AP Calculus AB Summer Assignment Name: Due Date: First day of school. The purpose of this assignment is to have you practice the mathematical skills necessary to be successful in Calculus AB. All of the skills covered in this packet are skills from Algebra 2 and Pre- Calculus. Each question was carefully selected and every one of the questions in this packet is extremely important, it is necessary that you master them. If you need to, you may use reference materials to assist you and refresh your memory (old notes, textbooks, online resources, etc.). While the graphing calculators will be used in class, there are no calculators allowed on this packet. You should be able to do everything without a calculator. AP Calculus AB is a fast paced course that is taught at the college level. There is a lot of material in the curriculum that must be covered before the AP exam in May. Therefore, we cannot spend a lot of class time re- teaching prerequisite skills. This is why you have this packet. Spend some time with it and make sure you are clear on everything covered in the packet so that you will be successful in Calculus. This assignment will be collected and graded. Be sure to show all appropriate work to support your answers. You may use loose paper to show any required work. In addition, there may be a quiz on this material during the first week of school. If you have major difficulties in doing this assignment or if you think it is too long perhaps you should consider not taking this course. Good Luck!
Name: Show all work no credit will be awarded for answers missing appropriate work. No calculators! Section I: Algebra Review 1. Solve xy + y + 1 = y for y. 2. Solve ln y = kt for y. 3. Solve ln (y 1) ln 2 = x + ln x for y 4. Factor: Simplify each expression. 5. (x 2 ) 3 1 x 3 6 6. x x x x 7 7. 5(x + h)2 5x 2 h 8. 1 x + 4 x 2 3 1 x
Simplify, using factoring of binomial expressions. Leave answers in factored form. Example: 9. (x 1) 3 (2x 3) (2x +12)(x 1) 2 10. (x 1)2 (3x 1) 2(x 1) (x 1) 4 Simplify by rationalizing the numerator. Example: 11. x + 9 3 x 12. x + h h x
Solve each equation or inequality for x over the set of real numbers. 13. 2x 4 + 3x 3 2x 2 = 0 14. 2x 7 x +1 = 2x x + 4 15. x 2 9 = x 1 16. 2x 3 =14 17. x 2 2x 8 < 0 [Your answer should be interval(s)] 18. 3x + 5 (x 1)(x 4 + 7) = 0 Solve each of the systems algebraically and graphically. 19. x + y = 8 20. y = x 2 3x 2x y = 7 y = 2x 6
Section II: Trigonometry Review Use your knowledge of the unit circle to evaluate each of the following. Leave your answers in radical form. DON T use your Calculator. 21. sin(30 ) 22. cos 2π 3 23. tan45 24. sin π 25. tanπ 26. csc 5π 6 6 27. cos( 90 ) 28. cos 3π 4 29. tan π 6 1 30. cos 1 2 31. sin 1 2 2 32. tan 1 (1) Solve each trigonometric equation for. 33. sin x = 3 2 34. tan2 x =1 35. cos x 2 = 2 2 36. 2sin2 x + sin x 1 = 0 Solve each exponential or logarithmic equation. 37. 5 x =125 38. 8 x +1 =16 x 39. 81 4 = x 3 2 40. 8 3 1 = x 41. log 2 32 = x 42. log x = 2 9 43. log 4 x = 3 44. log 3 (x + 7) = log 3 (2x 1)
Expand each of the following using the laws of logs. 45. log 3 5x 2 46. ln 5x y 2 Section III: Graphing Review I. Symmetry Even and Odd Functions Quick Review Example: Even Function Symmetric about the y axis for all x Example: Odd Function Symmetric about the origin (equivalent to a rotation of 180 degrees) for all x To determine algebraically if a function is even, odd, or neither, find to,, or neither. and determine if it is equal Example: Determine if is even or odd. Therefore, is an odd function.
Determine if the following functions are even, odd, or neither. 47. 48. 49. 50. II. Essential Graphs Sketch each graph. 51. 52. 53. 54. 55. 56. For each graph above, state the domain, range, x- intercept(s), y- intercept(s), and any asymptote(s).
Graphing Skill #1: You should be able to graph a function in a viewing window that shows the important features. You should be familiar with the built- in zoom options for setting the window such as zoom- decimal and zoom- standard. You should also be able to set the window conditions to values you choose. 57. Graph using the built in zoom- decimal and zoom- standard options. Draw each. 65. Find the appropriate viewing window to see the intercepts and the vertex defined by. Use the window editor to enter the x and y values. Window: Xmin = Xmax = Xscl = Ymin = Ymax = Yscl = 66. Find the appropriate viewing windows for the following functions: Xmin = Xmin = Xmin = Xmin = Xmax = Xmax = Xmax = Xmax = Xscl = Xscl = Xscl = Xscl = Ymin = Ymin = Ymin = Ymin = Ymax = Ymax = Ymax = Ymax = Yscl = Yscl = Yscl = Yscl =
Graphing Skill #2: You should be able to graph a function in a viewing window that shows the x- intercepts (also called roots and zeros). You should be able to accurately estimate the x- intercepts to 3 decimal places. Use the built- in root or zero command. [You should use your graphing calculator] 67. Find the x- intercepts of. Window [- 4.7, 4.7] x [- 3.1, 3.1] (Write intercepts as points) x- intercepts: 68. Find the x- intercepts of. x- intercepts: Graphing Skill #3: You should be able to graph two functions in a viewing window that shows the intersection points. Sometimes it is impossible to see all points of intersection in the same viewing window. You should be able to accurately estimate the coordinates of the intersection points to 3 decimal places. Use the built- in intersection command. 69. Find the coordinates of the intersection points for the functions:. Intersection points: 70. Find the coordinates of the intersection points of:. Intersection points:
Graphing Skill #4: You should be able to graph a function and estimate the local maximum or minimum values to 3 decimal places. Use the built- in max/min command. 71. Find the maximum and minimum values of the function. (Value means the y- value) Minimum value: Maximum value: 72. Find the maximum and minimum values of the function. 73. Find the x- intercepts, relative maximum, and relative minimum of 74. Find the coordinates of the intersection points for the functions and Section IV: Linear equations Write the equation for the line in both forms given a slope and a point. 75. m = 2/3 and P(3,5) 76. m = - 4/5 and P(1,2) Point- Slope: Point- Slope: Slope- Intercept: Slope- Intercept:
Write the equation for the line in both forms given two points. 77. P(2,2) and Q(4,2) 78. P(3,- 2) and Q(3,7) Point- Slope: Point- Slope: Slope- Intercept: Slope- Intercept: Section V: Limits Review Determine the limit by substitution. If substitution fails, factor or use the conjugate. 79. lim3x 4 x 2 80. lim x 1 (x 3 + 3x 2 2x 17) 81. lim x 2 x 2 4 x + 2 2 82. lim x 3 x 3 Determine the limit by any method. 83. lim x 2 x 2 3x + 2 x 2 4 1 84. lim 2 + x 1 2 x 0 x 85. lim x 0 sin(2x) x sin x 86. lim x 0 2x 2 x
x 87. Does lim x 0 x exist? Explain it algebraically and graphically. 88. Given the graph to the right, find the following limits. a. lim x 1 + f (x) = b. lim x 1 f (x) = c. lim x 1 f (x) = d. lim x 0 + f (x) = e. lim x 0 f (x) = f. lim x 0 f (x) = g. lim x 2 + f (x) = h. lim x 2 f (x) = 89. What is the limit as x approaches 2? a. b. c. lim f (x) = x 2 lim f (x) = x 2 lim f (x) = x 2
Section VI: Average Rate of Change Review 90. Find the average speed of a car that has traveled 350 miles in 7 hours. 91. Given the equation y = 4.9t 2, which represents the free fall equation in relationship to meter per second squared. Find the average from t = 0 to t = 3 and instantaneous speed at t = 3 seconds.