f(x)? 108

109 Area Between Curves 58 When finding the area between f(x) = x, g(x) = 1/2x, and h(x) =1/2x 1, what would be the area to the right of the y axis? A B C D 109

110 Area Between Curves 59 When finding the area between f(x) = x, g(x) = 1/2x, and h(x) =1/2x 1, what is the right bound of h(x) g(x)? 110

111 Area Between Curves 60 When finding the area between f(x) = x, g(x) = 1/2x, and h(x) =1/2x 1, what is the left bound of h(x) g(x)? 111

112 Area Between Curves 61 Find the area between f(x) = x, g(x) = 1/2x, and h(x) =1/2x

113 Volume: Disk Method Return to Table of Contents 113

114 Volume: Disk Method Another way to make a 3 D object is to take a region and rotate it about an axis. When this rectangle is rotated a cylinder is formed. We could use geometry, but can we use calculus? From the section on known cross sections: And since the cross sections are circles: 114

115 Volume: Disk Method Volume: Disk Method When rotating about a horizontal axis: When rotating about a vertical axis: 115

116 Volume: Disk Method Rotate about x axis from x=1 to x=4. 116

117 Volume: Disk Method Rotate about y=2 from x=1 to x=4. Since y=2 is a horizontal axis of rotation, integral is in terms of x. r=2 y Why? From the x axis to the curve is y and to axis of rotation is 2, we want upper minus lower. 117

118 Volume: Disk Method Rotate about y= 2 from x=1 to x=4. Since y= 2 is a horizontal axis of rotation, integral is in terms of x. 2 y r=y 2=y+2 Why? upper minus lower 118

119 Volume: Disk Method 62 Rotate y=2x 2 about the x axis over [0,5]. What is the lower bound? 119

120 Volume: Disk Method 63 Rotate y=2x 2 about the x axis over [0,5]. What is the upper bound? 120

121 Volume: Disk Method 64 Rotate y=2x 2 about the x axis over [0,5]. What is integral? A B C D 121

122 Volume: Disk Method 65 Rotate y=2x 2 about the x axis over [0,5]. What is the volume? 122

123 Volume: Disk Method 66 Rotate y=2x 2 about the line x=4 over [0,5]. What is the lower bound? 123

124 Volume: Disk Method 67 Rotate y=2x 2 about the line x=4 over [0,5]. What is the upper bound? 124

125 Volume: Disk Method 68 Rotate y=2x 2 about the line x=4 over [0,5]. What is the radius? 125

126 Volume: Disk Method 69 Rotate y=2x 2 about the line x=4 over [0,5]. What is integral? A B C D 126

127 Volume: Disk Method 70 Rotate y=2x 2 about the line x=4 over [0,5]. What is the volume? 127

128 Volume: Disk Method Rotate for x=0 to x=2 about the y axis. Since this is a vertical axis,the problem should be rewritten in terms of y: Rotate for y=0 to y=16 about the y axis

129 Volume: Disk Method Rotate for x=0 to x=2 about the x=2. Since this is a vertical axis, the problem should be rewritten in terms of y: 16 Rotate for y=0 to y=16 about the x=2. 129

130 Volume: Disk Method 71 Rotate about the y axis over. What is the lower bound? 130

131 Volume: Disk Method 72 Rotate about the y axis over. What is the upper bound? 131

132 Volume: Disk Method 73 Rotate about the y axis over. What is the radius? 132

133 Volume: Disk Method 74 Rotate about the y axis over. What is integral? A B C D 133

134 Volume: Disk Method 75 Rotate about the y axis over. What is the volume? 134

135 Volume: Disk Method 76 Rotate about the over. What is the lower bound? 135

136 Volume: Disk Method 77 Rotate about the over. What is the upper bound? 136

137 Volume: Disk Method 78 Rotate about the over. What is the radius? 137

138 Volume: Disk Method 79 Rotate about the over. What is integral? A B C D 138

139 Volume: Disk Method 80 Rotate about the over. What is the volume? 139

140 Volume: Washer Method Return to Table of Contents 140

141 Volume: Washer Method For the washer method, there is a gap between the region being rotated and the axis. Rotating this rectangle we get a tube, or a cylinder with a smaller cylinder taken away. Our cross section would be: R r 141

142 Volume: Washer Method Volume: Washer Method When rotating about a horizontal axis: Where R is the greater distance from the axis, not necessarily the upper function. When rotating about a vertical axis: *Caution: π can be factored out but not the square. 142

143 Volume: Washer Method Find the volume when f(x) and g(x) are rotated about the x axis [0,2]. 143

144 Volume: Washer Method Rotate the region bound by y=x 2, x=2, and y=0 about the y axis. Since a vertical axis of rotation, integration is done in terms of y. (2,4) Hint 144

145 Volume: Washer Method Rotate the region bound by x axis, y=x 2, x=1, and x=2 about y= 1. Since y= 1 is horizontal integration is done in terms of x. R=x 2 +1 r=1 145

146 Volume: Washer Method 81 Rotate the region between and about the x axis over [0,1]. What is the lower bound? 146

147 Volume: Washer Method 82 Rotate the region between and about the x axis over [0,1]. What is the upper bound? 147

148 Volume: Washer Method 83 Rotate the region between and about the x axis over [0,1]. What is integral? A B C D 148

149 Volume: Washer Method 84 Rotate the region between and about the x axis over [0,1]. What is the volume? 149

150 Volume: Washer Method 85 Rotate the region between and about the y axis over [0,1]. What is integral? A B C D 150

151 Volume: Washer Method 86 Rotate the region between and about the y axis over [0,1]. What is the volume? 151

152 Volume: Washer Method 87 Rotate the region between and about the y=1 over [0,1]. What is R? A B C D 152

153 Volume: Washer Method 88 Rotate the region between and about the y=1 over [0,1]. What is integral? A B C D 153

154 Volume: Washer Method 89 Rotate the region between and about the y=1 over [0,1]. What is the volume? 154

155 Volume: Washer Method 90 Rotate the region between and about the x= 1 over [0,1]. What is integral? A B C D 155

156 Volume: Washer Method 91 Rotate the region between and about the x= 1 over [0,1]. What is the volume? 156

157 Volume: Shell Method Return to Table of Contents 157

158 Volume: Shell Method Volume: Shell Method When rotating about a horizontal axis: When rotating about a vertical axis: 158

159 Volume: Shell Method Find the volume of the solid obtained by rotating the area under the graph of over about the y axis. 159

160 Volume: Shell Method Find the volume of the solid obtained by rotating the area between the graph of and about the y axis. 160

161 Volume: Shell Method Find the volume of the solid obtained by rotating the area under the graph of over about 161

162 Volume: Shell Method 92 Find the volume of the solid generated by rotating the area under the graph of from about the y axis. A B C D 162

163 Volume: Shell Method 93 Find the volume of the solid generated by rotating the area between the graphs of and about the y axis. A B C D 163

AP Calculus AB Integration

AP Calculus AB Integration Slide 1 / 175 Slide 2 / 175 AP Calculus AB Integration 2015-11-24 www.njctl.org Slide 3 / 175 Table of Contents click on the topic to go to that section Riemann Sums Trapezoid Approximation Area Under

More information

AP Calculus AB. Integration. Table of Contents

AP Calculus AB. Integration.  Table of Contents AP Calculus AB Integration 2015 11 24 www.njctl.org Table of Contents click on the topic to go to that section Riemann Sums Trapezoid Approximation Area Under a Curve (The Definite Integral) Antiderivatives

More information

AP Calculus AB. Slide 1 / 175. Slide 2 / 175. Slide 3 / 175. Integration. Table of Contents

AP Calculus AB. Slide 1 / 175. Slide 2 / 175. Slide 3 / 175. Integration. Table of Contents Slide 1 / 175 Slide 2 / 175 AP Calculus AB Integration 2015-11-24 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 175 Riemann Sums Trapezoid Approximation Area Under

More information

4.3. Riemann Sums. Riemann Sums. Riemann Sums and Definite Integrals. Objectives

4.3. Riemann Sums. Riemann Sums. Riemann Sums and Definite Integrals. Objectives 4.3 Riemann Sums and Definite Integrals Objectives Understand the definition of a Riemann sum. Evaluate a definite integral using limits & Riemann Sums. Evaluate a definite integral using geometric formulas

More information

Goal: Approximate the area under a curve using the Rectangular Approximation Method (RAM) RECTANGULAR APPROXIMATION METHODS

Goal: Approximate the area under a curve using the Rectangular Approximation Method (RAM) RECTANGULAR APPROXIMATION METHODS AP Calculus 5. Areas and Distances Goal: Approximate the area under a curve using the Rectangular Approximation Method (RAM) Exercise : Calculate the area between the x-axis and the graph of y = 3 2x.

More information

RAMs.notebook December 04, 2017

RAMs.notebook December 04, 2017 RAMsnotebook December 04, 2017 Riemann Sums and Definite Integrals Estimate the shaded area Area between a curve and the x-axis How can you improve your estimate? Suppose f(x) 0 x [a, b], then the area

More information

Integration. Copyright Cengage Learning. All rights reserved.

Integration. Copyright Cengage Learning. All rights reserved. 4 Integration Copyright Cengage Learning. All rights reserved. 1 4.3 Riemann Sums and Definite Integrals Copyright Cengage Learning. All rights reserved. 2 Objectives Understand the definition of a Riemann

More information

Chapter 6: The Definite Integral

Chapter 6: The Definite Integral Name: Date: Period: AP Calc AB Mr. Mellina Chapter 6: The Definite Integral v v Sections: v 6.1 Estimating with Finite Sums v 6.5 Trapezoidal Rule v 6.2 Definite Integrals 6.3 Definite Integrals and Antiderivatives

More information

y=5 y=1+x 2 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions

y=5 y=1+x 2 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions. Which of the following integrals correctly corresponds to the area of the shaded region in the figure to the right? (A) (B) (C) (D) (E)

More information

Objective SWBAT find distance traveled, use rectangular approximation method (RAM), volume of a sphere, and cardiac output.

Objective SWBAT find distance traveled, use rectangular approximation method (RAM), volume of a sphere, and cardiac output. 5.1 Estimating with Finite Sums Objective SWBAT find distance traveled, use rectangular approximation method (RAM), volume of a sphere, and cardiac output. Distance Traveled We know that pondering motion

More information

Calculus AB Topics Limits Continuity, Asymptotes

Calculus AB Topics Limits Continuity, Asymptotes Calculus AB Topics Limits Continuity, Asymptotes Consider f x 2x 1 x 3 1 x 3 x 3 Is there a vertical asymptote at x = 3? Do not give a Precalculus answer on a Calculus exam. Consider f x 2x 1 x 3 1 x 3

More information

Fundamental Theorem of Calculus

Fundamental Theorem of Calculus NCTM Annual Meeting and Eposition Denver, CO April 8, Presented by Mike Koehler Blue Valley North High School Overland Park, KS I. Approimations with Rectangles (Finding the Area Under Curves by Approimating

More information

Integrals. D. DeTurck. January 1, University of Pennsylvania. D. DeTurck Math A: Integrals 1 / 61

Integrals. D. DeTurck. January 1, University of Pennsylvania. D. DeTurck Math A: Integrals 1 / 61 Integrals D. DeTurck University of Pennsylvania January 1, 2018 D. DeTurck Math 104 002 2018A: Integrals 1 / 61 Integrals Start with dx this means a little bit of x or a little change in x If we add up

More information

7.1 Indefinite Integrals Calculus

7.1 Indefinite Integrals Calculus 7.1 Indefinite Integrals Calculus Learning Objectives A student will be able to: Find antiderivatives of functions. Represent antiderivatives. Interpret the constant of integration graphically. Solve differential

More information

AP Calculus AB 2nd Semester Homework List

AP Calculus AB 2nd Semester Homework List AP Calculus AB 2nd Semester Homework List Date Assigned: 1/4 DUE Date: 1/6 Title: Typsetting Basic L A TEX and Sigma Notation Write the homework out on paper. Then type the homework on L A TEX. Use this

More information

y=5 y=1+x 2 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions

y=5 y=1+x 2 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions. Which of the following integrals correctly corresponds to the area of the shaded region in the figure to the right? (A) (B) (C) (D) (E)

More information

AP Calculus AB Riemann Sums

AP Calculus AB Riemann Sums AP Calculus AB Riemann Sums Name Intro Activity: The Gorilla Problem A gorilla (wearing a parachute) jumped off of the top of a building. We were able to record the velocity of the gorilla with respect

More information

Calculus BC: Section I

Calculus BC: Section I Calculus BC: Section I Section I consists of 45 multiple-choice questions. Part A contains 28 questions and does not allow the use of a calculator. Part B contains 17 questions and requires a graphing

More information

Math 221 Exam III (50 minutes) Friday April 19, 2002 Answers

Math 221 Exam III (50 minutes) Friday April 19, 2002 Answers Math Exam III (5 minutes) Friday April 9, Answers I. ( points.) Fill in the boxes as to complete the following statement: A definite integral can be approximated by a Riemann sum. More precisely, if a

More information

Foothill High School. AP Calculus BC. Note Templates Semester 1, Student Name.

Foothill High School. AP Calculus BC. Note Templates Semester 1, Student Name. Foothill High School AP Calculus BC Note Templates Semester 1, 2011-2012 Student Name Teacher: Burt Dixon bdixon@pleasanton.k12.ca.us 2.1 Limits Chap1-2 Page 1 Chap1-2 Page 2 Chap1-2 Page 3 Chap1-2 Page

More information

t dt Estimate the value of the integral with the trapezoidal rule. Use n = 4.

t dt Estimate the value of the integral with the trapezoidal rule. Use n = 4. Trapezoidal Rule We have already found the value of an integral using rectangles in the first lesson of this module. In this section we will again be estimating the value of an integral using geometric

More information

Honors Calculus Curriculum Maps

Honors Calculus Curriculum Maps Honors Calculus Curriculum Maps Unit of Study: Prerequisites for Calculus Unit of Study: Limits and Continuity Unit of Study: Differentiation Unit of Study: Applications of Derivatives Unit of Study: The

More information

INTRO TO LIMITS & CALCULUS MR. VELAZQUEZ AP CALCULUS

INTRO TO LIMITS & CALCULUS MR. VELAZQUEZ AP CALCULUS INTRO TO LIMITS & CALCULUS MR. VELAZQUEZ AP CALCULUS WHAT IS CALCULUS? Simply put, Calculus is the mathematics of change. Since all things change often and in many ways, we can expect to understand a wide

More information

Curriculum and Pacing Guide Mr. White AP Calculus AB Revised May 2015

Curriculum and Pacing Guide Mr. White AP Calculus AB Revised May 2015 Curriculum and Pacing Guide Mr. White AP Calculus AB Revised May 2015 Students who successfully complete this course will receive one credit AP Calculus AB and will take the AP Calculus AB Exam. 1. The

More information

Steps for finding area using Summation

Steps for finding area using Summation Steps for finding area using Summation 1) Identify a o and a 0 = starting point of the given interval [a, b] where n = # of rectangles 2) Find the c i 's Right: Left: 3) Plug each c i into given f(x) >

More information

2007 AP Calculus AB Free-Response Questions Section II, Part A (45 minutes) # of questions: 3 A graphing calculator may be used for this part

2007 AP Calculus AB Free-Response Questions Section II, Part A (45 minutes) # of questions: 3 A graphing calculator may be used for this part 2007 AP Calculus AB Free-Response Questions Section II, Part A (45 minutes) # of questions: 3 A graphing calculator may be used for this part 1. Let R be the region in the first and second quadrants bounded

More information

GENERAL TIPS WHEN TAKING THE AP CALC EXAM. Multiple Choice Portion

GENERAL TIPS WHEN TAKING THE AP CALC EXAM. Multiple Choice Portion GENERAL TIPS WHEN TAKING THE AP CALC EXAM. Multiple Choice Portion 1. You are hunting for apples, aka easy questions. Do not go in numerical order; that is a trap! 2. Do all Level 1s first. Then 2. Then

More information

AP Calculus AB Course Syllabus

AP Calculus AB Course Syllabus AP Calculus AB Course Syllabus Grant Community High School Mr. Rous Textbook Finney, Ross L., Franklin D. Demana, Bert K. Waits, and Daniel Kennedy. Calculus Graphical, Numerical, Algebraic, Fourth Addition,

More information

Radnor High School Course Syllabus Advanced Placement Calculus BC 0460

Radnor High School Course Syllabus Advanced Placement Calculus BC 0460 Radnor High School Modified April 24, 2012 Course Syllabus Advanced Placement Calculus BC 0460 Credits: 1 Grades: 11, 12 Weighted: Yes Prerequisite: Recommended by Department Length: Year Format: Meets

More information

MA 137 Calculus 1 with Life Science Applications. (Section 6.1)

MA 137 Calculus 1 with Life Science Applications. (Section 6.1) MA 137 Calculus 1 with Life Science Applications (Section 6.1) Alberto Corso alberto.corso@uky.edu Department of Mathematics University of Kentucky December 2, 2015 1/17 Sigma (Σ) Notation In approximating

More information

Practice problems from old exams for math 132 William H. Meeks III

Practice problems from old exams for math 132 William H. Meeks III Practice problems from old exams for math 32 William H. Meeks III Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These practice tests are

More information

Burlington County Institute of Technology Curriculum Document

Burlington County Institute of Technology Curriculum Document Burlington County Institute of Technology Curriculum Document Course Title: Calculus Curriculum Area: Mathematics Credits: 5 Credits per course Board Approved: June 2017 Prepared by: Jessica Rista, John

More information

Math 123 Elem. Calculus Fall 2014 Name: Sec.: Exam 4 Bonus Questions

Math 123 Elem. Calculus Fall 2014 Name: Sec.: Exam 4 Bonus Questions Math 13 Elem. Calculus Fall 01 Name: Sec.: Exam Bonus Questions The questions below are bonus questions. You should write your answers on this page. BOTH THE STEPS YOU SHOW (YOUR WORK) AND YOUR FINAL ANSWER

More information

Chapter 4 Integration

Chapter 4 Integration Chapter 4 Integration SECTION 4.1 Antiderivatives and Indefinite Integration Calculus: Chapter 4 Section 4.1 Antiderivative A function F is an antiderivative of f on an interval I if F '( x) f ( x) for

More information

INTEGRATION: AREAS AND RIEMANN SUMS MR. VELAZQUEZ AP CALCULUS

INTEGRATION: AREAS AND RIEMANN SUMS MR. VELAZQUEZ AP CALCULUS INTEGRATION: AREAS AND RIEMANN SUMS MR. VELAZQUEZ AP CALCULUS APPROXIMATING AREA For today s lesson, we will be using different approaches to the area problem. The area problem is to definite integrals

More information

v(t) v(t) Assignment & Notes 5.2: Intro to Integrals Due Date: Friday, 1/10

v(t) v(t) Assignment & Notes 5.2: Intro to Integrals Due Date: Friday, 1/10 Assignment & Notes 5.2: Intro to Integrals 1. The velocity function (in miles and hours) for Ms. Hardtke s Christmas drive to see her family is shown at the right. Find the total distance Ms. H travelled

More information

All work must be shown in this course for full credit. Unsupported answers may receive NO credit.

All work must be shown in this course for full credit. Unsupported answers may receive NO credit. AP Calculus 6.. Worksheet Estimating with Finite Sums All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. Suppose an oil pump is producing 8 gallons per hour

More information

Calculus II (Fall 2015) Practice Problems for Exam 1

Calculus II (Fall 2015) Practice Problems for Exam 1 Calculus II (Fall 15) Practice Problems for Exam 1 Note: Section divisions and instructions below are the same as they will be on the exam, so you will have a better idea of what to expect, though I will

More information

Topic Outline AP CALCULUS AB:

Topic Outline AP CALCULUS AB: Topic Outline AP CALCULUS AB: Unit 1: Basic tools and introduction to the derivative A. Limits and properties of limits Importance and novelty of limits Traditional definitions of the limit Graphical and

More information

MATH 1014 Tutorial Notes 8

MATH 1014 Tutorial Notes 8 MATH4 Calculus II (8 Spring) Topics covered in tutorial 8:. Numerical integration. Approximation integration What you need to know: Midpoint rule & its error Trapezoid rule & its error Simpson s rule &

More information

AP Calculus Curriculum Guide Dunmore School District Dunmore, PA

AP Calculus Curriculum Guide Dunmore School District Dunmore, PA AP Calculus Dunmore School District Dunmore, PA AP Calculus Prerequisite: Successful completion of Trigonometry/Pre-Calculus Honors Advanced Placement Calculus is the highest level mathematics course offered

More information

MATH 2053 Calculus I Review for the Final Exam

MATH 2053 Calculus I Review for the Final Exam MATH 05 Calculus I Review for the Final Exam (x+ x) 9 x 9 1. Find the limit: lim x 0. x. Find the limit: lim x + x x (x ).. Find lim x (x 5) = L, find such that f(x) L < 0.01 whenever 0 < x

More information

Notes about changes to Approved Syllabus # 43080v2

Notes about changes to Approved Syllabus # 43080v2 Notes about changes to Approved Syllabus # 43080v2 1. An update to the syllabus was necessary because of a county wide adoption of new textbooks for AP Calculus. 2. No changes were made to the Course Outline

More information

CALCULUS AP AB Q401.Chapter 5B Lesson 1: Fundamental Theorem (Part 1) Fundamental Theorem of Calculus (Part I)

CALCULUS AP AB Q401.Chapter 5B Lesson 1: Fundamental Theorem (Part 1) Fundamental Theorem of Calculus (Part I) CALCULUS AP AB Q401.Chapter 5B Lesson 1: Fundamental Theorem (Part 1) Fundamental Theorem of Calculus (Part I) CALCULUS AP AB Q401.Chapter 5B Lesson 1: Fundamental Theorem (Part 1) APPLICATION (1, 4) 2

More information

L. Function Analysis. ). If f ( x) switches from decreasing to increasing at c, there is a relative minimum at ( c, f ( c)

L. Function Analysis. ). If f ( x) switches from decreasing to increasing at c, there is a relative minimum at ( c, f ( c) L. Function Analysis What you are finding: You have a function f ( x). You want to find intervals where f ( x) is increasing and decreasing, concave up and concave down. You also want to find values of

More information

Math 106 Answers to Exam 3a Fall 2015

Math 106 Answers to Exam 3a Fall 2015 Math 6 Answers to Exam 3a Fall 5.. Consider the curve given parametrically by x(t) = cos(t), y(t) = (t 3 ) 3, for t from π to π. (a) (6 points) Find all the points (x, y) where the graph has either a vertical

More information

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus Objectives Evaluate a definite integral using the Fundamental Theorem of Calculus. Understand and use the Mean Value Theorem for Integrals. Find the average value of

More information

DEFINITE INTEGRALS & NUMERIC INTEGRATION

DEFINITE INTEGRALS & NUMERIC INTEGRATION DEFINITE INTEGRALS & NUMERIC INTEGRATION Calculus answers two very important questions. The first, how to find the instantaneous rate of change, we answered with our study of the derivative. We are now

More information

Answer Key for AP Calculus AB Practice Exam, Section I

Answer Key for AP Calculus AB Practice Exam, Section I Answer Key for AP Calculus AB Practice Exam, Section I Multiple-Choice Questions Question # Key B B 3 A 4 E C 6 D 7 E 8 C 9 E A A C 3 D 4 A A 6 B 7 A 8 B 9 C D E B 3 A 4 A E 6 A 7 A 8 A 76 E 77 A 78 D

More information

A.P. Calculus BC Summer Assignment 2018 I am so excited you are taking Calculus BC! For your summer assignment, I would like you to complete the

A.P. Calculus BC Summer Assignment 2018 I am so excited you are taking Calculus BC! For your summer assignment, I would like you to complete the A.P. Calculus BC Summer Assignment 2018 I am so excited you are taking Calculus BC! For your summer assignment, I would like you to complete the attached packet of problems, and turn it in on Monday, August

More information

5.5 Volumes: Tubes. The Tube Method. = (2π [radius]) (height) ( x k ) = (2πc k ) f (c k ) x k. 5.5 volumes: tubes 435

5.5 Volumes: Tubes. The Tube Method. = (2π [radius]) (height) ( x k ) = (2πc k ) f (c k ) x k. 5.5 volumes: tubes 435 5.5 volumes: tubes 45 5.5 Volumes: Tubes In Section 5., we devised the disk method to find the volume swept out when a region is revolved about a line. To find the volume swept out when revolving a region

More information

I. Horizontal and Vertical Tangent Lines

I. Horizontal and Vertical Tangent Lines How to find them: You need to work with f " x Horizontal tangent lines: set f " x Vertical tangent lines: find values of x where f " x I. Horizontal and Vertical Tangent Lines ( ), the derivative of function

More information

a) rectangles whose height is the right-hand endpoint b) rectangles whose height is the left-hand endpoint

a) rectangles whose height is the right-hand endpoint b) rectangles whose height is the left-hand endpoint CALCULUS OCTOBER 7 # Day Date Assignment Description 6 M / p. - #,, 8,, 9abc, abc. For problems 9 and, graph the function and approximate the integral using the left-hand endpoints, right-hand endpoints,

More information

. CALCULUS AB. Name: Class: Date:

. CALCULUS AB. Name: Class: Date: Class: _ Date: _. CALCULUS AB SECTION I, Part A Time- 55 Minutes Number of questions -8 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION Directions: Solve each of the following problems, using

More information

AP Calculus AB. Limits & Continuity.

AP Calculus AB. Limits & Continuity. 1 AP Calculus AB Limits & Continuity 2015 10 20 www.njctl.org 2 Table of Contents click on the topic to go to that section Introduction The Tangent Line Problem Definition of a Limit and Graphical Approach

More information

Chapter 5 Integrals. 5.1 Areas and Distances

Chapter 5 Integrals. 5.1 Areas and Distances Chapter 5 Integrals 5.1 Areas and Distances We start with a problem how can we calculate the area under a given function ie, the area between the function and the x-axis? If the curve happens to be something

More information

Distance and Velocity

Distance and Velocity Distance and Velocity - Unit #8 : Goals: The Integral Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite integral and

More information

Math 121 (Lesieutre); 9.1: Polar coordinates; November 22, 2017

Math 121 (Lesieutre); 9.1: Polar coordinates; November 22, 2017 Math 2 Lesieutre; 9: Polar coordinates; November 22, 207 Plot the point 2, 2 in the plane If you were trying to describe this point to a friend, how could you do it? One option would be coordinates, but

More information

The Fundamental Theorem of Calculus Part 3

The Fundamental Theorem of Calculus Part 3 The Fundamental Theorem of Calculus Part FTC Part Worksheet 5: Basic Rules, Initial Value Problems, Rewriting Integrands A. It s time to find anti-derivatives algebraically. Instead of saying the anti-derivative

More information

The Integral of a Function. The Indefinite Integral

The Integral of a Function. The Indefinite Integral The Integral of a Function. The Indefinite Integral Undoing a derivative: Antiderivative=Indefinite Integral Definition: A function is called an antiderivative of a function on same interval,, if differentiation

More information

We saw in Section 5.1 that a limit of the form. arises when we compute an area.

We saw in Section 5.1 that a limit of the form. arises when we compute an area. INTEGRALS 5 INTEGRALS Equation 1 We saw in Section 5.1 that a limit of the form n lim f ( x *) x n i 1 i lim[ f ( x *) x f ( x *) x... f ( x *) x] n 1 2 arises when we compute an area. n We also saw that

More information

PLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 2. (a) (b) (c) (d) (e) (a) (b) (c) (d) (e) (a) (b) (c) (d) (e)...

PLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 2. (a) (b) (c) (d) (e) (a) (b) (c) (d) (e) (a) (b) (c) (d) (e)... Math 55, Exam III November 5, The Honor Code is in effect for this examination. All work is to be your own. No calculators. The exam lasts for hour and 5 min. Be sure that your name is on every page in

More information

(a) The best linear approximation of f at x = 2 is given by the formula. L(x) = f(2) + f (2)(x 2). f(2) = ln(2/2) = ln(1) = 0, f (2) = 1 2.

(a) The best linear approximation of f at x = 2 is given by the formula. L(x) = f(2) + f (2)(x 2). f(2) = ln(2/2) = ln(1) = 0, f (2) = 1 2. Math 180 Written Homework Assignment #8 Due Tuesday, November 11th at the beginning of your discussion class. Directions. You are welcome to work on the following problems with other MATH 180 students,

More information

Sudoku Puzzle A.P. Exam (Part B) Questions are from the 1997 and 1998 A.P. Exams A Puzzle by David Pleacher

Sudoku Puzzle A.P. Exam (Part B) Questions are from the 1997 and 1998 A.P. Exams A Puzzle by David Pleacher Sudoku Puzzle A.P. Exam (Part B) Questions are from the 1997 and 1998 A.P. Exams A Puzzle by David Pleacher Solve the 4 multiple-choice problems below. A graphing calculator is required for some questions

More information

1 Chapter 1: Areas, Volumes, and Simple Sums

1 Chapter 1: Areas, Volumes, and Simple Sums Syllabus Summary This is a living list and will be updated throughout the semester. In this list I summarize the material from the syllabus indicating which material is very important, of normal importance,

More information

AP Calculus AB. Review for Test: Applications of Integration

AP Calculus AB. Review for Test: Applications of Integration Name Review for Test: Applications of Integration AP Calculus AB Test Topics: Mean Value Theorem for Integrals (section 4.4) Average Value of a Function (manipulation of MVT for Integrals) (section 4.4)

More information

6.1 Area Between Curves. Example 1: Calculate the area of the region between the parabola y = 1 x 2 and the line y = 1 x

6.1 Area Between Curves. Example 1: Calculate the area of the region between the parabola y = 1 x 2 and the line y = 1 x AP Calculus 6.1 Area Between Curves Name: Goal: Calculate the Area between curves Keys to Success: Top Curve Bottom Curve (integrate w/respect to x or dx) Right Curve Left Curve (integrate w/respect to

More information

See animations and interactive applets of some of these at. Fall_2009/Math123/Notes

See animations and interactive applets of some of these at.   Fall_2009/Math123/Notes MA123, Chapter 7 Word Problems (pp. 125-153) Chapter s Goal: In this chapter we study the two main types of word problems in Calculus. Optimization Problems. i.e., max - min problems Related Rates See

More information

Sample Questions PREPARING FOR THE AP (AB) CALCULUS EXAMINATION. tangent line, a+h. a+h

Sample Questions PREPARING FOR THE AP (AB) CALCULUS EXAMINATION. tangent line, a+h. a+h Sample Questions PREPARING FOR THE AP (AB) CALCULUS EXAMINATION B B A B tangent line,, a f '(a) = lim h 0 f(a + h) f(a) h a+h a b b f(x) dx = lim [f(x ) x + f(x ) x + f(x ) x +...+ f(x ) x ] n a n B B

More information

Math 42: Fall 2015 Midterm 2 November 3, 2015

Math 42: Fall 2015 Midterm 2 November 3, 2015 Math 4: Fall 5 Midterm November 3, 5 NAME: Solutions Time: 8 minutes For each problem, you should write down all of your work carefully and legibly to receive full credit When asked to justify your answer,

More information

Math 1526 Excel Lab 2 Summer 2012

Math 1526 Excel Lab 2 Summer 2012 Math 1526 Excel Lab 2 Summer 2012 Riemann Sums, Trapezoidal Rule and Simpson's Rule: In this lab you will learn how to recover information from rate of change data. For instance, if you have data for marginal

More information

The Princeton Review AP Calculus BC Practice Test 1

The Princeton Review AP Calculus BC Practice Test 1 8 The Princeton Review AP Calculus BC Practice Test CALCULUS BC SECTION I, Part A Time 55 Minutes Number of questions 8 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION Directions: Solve each

More information

HUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK

HUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK HUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK COURSE / SUBJECT A P C a l c u l u s ( A B ) KEY COURSE OBJECTIVES/ENDURING UNDERSTANDINGS OVERARCHING/ESSENTIAL SKILLS OR QUESTIONS Limits and Continuity Derivatives

More information

4.9 APPROXIMATING DEFINITE INTEGRALS

4.9 APPROXIMATING DEFINITE INTEGRALS 4.9 Approximating Definite Integrals Contemporary Calculus 4.9 APPROXIMATING DEFINITE INTEGRALS The Fundamental Theorem of Calculus tells how to calculate the exact value of a definite integral IF the

More information

Final Exam Solutions

Final Exam Solutions Final Exam Solutions Laurence Field Math, Section March, Name: Solutions Instructions: This exam has 8 questions for a total of points. The value of each part of each question is stated. The time allowed

More information

ANOTHER FIVE QUESTIONS:

ANOTHER FIVE QUESTIONS: No peaking!!!!! See if you can do the following: f 5 tan 6 sin 7 cos 8 sin 9 cos 5 e e ln ln @ @ Epress sin Power Series Epansion: d as a Power Series: Estimate sin Estimate MACLAURIN SERIES ANOTHER FIVE

More information

Calculus Honors Curriculum Guide Dunmore School District Dunmore, PA

Calculus Honors Curriculum Guide Dunmore School District Dunmore, PA Calculus Honors Dunmore School District Dunmore, PA Calculus Honors Prerequisite: Successful completion of Trigonometry/Pre-Calculus Honors Major topics include: limits, derivatives, integrals. Instruction

More information

AP Calculus AB. Limits & Continuity. Table of Contents

AP Calculus AB. Limits & Continuity.   Table of Contents AP Calculus AB Limits & Continuity 2016 07 10 www.njctl.org www.njctl.org Table of Contents click on the topic to go to that section Introduction The Tangent Line Problem Definition of a Limit and Graphical

More information

M152: Calculus II Midterm Exam Review

M152: Calculus II Midterm Exam Review M52: Calculus II Midterm Exam Review Chapter 4. 4.2 : Mean Value Theorem. - Know the statement and idea of Mean Value Theorem. - Know how to find values of c making the theorem true. - Realize the importance

More information

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA CALCULUS AB SECTION I, Part A Time 55 minutes Number of questions 8 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directions: Solve each of the following problems,

More information

Brunswick School Department: Grades Essential Understandings

Brunswick School Department: Grades Essential Understandings Understandings Questions Knowledge Vocabulary Skills Mathematics The concept of an integral as the operational inverse of a derivative and as a summation model is introduced using antiderivatives. Students

More information

Energy Flow in Technological Systems. December 01, 2014

Energy Flow in Technological Systems. December 01, 2014 Energy Flow in Technological Systems Scientific Notation (Exponents) Scientific notation is used when we are dealing with very large or very small numbers. A number placed in scientific notation is made

More information

Fairfield Public Schools

Fairfield Public Schools Mathematics Fairfield Public Schools AP Calculus BC AP Calculus BC BOE Approved 04/08/2014 1 AP CALCULUS BC Critical Areas of Focus Advanced Placement Calculus BC consists of a full year of college calculus.

More information

Level 1 Calculus Final Exam Day 1 50 minutes

Level 1 Calculus Final Exam Day 1 50 minutes Level 1 Calculus Final Exam 2013 Day 1 50 minutes Name: Block: Circle Teacher Name LeBlanc Normile Instructions Write answers in the space provided and show all work. Calculators okay but observe instructions

More information

AB 1: Find lim. x a.

AB 1: Find lim. x a. AB 1: Find lim x a f ( x) AB 1 Answer: Step 1: Find f ( a). If you get a zero in the denominator, Step 2: Factor numerator and denominator of f ( x). Do any cancellations and go back to Step 1. If you

More information

MATH 1241 Common Final Exam Fall 2010

MATH 1241 Common Final Exam Fall 2010 MATH 1241 Common Final Exam Fall 2010 Please print the following information: Name: Instructor: Student ID: Section/Time: The MATH 1241 Final Exam consists of three parts. You have three hours for the

More information

Topic Subtopics Essential Knowledge (EK)

Topic Subtopics Essential Knowledge (EK) Unit/ Unit 1 Limits [BEAN] 1.1 Limits Graphically Define a limit (y value a function approaches) One sided limits. Easy if it s continuous. Tricky if there s a discontinuity. EK 1.1A1: Given a function,

More information

Sample Questions PREPARING FOR THE AP (BC) CALCULUS EXAMINATION. tangent line, a+h. a+h

Sample Questions PREPARING FOR THE AP (BC) CALCULUS EXAMINATION. tangent line, a+h. a+h Sample Questions PREPARING FOR THE AP (BC) CALCULUS EXAMINATION B B A B tangent line,, a f '(a) = lim h 0 f(a + h) f(a) h a+h a b b f(x) dx = lim [f(x ) x + f(x ) x + f(x ) x +...+ f(x ) x ] n a n B B

More information

Multiple Choice Answers. MA 110 Precalculus Spring 2016 Exam 1 9 February Question

Multiple Choice Answers. MA 110 Precalculus Spring 2016 Exam 1 9 February Question MA 110 Precalculus Spring 2016 Exam 1 9 February 2016 Name: Section: Last 4 digits of student ID #: This exam has eleven multiple choice questions (five points each) and five free response questions (nine

More information

All work must be shown in this course for full credit. Unsupported answers may receive NO credit.

All work must be shown in this course for full credit. Unsupported answers may receive NO credit. AP Calculus 5. Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. Suppose an oil pump is producing 8 gallons per hour for the first 5 hours of

More information

1.1 Radical Expressions: Rationalizing Denominators

1.1 Radical Expressions: Rationalizing Denominators 1.1 Radical Expressions: Rationalizing Denominators Recall: 1. A rational number is one that can be expressed in the form a, where b 0. b 2. An equivalent fraction is determined by multiplying or dividing

More information

AP Calculus AB Course Description and Syllabus

AP Calculus AB Course Description and Syllabus AP Calculus AB Course Description and Syllabus Course Objective: This course is designed to prepare the students for the AP Exam in May. Students will learn to use graphical, numerical, verbal and analytical

More information

Pre-Calculus Vectors

Pre-Calculus Vectors Slide 1 / 159 Slide 2 / 159 Pre-Calculus Vectors 2015-03-24 www.njctl.org Slide 3 / 159 Table of Contents Intro to Vectors Converting Rectangular and Polar Forms Operations with Vectors Scalar Multiples

More information

CALCULUS AP BC Q301CH5A: (Lesson 1-A) AREA and INTEGRAL Area Integral Connection and Riemann Sums

CALCULUS AP BC Q301CH5A: (Lesson 1-A) AREA and INTEGRAL Area Integral Connection and Riemann Sums CALCULUS AP BC Q301CH5A: (Lesson 1-A) AREA and INTEGRAL Area Integral Connection and Riemann Sums INTEGRAL AND AREA BY HAND (APPEAL TO GEOMETRY) I. Below are graphs that each represent a different f()

More information

Welcome to. Elementary Calculus with Trig II CRN(13828) Instructor: Quanlei Fang. Department of Mathematics, Virginia Tech, Spring 2008

Welcome to. Elementary Calculus with Trig II CRN(13828) Instructor: Quanlei Fang. Department of Mathematics, Virginia Tech, Spring 2008 Welcome to Elementary Calculus with Trig II CRN(13828) Instructor: Quanlei Fang Department of Mathematics, Virginia Tech, Spring 2008 1 Be sure to read the course contract Contact Information Text Grading

More information

a) rectangles whose height is the right-hand endpoint b) rectangles whose height is the left-hand endpoint

a) rectangles whose height is the right-hand endpoint b) rectangles whose height is the left-hand endpoint # Day Date Assignment Description 36 M /3 p. 32-32 #, 2, 8, 2, 29abc, 3abc. For problems 29 and 3, graph the function and approximate the integral using the left-hand endpoints, right-hand endpoints, and

More information

Derivatives and Rates of Change

Derivatives and Rates of Change Sec.1 Derivatives and Rates of Change A. Slope of Secant Functions rise Recall: Slope = m = = run Slope of the Secant Line to a Function: Examples: y y = y1. From this we are able to derive: x x x1 m y

More information

Final Exam Review Exercise Set A, Math 1551, Fall 2017

Final Exam Review Exercise Set A, Math 1551, Fall 2017 Final Exam Review Exercise Set A, Math 1551, Fall 2017 This review set gives a list of topics that we explored throughout this course, as well as a few practice problems at the end of the document. A complete

More information

Turn off all noise-making devices and all devices with an internet connection and put them away. Put away all headphones, earbuds, etc.

Turn off all noise-making devices and all devices with an internet connection and put them away. Put away all headphones, earbuds, etc. NAME: FINAL EXAM INSTRUCTIONS: This exam is a closed book exam. You may not use your text, homework, or other aids except for a 3 5 notecard. You may use an allowable calculator, TI 83 or 84 to perform

More information

AP Calculus AB. Course Overview. Course Outline and Pacing Guide

AP Calculus AB. Course Overview. Course Outline and Pacing Guide AP Calculus AB Course Overview AP Calculus AB is designed to follow the topic outline in the AP Calculus Course Description provided by the College Board. The primary objective of this course is to provide

More information