Mr. Castle

Transcription

1 AP CALCULUS SUMMER ASSIGNMENT Dear Prospective AP Calculus Student, Welcome to AP Calculus! This is a rigorous yet rewarding math course that will challenge you and develop your critical thinking and problem solving skills. Calculus is the beginning of the fusion of all of your previous math courses: Algebra I, Geometry, Algebra II, and Pre-Calculus. Problem solving in AP Calculus will be embedded with many skills, methods, and problem solving techniques which should have been mastered in the prerequisite courses listed above. Additionally, you should epect to allot a certain amount of time each night for homework and studying and for the first time in your mathematics career, you might need to attend tutoring with relative frequency. This class is designed to prepare you for the AP Calculus AB eam, which is the equivalent to your first semester of university level Calculus. Calculus will enhance your critical thinking skills and often require you to think outside of your comfort zone. For some of you, this is a natural skill, while for others, it is uncomfortable. There are many problems in this packet that are unfamiliar and it will be tempting to skip them. However, many of these questions are well within your capability despite their unfamiliarity so I encourage you to just go for it see what happens. Once you start working, you may find the concept is much less complicated than you anticipated. Please feel free to form study groups, collaborate and work together. An important distinction should be made between collaboration/group work and cheating. Working together is acceptable and absolutely encouraged, but it does mean that each member of the group is responsible for working out each problem, though it may be with the assistance of a classmate. An eample of cheating in a group work task would be taking answers from another student s work without having investigated and worked the problem personally. Cheating is taken very seriously and is completely unacceptable under all circumstances. This packet should be completed without the use of your calculator, unless indicated. You will learn to use your calculator as a resource this year, but you will also be required to be algebraically proficient without it. Please report all answers to four decimal places, if possible. In order to ensure that you have completed this packet and have mastered the included topics, the material will be tested within the first weeks of school. This will provide you with time to ask questions and be appropriately prepared if you do need etra help. This packet will be collected for a grade on the first day of class, and it is epected that appropriate supporting work is shown for all problems. Enjoy your summer! Do a little bit of studying, as a little bit of review will save you much time in the coming school year. If you need additional assistance, feel free to me at the address listed below. I am so looking forward to meeting you (or if you ve been in my PreCal class seeing you again) in the Fall! I will make myself available during the summer by appointment at Panera Bread in Mesquite. My cell phone number is You may or tet me with any questions you have or to set up an appointment for a tutoring session. Mr. Castle rcastle@mesquiteisd.org

2 Name: AP Calculus/Period Date Summer Calculus Assignment 05 Directions: Work all problems on a SEPARATE SHEET OF NOTEBOOK PAPER. I. Simplify the following epressions ( h) h. 5( h) 5 h 5. ( ) ( ) 6( ) ( ) 6. ( ) ( ) u u u 8. 6 a a 9. b a a b 0.. Evaluate f (9) for the function given by numerator and simplifying. f( ) 5 9 by multiplying by the conjugate of the. Evaluate h () for the function given by of the function. h ( ) by forming a common denominator in the numerator II. Solve the following equations ( )( 7). ( )( ) ( 5)( ) ln ( ) ln ( 9 0) Solve for y : y y y 5y y

3 III. Using the given information, write an equation of the line in point-slope form. 9. slope = 5, passing through the point (, ). 0. Passing through the points (,) and (, ). Passing through the origin and perpendicular to y 7. Passing through the point (,) and parallel to the line y. IV. Use the table to answer the following questions.. g( f ()). f( g ()) 5. f( f ()) f (5) g () f( ) g ( ) g ( f ()) g( f ()) 0. Given that h( ) f ( ) g( ) 6, find h (). V. Use the graph of f( ) shown to the right to answer the following questions. ( f( ) is tangent to the -ais at, has a zero at, and a relative minimum at ). Write a transformation of f( ) that will result in a relative minimum at the point (5, 5) (, 0). What is the minimum value of the function over the interval [,]?. Where does the minimum value occur? (-, 0). Which of the following equations would have a relative minimum value of 8? (A) g( ) f ( ) (B) h( ) f ( ) 8 (C) m( ) f ( ) 8 (D) k( ) f ( ) (-,-8) (,-)

4 5. (Calc) If f ( ), find the solution to the equation and then solving the equation using your calculator. f ( ) f ( ) by finding f ( ) algebraically VI. Find the average rate of change of the given function over the given interval. Hint: Average rate of change f ( b) f ( a) of a function f( ) over an interval [ ab, ] is given by. b a 6. f ( ) on [0,8] 7. on [,] f ( ) 8. f ( ) sin on, 9. f ( ) cos on, 6 6 VII. Given the following functions, state any points of discontinuity and write the equation for all vertical and horizontal asymptotes. Classify discontinuities as removable or non-removable. 0. f( ). f( ) 5 6. Given the function may be true) ( a )( b) h ( ), which of the following statements are true? (Multiple statements ( b)( c) (A) The function h ( ) has a vertical asymptote at c. (B) The function h ( ) has a vertical asymptote at b (C) The function h ( ) has a horizontal asymptote at y a.. Given f ( ), find the value of that gives a slope of zero for the secant line passing through the two points (, f ( )) and (, f ( )). You can do this!! Use. Let a f( ) b( c)( d) y y m. Find the values of constants a, b, c, and d such that f( ) has -intercepts at (,0) and (,0), vertical asymptotes at 9 and, and a horizontal asymptote at VIX. Find the inverse of the following functions. y. 5. y e 6. y 7. y tan 8. y 9. y 50. y ln

5 X. Solve the following problems. 5. The rate at which water is filling and draining from a tank starting from time t 0 is represented by the given graph. A positive rate means that water is entering the tank, while a negative rate means that the water is leaving the tank. State the interval(s) on which each of the following is true. (a) The volume of the water is constant. (b) The volume of the water is decreasing. (c) The volume of the water is increasing the fastest. 5. A swimming pool can hold a maimum of 60 gallons of water. The pool develops a leak and is losing water at a constant rate. After hours, the pool has 5 gallons of water in it. (a) Write a function for gt (), the total number of gallons of water in the pool at time t, measured in hours. (b) Find g (0) and eplain the meaning of the answer in correct units. (c) If the leak is fied after 0 hours and the owner immediately begins to refill the pool at a rate of gallons of water per hour, write a piecewise function for pt (), the total number of gallons of water that is in the pool for any time t, measured in hours. 5. Consider the curve. The slope of the curve at any point on the graph ( y, ) can be found y 7 y y using m. Find the point on the graph at which the slope of the graph is equal to zero. You can do this. 8y Think!!! 5. Consider the curve using. The slope of the curve at any point on the graph ( y, ) can be found y y m. Find the point on the graph at which the slope of the graph is undefined. You can do this!! 6y XI. Trigonometry Review. You need to know the values of the unit circle by memory. There are several techniques you should have learned in PreCal to help you figure out what the value is. Look over your notes! We will have quizzes over the unit circle beginning the nd class meeting. 55. sin 56. cos sin sin 59. cos tan 6. tan 6 6. cos0 6. tan 6. 5 sin

6 XII. Particle Motion. Note: The rate of change (derivative) of position is known as velocity and the rate of change (derivative) of velocity is known as acceleration. 65. A particle moving in a straight line has a velocity given by v( t) sin( t). What is the average acceleration of the particle over the interval t? The function v( t) sin( t) represents the velocity in feet per second at any time t 0 of a particle s movement along a horizontal line. Find the first two time values at which the particle is at rest. Note: a particle is at rest when vt ( ) (Calc) A particle moves along a horizontal ais so that its velocity in feet per second at any time t 0 is t given by vt ( ) cos. When the velocity is negative, the particle is moving to the left. Find the intervals for 0 t 0 when is the particle moving to the left. XIII. Use the figure to find the limit. 68. lim f 70. lim f 7. lim f 69. lim f 7. lim f 0 7. lim f 5 XIV. Find the first derivative of the function. Simplify. 67. f f 69. f f 7. f f 7 ( ) 7. f( ) 7. ( )( ) 75. f ( ) y 76. f ( ) tan( ) 77. f ( ) cos ( ) 78. f ( ) e 79. y 80. y 6 ( 9 ) 8. y ( ) ( ) 8. f ( ) sin( )cos 8. g() t t t e 8. y cos(cos(cos ))) 85. f ( ) sin 86. y sin 87. y 88. f ( ) tan(cos ) 89. y ln( ) 90. g 5 ( ) 7 9. f ( ) ln(sin ) 9. y sin 9. y sin ( )

7 9. Find the equation of the line tangent to the graph of f ( ) 7 at. 95. Find the equation of the line tangent to the graph of f ( ) sin( ) at XV. On problems 8-8, (a) find the intervals where the function is increasing and decreasing. (b) find the local maimums and minimums. (c) find the intervals where the function is concave up and concave down. (d) find the points of inflection f ( ) 97. f ( ) 98. f ( ) sin, 0 XVI. The graph of f '( ) is given. Use the graph to answer the following. Justify your answers using a sentence. 99. What are the critical points of f? Justify your answer. 00. Determine whether f has a local maimum, local minimum, or neither at each critical point. Justify your answer. 0. At what value(s) of does f( ) have a point of inflection? Give a reason for your answer. Graph of f