Summer Review Packet AP Calculus

Transcription

1 Summer Review Packet AP Calculus ************************************************************************ Directions for this packet: On a separate sheet of paper, show your work for each problem in this packet, and circle your answer. This is not an option! ************************************************************************ While you will possibly be able to solve or simplify many of these on your own, others may require assistance from outside sources. You are strongly encouraged to:. search the internet;. work with others (form a study group!); or. find other resources to solve these problems (suggestions include possibly checking out a Precalculus book from the high school, looking back over any notes you have kept from the past two or three years, etc.) All of you have a calculator that either the school has issued to you for this school year, or that you personally own. If you have a school-issued calculator, please keep this for the summer, and bring back at the beginning of the school year in August. If you own your own calculator, you are free to use it as necessary to complete your packet. You are epected to finish as much of this as possible this summer. We will be spending approimately the first week of school going over any questions you may have. This work will receive multiple grades for correct completion, as well as one quiz grade. GOOD LUCK! Mrs. O Rear

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3 Are You Ready for Calculus? I) Simplify the following fractions:. + = + y. =. = + y h = II) Factor each epression completely: ( ) ( ) ( )( ) 4. ( ) ( ) ( ) ( ) III) Solve the following equations for : = = 5

4 7. = = = 0 0. ( )( + 0) = 0. + = 4. y = + z. + 4 = 4. log 5 = 5. = 6. 9 log 8 = 7. = 8. log = = 0 0. = 4. = 8. + = 0. = 4. ( ) + 9 = = 0 6. ( + )( + ) = = 0 8. = y z = z + w 40. = ( ) ( 5) + ( ) ( 5) = 0

5 IV) Show that each equation is true by simplifying the following: 4. + y = y y 4. + h = h + h + V) Which of the following epressions equals log 4? 44. log 45. log8 log 46. log8 log 47. log 4 + log log log 49. ( log ) VI) Write an equation of a line based on the given information: 50. Find the equation of the line that has a slope of 5 and passes through the point (, -4). 5. Find the equation of the line that passes through the points (4, ) and (, -). 5. Find the equation of the line that passes through the point (-, ) and is parallel to the line 4 + y = 5. Find the equation of the line that has a slope of zero and passes through the point (-5, ). 54. Find the equation of the line that passes through the origin and is perpendicular to the line + 4y = Find the equation of the line that has an undefined slope and passes through the point (4, -5). 56. Find the equation of the line that has an -intercept of 5 and a y-intercept of. 57. Find the equation of the vertical line that passes through the point (, ). 58. Find the equation of the horizontal line that passes through the point (, -5). VII) Find the intercepts:

6 59. Find the and y intercepts of the line that passes through the point (, -5) and is perpendicular to the line y =. 60. Find the and y intercepts of y = Find the and y intercepts of + = Find the and y intercepts of + y = 9 VIII) Write the equation of the following graphs: IX) Given the slope, sketch the following lines: 69. Sketch a line with a slope of. 70. Sketch a line with a slope of -/.

7 X) Sketch the following graphs: 7. y = y = 74. y = y = 75. = 76. y = y = ln 78. y = 79. y = 80. y = 8. y = + 8. y = + 8. = + y 84. y = ( + ) 85. = + y 86. y ( ) = y = 88. y = ( + ) 89. y = log 90. y = e 9. y = 9. + y = y = y = y = 0 for for for < < y = for < for

8 XI) Rewrite the following functions without absolute values: 97. y = 98. y = 99. y = y = XII) Find the domain and range of each function: 0. y = 0. y = 0. y = y = y = y = y = log 08. y = 09. y = y = for for < > XIII) Find the inverse of each function:. f ( ) = +. f ( ) = f =. ( ) XIV) Find the compositions of the function if: f ( ) = +, g( ) =, and h( ) = + 4. f() = 5. f(h()) = 6. f(g()) = 7. g(g()) =

9 8. g(f()) = 9. h(g()) = h. f ( + h) = 0. ( ) =. h(p) =. g ( t + h) g( t) h XV) Solve the simultaneous equations: 4. + y = 8 + y = 5 5. y = y = 9 XVII) What do the following mean if: 6. a graph is in the first quadrant 7. f() = 5 8. an epression is a function 9. a zero of a function is 4 0. y is directly proportional to (give an eample). the coefficient of the third term is 5 (give an eample). a function only has one root. a function is a polynomial 4. two triangles are similar XVIII) What are the following formulas: 5. quadratic formula 6. Pythagorean theorem 7. the hypotenuse of a isosceles right triangle with a leg of length 8. the hypotenuse of a right triangle with shortest leg having a length of

10 9. the volume of a sphere 40. the volume of a cylinder 4. the volume of a cone 4. the volume of a bo with a square base 4. the surface area of a sphere 44. the surface area of a cylinder with no top 45. the area of a triangle 46. the area of a trapezoid 47. the cross section through the center of a sphere 48. the volume of a prism that has an equilateral triangle with side length of and height of length y XIX) Solve by using similar triangles: 49. Water is dripping out of a conical figure that has a diameter of 8 inches and a height of inches. When the depth of the water is 8 inches, what is the radius of the water? XX) Describe the symmetry of the following functions (y-ais, -ais, or origin): 50. y = + 5. y = + 5. y = 5. y = y = sin 55. y = cos XXI) Find the equations for the horizontal and vertical asymptotes of each function: y = 6. y = 6. y = y = 9 +

11 XXII) Write the following without sigma notation: n n= XXIII) Eponent rules: Which of the following are true? 7 n= = 67. = y = + y = ( ) 5 5 y = y 8 7. ( ) 5 = 5 5 w 7. = w = t+ s t 7. ( ) s = 75. ( ) 4 = = 77. = = 79, = y + = + y 8. = 8. ln e = 8. ln e = = 5 ln ln4 e 85. ln = ( ln ) XXIV) Using the graphing calculator: 86. Graph y =. + on this -y plane:

12 87. Find the roots of the equation above. 88. Find the points of intersection for the graphs y = + and y = Find the maimum value for the graph ( ) = + 4 f f ( ) = + 4. On what intervals for is f increasing? XXV) What are the following trigonometric identities: 9. sec = 9. csc = 9. tan = 94. cot = 95. cos = 96. sec = 97. cot + = 98. (half angle) cos = 99. (half angle) sin = 00. (double angle) sin = 0. (double angle) = cos 0. sin ( a + b) = 0. cos ( a + b) = XXVI) Find the period of the following functions: 04. y = 4 sin ( ) y == tan( π) 06. y = cos ( ) 07. y = sec ( 6) +

13 XXVII) Evaluate the following epressions: π 08. sin = 09. cos = 6 0. tan ( 0 ) =. cos ( 0) = 5π cos. csc = 6. ( 45 ) = π sec 5. cot = 4. ( 80 ) = π 6. sin = 7. tan = 5π π 8. sin = 9. cot = 6 π 0. = sin. cot ( ) =. = 4 sec. tan ( ) = π sec 5. sec = 6 csc sec sec 0 4. ( 80 ) = ( ) = 6. sin ( ) 7. ( ( )) = XXVIII) Sketch one period of the following trigonometric graphs: 8. y = sin 9. y = cos 0. y = tan. y = sec. y = csc. y = cot

14 4. y = sin( ) 5. y = cos ( 4) 6. y cos ( ) = 7. y = sin 8. y = cos( π) 9. y = cos ( π) XXIX) Solve the following trigonometric equations for the given domain: sin = on [ 0, π ] 40. cos 4. sin = on [ 0, π ] 4. sin = for all 4. cos 4 = for all 44. tan = 0 on π, π 45. sec π = for all cot on [ 0,π ] 46. = cot a = 0 for all 48. sin cos = 0 for all sin on [ 0, π ] sin = 0