AP Calculus Review Assignment Answer Sheet 1 Name: Date: Per. Harton Spring Break Packet 015 This is an AP Calc Review packet. As we get closer to the eam, it is time to start reviewing old concepts. Use the answer sheet provided for all answers and then when appropriate show work. Assignment: First 60 problems Due Date: April 14, 015 Late work will receive reduced credit. This assignment will be graded. Therefore, this is a project. A lot of you will skip the reading material and see if you can work the problems. If you can work the problems, then do, but, I do you want you to read the reading material. This is an important part of the review, going over old concepts.
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AP Review Packet: 7 Essential AP Calculus Concepts 1. Limits A. Finite Limits - Try direct substitutions first, if that fails, try to factor or use L Hopital s Rule. 1. lim 5 4. 8 lim 4. lim sin 4. lim 0 5. 1 cos lim 0 B. Limits to Infinity - These are also called horizontal asymptotes. If the degree of the numerator is smaller than that of the denominator, then the lim 0. If the degree of the numerator is equal to that of the denominator, then the lead coefficient lim. lead coefficient If the degree of the numerator is larger than that of the denominator, then the lim Does Not Eist Be careful of limits going to. Sometimes they don t follow the patterns above! When you have e, remember that it dominates any polynomial and you need to look at find a simple left-end and right-end model to evaluate. 6. lim 4 7. 5 8 lim 8. lim 4 4 9. 4 lim 4 5 4 10. lim e 4 11. lim e 4 C. Limits from a Picture Look at the graph and find the intended height of the function. Remember, left and right hand limits must agree for the limit to eist. 1. lim f( ) f () 1. lim f( ) 14. lim f( ) lim f( ) lim f( ) f () AP Calculus 07-08 AP Review Packet: Page 1 of 15
. Types of Discontinuity Jump Discontinuity (Non-Removable) Function is Undefined (Removable) Limit doesn t equal value (Removable). Types of Non-Differentiability The slope on left approach must agree with slope on right approach. Not continuous Corner/cusp Vertical tangent 15. For what values of a and b is the function a, g ( ) b 8, continuous and differentiable? 4. Intermediate Value Theorem 16. f( ) is continuous on [, 4] and differentiable on (,4). If f () 6 and f () and f (4) 6 : a. How many solutions are guaranteed? b. Does f( ) for a value of in the interval [, 4]? c. Can f( ) 8 for some value of in the interval [, 4]? AP Calculus 07-08 AP Review Packet: Page of 15
5. Basic Differentiation Rules d n d n n1 Differentiation Rules d sin cos d d sin d cos tan sec d sec sec tan cot csc d d d d d csc cot ln d csc d d 1 u du d u d e e u du u d The Product Rule d f ( ) g ( ) f ( ) g '( ) g ( ) f '( ) d 17. f ( ) ( ) ( 1), f '( ) The Quotient Rule d f ( ) g( ) f '( ) f ( ) g '( ) d g ( ) g ( ) 18. ( ) f ( ), f '( ) 1 The Chain Rule d f ( g ( )) f '( g ( )) g '( ) d 19. f f ( ) sin( ), '( ) If f ( ), and a = 10 find f '( a) 0. 1 f ' Derivative of an Inverse 1 1 ( a) 1 f '( b) f f : ( a, b) : ( b, a) 1. Find -1 ( f )'() if for 0. f ( ) AP Calculus 07-08 AP Review Packet: Page of 15
6. Derivative Rules From Tabular Data 5 f( ) 5 f '( ) 5 g ( ) 5 g'( ) 5 f( ). If h( ) f ( ) g( ), find h '().. If h( ) f ( g( )), find h '(). 4. If h ( ) find h '(). g ( ) 7. Equation of a Tangent Line The derivative indicates the instantaneous rate of change, or the slope of a line tangent to a curve. The equation of a line tangent to ( ) y b f a a. f at the point (a, b) is given by: '( )( ) A normal line is perpendicular to a tangent line through the same point. 5. Find the equation of the line tangent to f ( ) at (, 8). 6. Find the equation of the line normal to f ( ) at (, 8). 7. For what value does the graph f ( ) 4 have a tangent with a slope of 6? 8. Find the equation of a line tangent to f ( ) 5 that is parallel to 6y 8. 9. At what point does f ( ) have a slope of 4? 0. Find the equation of a line tangent to approimation to estimate f (1.1). f through the point (1,6) and use your local linear ( ) 8. Slope and Relative Error Draw in a tangent line at the indicated point and fill in the blank for each picture. 1. inc & ccd: linear appro actual value. inc & ccu: linear appro actual value. dec & ccd: linear appro actual value 4. dec & ccu: linear appro actual value AP Calculus 07-08 AP Review Packet: Page 4 of 15
9. The Mean Value Theorem for Derivatives f '( c) f ( b) f ( a) b a The slope of the tangent at c equals the slope of the secant through a and b. a c b 5. If f is continuous and differentiable for all values on (,5) and f(), f() 4 and f (5), then on the interval (,5) a. What can be said about the slope? b. How many etrema? 6. Find the average rate of change for f on [-, 0]. (slope of the secant line) ( ) 4 7. For what value c, such that 0 < c <, is the instantaneous rate of change for average rate of change over the interval [0, ]? f ( ) equal to the 8. Find the average velocity of a particle on the interval [1, ] if the particle s position is given by s( t) t 5t, where t is measured in seconds and s(t) in feet. b. At what time, c, does the particle reach its average velocity? 10. Etreme Value Theorem Etrema on a closed interval [a, b]: Test f ( a), f ( b ) and f( c #) between a and b. 9. Find the absolute etrema for f ( ) 8 on [-, 0]. 40. Find the any absolute etrema for f ( ) 6 on [0, 4]. AP Calculus 07-08 AP Review Packet: Page 5 of 15
11. Slope and Concavity Critical Points occur where f '( ) 0 or is undefined. Increasing Intervals occur where f '( ) 0. Decreasing Intervals occur where f '( ) 0. Minimums occur where Maimums occur where f ' changes from neg to pos. f ' changes from pos to neg. Points of inflection occur where f "( ) 0 or is undefined AND has a sign change. Intervals that are concave up occur where f "( ) 0 (or when f ' is inc). Intervals that are concave down occur where f "( ) 0 (or when f ' is dec). 41. Find all critical points, indicate increasing/decreasing intervals and all min s and ma s for f ( ) 5. 1 1 4. Find the location for all local etrema for f ( ) 1. 4. Find all points of inflection and indicate the concavity over each interval for 1 1. 0 6 5 f ( ) nd Derivative Test: For any critical point c, If f "( c) 0 the function is concave up, indicating a min. If f "( c) 0 the function is concave down, indicating a ma. 44. If f ''() and f '() 0, what can you conclude about f ()? 1. Curve Sketching Slope become y values (focus on slopes) f f f( a) is the height of f at a. '( a) is the slope of the f at a. "( a) is the concavity of f at a. y values become slopes (focus on y-values) Pay attention to what the given graph represents!!!!!!!! AP Calculus 07-08 AP Review Packet: Page 6 of 15
45. Given f : a) graph f ' b) Graph f " f f 46. Given f ' : a) graph f b) Graph f " f f 47. Given f ": a. Graph f ' f 1. Slope and Concavity From Graph Answer the following based on the given graph of f. 48. At what point is f '( ) 0 and f "( ) 0? 49. At what point is f( ) 0 and f '( ) 0? A B C D 50. If g( ) f '( ), at what point is g ( ) 0 and g'( ) 0? Answer the following based on the given graph of f '. 51. What are the critical numbers for f? 5. Where does f have relative ma s? 5. What are the locations for points of inflection on f? AP Calculus 07-08 AP Review Packet: Page 7 of 15
Answer the following based on the given graph of f ". 54. What are the locations for points of inflection on f? 55. What are the critical numbers for f? Use the given graph of g ' to answer the following on the interval [-, ]: 56. Find the values for which g has local etrema. 57. State the intervals on which g is increasing. 58. Find the values for which g has a point of inflection. 59. State the location and classify the etrema on g ". 60. Sketch a graph of g. 14. Curve Sketching 61. Let f be the function that is even and continuous on the closed interval [-6, 6]. The function f and its derivatives have the properties indicated on the table below. 0 0 < < < < 4 4 4 < < 6 f( ) - Negative 0 Positive Positive f '( ) Undefined Positive 1 Positive Undefined Negative f "( ) Undefined Negative 0 Positive Undefined Positive a. Identify all -coordinates at which f attains local etrema and classify etrema as minimum or maimum. Justify your answer. b. Identify all -coordinates of points of inflection. Justify your answer. c. Sketch a graph of f. AP Calculus 07-08 AP Review Packet: Page 8 of 15
15. Implicit Differentiation 6. If dy y 10, then when, d? dy 6. Find y f ( ) by solving the differential equation y with the initial condition (0) d f. dy 64. Consider the differential equation 1 d. a. Sketch a slope field for the given differential equation at the twelve indicated points. b. Sketch the solution curve through the point (1, 0). dy c. Find y f ( ) by solving the differential equation 1 d with the initial condition f (1) 0. 16. Indefinite Integrals means find the antiderivative. Remember your + C 5 65. d 66. d 67. 5d 69. 9 sec 68. cos d d 70. csc cot cot e d 17. Riemann Sums Find a cumulative area of rectangles. 71. Use a right-hand Riemann sum with four equal subdivisions to estimate the integral ( ) 0 d. 7. Use a left-hand Riemann sum with four equal subdivisions to estimate the integral ( ) 0 d. 7. Use a mid-point Riemann sum with four equal subdivisions to estimate the integral d. 0 4 74. Use a trapezoidal Riemann sum with three subdivisions of length, and 1 respectively to estimate the 6 integral ( 6 ) d. 0 AP Calculus 07-08 AP Review Packet: Page 9 of 15
18. The Fundamental Theorem of Calculus Definite Integrals represent the area under the graph up to the -ais. b f ( ) F ( ) ( ) a Positive Area Negative Area Negative Area Positive Area 75. Evaluate ( 4 ) d 76. Evaluate 1 ( 1 ) d 77. Find the value of and k that divides the area between the -ais, 4, and y into two regions of equal area. 19. Properties of Definite Integrals 78. If 5 g( ) d 10, then find: 5 a. g( ) 4 d b. 5 7 g( ) d c. 0 g( ) d 0. The Mean Value Theorem for Integrals Area = width length Avg. Height a b 79. Find the average value of f 8 ( ) on [1, ]. 80. Find the average velocity of a projectile on the interval [1, 4] for v( t) t 8t, where t is measured in seconds and vt () in cm per second. b. At what time on [1, 4] does the projectile reach its average velocity on [1, 4]? (A calculator may be used for question 4b.) AP Calculus 07-08 AP Review Packet: Page 10 of 15
1. The Second Fundamental Theorem of Calculus d f ( t) dt f ( )* d d a Taking the derivative of an integral cancel each other out. 81. Evaluate d 8. Evaluate the derivative of F( ) ( ) t dt d t 1 dt 0 8. Find f '() if f ( ) t dt.. Functions Defined as Integrals 84. Fill in the table and sketch a graph of g ( ) if g ( ) ( ) 0 f d -4-0 4 85. The graph of function g shown to the right consists of three line segments. Let h be defined as h( ) g( t) dt. 0 a. Find h (6), h'(6) and h "(6). b. For what values of in the open interval (0, 1) is h increasing? c. For what values of in the open interval (0, 1) is h concave down? d. Sketch a graph of h. AP Calculus 07-08 AP Review Packet: Page 11 of 15
. Area Between Two Curves BE CAREFUL WHEN DEALING WITH AREA! When finding area between two curves by A Top Bottom d the answer is always positive. For Accumulated Rates and Definite Integrals - If the AP Eam asks for the area between a function and the -ais, determine the areas of regions above and below the ais separately and add them together. High y Left Right Low y If one function is above the other function on the entire interval, then If one function is to the right of the other function on the entire interval, then 86. Find a value of, that divides the area bounded by the -ais and the function sectors of equal area. y 6 into two 5 87. Set up an integral epression and use it to find the area between the -ais and y cos on, 4 6. 4. Finding Volumes chord Chord = side = diameter SLABS Look for the word cross sections in the problem. 1 chord Isos rt. Tri (sitting on hyp): 1 Isos rt. Tri (sitting on side): ( chord ) 1 chord Semicircle: Square: ( chord ) Rectangle: ( chord )( given height ) Eq. Tri: 4 ( chord ) AP Calculus 07-08 AP Review Packet: Page 1 of 15
DISCS NO space between shape and the ais of rotation. r r = top bottom r r = right left WASHERS ANY space between shape and the ais of rotation. R r r R 88. Find the volume of the solid whose base is bounded by are semicircles. f ( ), 0 and y 4 and whose cross sections 89. Find the volume of the solid formed by f ( ),, and y 0 about: a. the y-ais b. the -ais c. the line y 4 d. the line 90. Set up an integral epression and use it to find the volume of a three dimensional solid whose base is bounded by the functions y e, y and 0 if every cross section perpendicular to the -ais is an equilateral triangle. 91. Set up an integral epression and use it to find the volume if the region bounded by the functions y and 0 is rotated about the y-ais. y, e 9. Set up an integral epression and use it to find the volume if the region bounded by the functions y and 0 is rotated about the -ais. y e, AP Calculus 07-08 AP Review Packet: Page 1 of 15
5. Position, Velocity, Acceleration The graph above represents the velocity of a particle traveling on the -ais, where t is time in seconds and vt () is velocity in ft/sec. The position at t 0 is 5 units to the right of the origin. 9. What is the acceleration at t 1? Justify your answer & include appropriate units. 94. What is the displacement of the particle from t 0 to t 8? Justify your answer & include appropriate units. 95. What is the particle s position at t 8? Justify your answer & include appropriate units. 96. What is the total distance traveled from t 0 to t 8? Justify your answer & include appropriate units. 97. When does the particle return to its original position? Justify your answer & include appropriate units. 98. On what interval is the particle decelerating? Justify your answer & include appropriate units. 6. Related Rates 99. A container has the shape of an open right cone, as shown in the figure. The height of the container is 10 cm and the diameter of the opening is 10 cm. Water in the container is evaporating so that its depth h is changing at the constant rate of cm / hr 10. 10 cm a. Find the volume V of water in the container when h 5 cm. Indicate units of measure. r 10 cm b. Find the rate of change of the volume of water in the container, with respect to time, when h 5 cm. Indicate units of measure. c. Show that the rate of change of the volume of water in the container due to evaporation is directly proportional to the eposed surface area of the water. What is the constant of proportionality? h AP Calculus 07-08 AP Review Packet: Page 14 of 15
7. Adding and Removing Free Response Problems 100. A tank contains 15 gallons of heating oil at time 0 t. During the time interval 0 t 1 hours, heating oil is pumped into the tank at a rate 10 Ht ( ) gallons per hour. 1 ln t 1 During the same time interval, heating oil is removed from the tank at the rate t Rt ( ) 1sin gallons per hour. 47 a. How many gallons of heating oil are pumped into the tank during the time interval 0 t 1 hours? b. Is the level of heating oil in the tank rising or falling at time t 6 hours? Give a reason for your answer. c. How many gallons of heating oil are in the tank at time t 1 hours? d. At what time t, for 0 1 t, is the volume of heating oil in the tank the least? Show the analysis that leads to your conclusion. AP Calculus 07-08 AP Review Packet: Page 15 of 15