Regina High School AP Calculus 018-19 Miss Moon Going into AP Calculus, there are certain skills that have been taught to you over the previous years that we assume you have. If you do not have these skills, you will find that you will consistently get problems incorrect next year, even though you understand the calculus concepts. It is frustrating for students when they are tripped up by the algebra and not the calculus. This summer packet is intended to help you brush up and possibly relearn these topics. We assume that you have the basic skills in algebra. Being able to solve equations, work with algebraic expressions, and basic factoring, for example, should be a part of you. If not, you would not be going on to AP Calculus. The topics covered in the packet are skills that are used continually in AP Calculus. The attached summer packet is for all students enrolled in AP Calculus AB in the fall. This packet will be collected on the first day back to school and will count as a test grade. Show all work on a separate paper and put your final answers on the answer sheets provided. NO work written in the question packet will be checked. Rather than give you a textbook to remind you of the techniques necessary to solve the problem, I have given you several websites that have full instructions on the techniques. If and when you are unsure of how to attempt these problems, examine these websites. Do not fake your way through these problems. As stated, students are notoriously weak in them, even students who have achieved well prior to AP Calculus, use the websites. Realize also that certain concepts are interrelated. Domain, for example, may require you to be expert at working with inequalities. Solving quadratics equations may involve techniques used in fractional equations. Again, this packet is due and will be graded on the first day back to school in the fall. You need to get off to a good start so spend some quality time on this packet this summer. Do not rely on the calculator. Use the calculator only on the problems where calculator use us indicated. Half of the AP exam next year is taken without the calculator. It is a mistake to decide to do this now. Let it go until midsummer. We want these techniques to be relatively fresh in your mind in the fall. Also, do not wait to do them at the very last minutes, the assignment is lengthy and will take some time. If you have any questions about any of these problems or techniques used in solving them, you may contact me at amoon@rhssaddles.net or through the RemindMe App. Material will also be available on google classroom page. Code:k7qlki) Have a great summer! 1
Below are listed topics in the review. You can certainly do Google searches for any of these topics. But we have given you several sites that will cover pretty much all of these topics. http://www.purplemath.com/modules/index.htm Beginning Algebra Topics Exponents Negative ad Fractional Exponents Intermediate Algebra Topics Domain Solving Inequalities: Absolute Value Solving Inequalities: Quadratics Special Factoring Formulas Function Transformation Factor Theorem Even and Odd Functions Solving Quadratic Equations and Quadratic Formula Advanced Algebra Topics Asymptotes Complex Fractions Composition of Functions Solving Rational fractional) Equations Trig Information Basic Right Angle Trig Trig Equations Limits You can also access Khan Academy using your student account through our classroom. Follow the directions on Google Classroom to access the section for our class.
Complex Fractions When simplifying complex fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common denominator of all the denominators in the complex fraction. 1. 3 7 x + 1 1 x 3 3. x 13x 3 x 4 x + x + 3 x 4. 13 x + 5 + 3 x 3 4. x x + 1 1 x x x + 1 + 1 x 5. 5 a a 5 + a Functions To evaluate a function for a given value, simply plug the value into the function for x. Let fx) = x + 1 and gx) = x 1 6. f3) = 8. ft + ) = 10. g[fm + 1)] = 7. g ) = 9. f[g 1)] = 11. fx + h) fx) h = Let fx) = cos x π ) π ) 1. f = 13. f = 3 Let fx) = x, gx) = x + 5 and hx) = x 1 14. h[f 3)] = 15. f[gx + 1)] = 16. g[hx 4 )] = Find difference quotient for the given functions of f, fx + h) fx). h 17. fx) = 8x + 5 18. fx) = 7 3x 3
Find the x and y intercepts for the functions below. 19. y = 3x + 5 1. y = x 9 x 0. y = x 6x + 8. y = x 3 4x Find the solutions for the systems below. 3. { x y = 10 3x + y = 1 4. { y = x 3 6x y = 3x 5. { x 4y 0x 64y 17 = 0 16x + 4y 30x + 64y + 1600 = 0 6. Write thin interval notation, set notation and graph in the space below. a) < x 4 b) 1 x < 7 c) x 8 Solve each equation. State your answer in BOTH interval notation and graphically. 7. 3x 1 0 8. 4 3x + 4 < 4 9. x x 3 > 5 Domain and Range Find the domain and range of each function. Wrtie your answer in INTERVAL notation. 30. fx) = x 5 31. fx) = x + 3 3. fx) = 3 sin x 33. fx) = x 1 Find the inverse for each function. 34. fx) = 3x 1 35. fx) = x 7 Prove f and g are inverses of each other. Show they equal the identity function) 36. fx) = x3 and gx) = 3 x 37. fx) = 9 x, x 0 and gx) = 9 x 4
Equations of a Line 38. Use slope-intercept form to find the equation of the line having a slope of 8 and a y-intercept of 5. 39. Determine the equation of a line passing through the point 8, 3) with an undefined slope. 40. Determine the equation of a line passing through the point 4, 3) with a slope of 0. 41. Use point-slope form to find the equation of the line passing through the point 0, 9) with a slope of 3. 4. Find the equation of a line passing through the point, 8) and parallel to the line y = 5 6 x 1. 43. Find the equation of a line perpendicular to the y-axis pasing through the point 4, 9). 44. Find the equation of a line passing hrough the points 3, 6) and 1, ). 45. Find the equation of a line with an x-intercept, 0) and a y-intercept 0, 3). 46. Convert to Degrees: *YOU CAN USE A CALCULATOR ON 46 and 47* a) 11π 6 47. Convert to Radians: b) 4π 5 c).63 radians a) 45 b) 19 c) 137 Angles in Standard Position 48. Sketch the angle in standard position. a) 11π 6 b) 30 c) 5π 3 d) 1.8radians Reference Triangles 49. Sketch the angle in standard position. Draw the reference triangle and label the sides, if possible. a) π 3 b) 5 c) π 4 d) 30 5
Unit Circle 50. Evaluate each sine or cosine below using your knowledge of the unit circle. a) sin 180 c) sin 90) e) sin 70 b) cos 70 d) cos π f) cos 180 Graphing Trig Functions Graph two complete periods of the function. 51. fx) = 5 sin x 5. fx) = sin x 53. fx) = cos x π ) 4 54. fx) = cos x π ) 4 Trigonometric Equations Solve each of the equations for 0 x π. Isolate the variable, sketch a reference triangle, find all the solutions within the given domain, 0 x < π. Remember to double the domain when solving for a double angle. Use trig identities, if needed, to rewrite the trig functions. 55. sin x = 1 3 59. sin x = 56. cos x = 3 60. cos x 1 cos x = 0 57. cos x = 1 61. 4 cos x 3 = 0 58. sin x = 1 6. sin x + cos x cos x = 0 Inverse Trigonometric Functions Find the value without a calculator. Remember inverse functions have domain restrictions) 63. tan arccos ) 3 65. sin arctan 1 ) 5 64. sec sin 1 1 13 ) 66. sin sin 1 7 ) 8 6
Circle and Ellipses Graph the circles and ellipses below: 67. x + y = 16 68. x + y = 5 69. x 1) + y + 5) = 4 70. x 1 + y 9 = 1 Limits Finding limits numerically. 71. Complete the table and use the results to estimate the limit. lim x 3 x 3 x x 3 x.9.99.999 3.001 3.01 3.1 7
7. Complete the table and use the results to estimate the limit. 4 x 3 lim x 5 x + 5 x -5.1-5.01-5.001-4.999-4.99-4.9 Finding limits graphically. Find each limit graphically. Use your calculator to assist in graphing. 73. lim x 0 cos x 74. lim x 5 x 5 75. lim x 1 fx) where fx) = { x + 3, x 1, x = 1 Evaluate Limits Analytically Solve by direct substitution whenever possible. If needed, rearrange the expression so that you can do direct substitution. 76. lim x 4x + 3) 80. lim x 3 3 x x 9 77. lim x 0 x + 4 x 1 ) 78. lim x 1 Hint: Factor & Simplify. x 1 81. lim x 1 x + x + x + 1 8. lim x π cos x 79. lim x 0 x + 1 1 x Hint: Rationalize Numerator 83. lim x 3 x + x 6 x + 3 84. lim x 0 x + h) x h 8
One-Sided Limits Find the limit if it exists. First, try to solve for the overall limit. If an overall limit exists, then the one-sided limit will be the same as the overall limit. If not, use the graph and/or a table of values to evaluate one-sided limits. 85. lim x 5 x 5 x 5 x 87. lim x 3 x 9 86. lim x 10 + x 10 x 10 88. lim x 5 3 ) x + 5 Vertical Asymptotes Determine the vertical asymptotes for the function. Set the denominator equal to zero to find the x-value for which the function is undefined. That will be the vertical asymptote. 89. fx) = 1 x 90. fx) = x x 4 91. fx) = + x x 1 x) Horizontal Asymptotes Determine the horizontal asymptote using the three cases: Case I - Degrees of the numerator is less then the degree of the denominator. The asymptote is y = 0. Case II - Degree of the numerator is the same as the degree of the denominator. The asymptote is the ratio of the leading coefficients. Case III - Degree of the numerator is greater then the degree of the denominator. There is no horizontal asymptote. The function increases without bound. If the degree of the numerator is exactly 1 more than the degree of the denominator, then there exist a slant asymptote, which is determined by long division. Determine all Horizontal Asymptotes. 9. fx) = x x + 1 x 3 + x 7 93. fx) = 5x3 x + 8 4x 3x 3 + 5 94. fx) = 4x5 x 7 Determine each limit as x goes to infinity RECALL: This is the same process you used to find Horizontal Asymptotes for a rational function. In a nutshell :) Find the highest power of x. How many of the type of x do you have in the numerator? How many of the type of x do you have in the denominator? The ratio is you limit! 95. lim x x 5 + 4x 3 5x + x ) x 5 ) 96. lim x 3 5x + 3x 97. lim x 7x + 6 x 3 3 + 14x + x ) 9
Limits to Infinity A rational function does not have a limit if it goes to ±, however, you can state the direction the limit is headed if both the left and right hand side fo in the same direction. Determine each limit if it exist. If the limit approaches or, please state which one the limit approaches. 98. lim x 1 1 x + 1 99. lim x 1 + x 1 x 100. lim x 0 sin x Problem Solving Calculus requires student to model a variety of situations using basic functions. Write a model for each of the following situations and solve accordingly. A calculator may be used on this section. 101. A rancher has 00 feet of fencing with which to enclose two adjacent rectangular corrals, as shown. What dimensions should be used to that the enclose area will have a maximum area. 10. A rectangle is inscribed inside a semi-circle with intercepts at 8, 0), 0, 8) and 8, 0). Find a function that models the area of the rectangle in terms of half of the base, x, of the rectangle, as show in the figure below. Determine what x-value will produce an area of 30 square units. 103. A camera is mounted at a point 300 ft from the base of a rocket launching pad. The rocket rises vertically when launched. Express the distance, x, traveled by the rocket as a function of the camera elevation angle, θ. Find the distance traveled by the rocket when the angle of elevation is 0 degrees. 10