Overlake School Summer Math Packet AP Calculus AB

Transcription

1 Overlake School Summer Math Packet AP Calculus AB Name: Instructions 1. This is the packet you should be doing if you re entering AP Calculus AB in the Fall. 2. You may (and should) use your notes, textbook, the internet, friends, family, and other resources to help you complete this packet. The packet is meant as a review, not as a test. 3. You should work on this packet in a place where you can concentrate without distractions, and work at least an hour at a time as opposed to in small intervals. 4. Once you are finished, go back and highlight any problems you had a lot of difficulty completing or have questions about. Then answer the Post-Packet Questions below. Expectations 1. This packet will be collected on the first day of class. It will count similar to a homework assignment towards your first semester grade. 2. Expect a quiz on the material covered by this packet during the first week of the semester. If there are any topics that you feel rusty on, make sure to review them before the semester starts or with your teacher during the first week of class. 3. This packet should take you no more than 5 hours. 4. If, due to extenuating circumstances, you are unable to do this packet, please contact Wing L. Mui, math department chair, at wmui@overlake.org before the beginning of the school year. Post-Packet Questions Answer these questions after you re done with the packet so your teacher can get a better sense of your progress. 1. What resources did you use to do this packet? Check all that apply. notes from last year textbook from last year the internet friends and family a tutor other: 2. How long did it take you to finish the packet? (t = the time it took for you to finish the packet in hours) t < 1 1 < t < 3 3 < t < 5 t > 5

2 Do as much as possible without a calculator. A. Geometry. You may need to look up the necessary area and volume formulas. It is in your best interest to know the geometric area and/or volume formulas for circles, spheres, cones, prisms, and pyramids. 1. Determine the volume and surface area of a sphere with radius 3 inches. 2. a) Find the volume of a cone with a base radius of 4 feet and height of 8 feet. b) In terms of the base radius r, find the volume of a cone whose height is twice its base radius. c) In terms of the height h, find the volume of a cone whose height is three times its base radius. 3. Find the area of the trapezoid shown here Determine formula for the area of a circle in terms of the side length, s, of the inscribed square. 5. Find the volume of a washer with outer radius of 18 ft., inner radius of 15 ft., and height of 3 ft.

3 B. Graphing. Sketch graphs of the following. 6. Sketch the graphs of: f(x) = 2sinx g(x) = cos2x + 1 h(x) = tan x 3 7. y = ln(x + 5) 8. y = x2 +8x+15 x+3 9. y = x x y = 1 x y = 1 (x+4) y = x y = 2 x y = x 2 + 3x Draw the graph then state the domain and range for the given function. y = 4 x 2

4 16. Draw the graph for each piecewise function. a. b. C. Trigonometry 17. Evaluate. a. sin ( /3) b. cos( /6) c. tan ( /2) d. tan ( /4) e. cos (0) f. sin (2 /3) g. cos (4 /3) h. tan (3 ) i. sec (0) j. csc( ) k. cot (2 ) l. sin (4 /3) m. cos(11 /6) n. tan (-3 /4) o. cos(- ) p. arcsin(0.5) q. tan -1 ( 3 ) r. cos -1 (- 1 2 ) Solve where, 0 < θ < sinθ = cosθ = tanθ = cosθ = 3 2

5 Solve where, 0 < θ < cos 2 θ 2cosθ = sin 2 (3x) 3 = 0 D. Limits 24. Use the graph to determine each limit or function value. 25. Sketch a single function with domain that possesses all of the characteristics listed. a. lim x 3 f(x) = 5 b. lim x 3 + f(x) = 2 c. lim x 2 f(x) = d. n. e. but f(2) = 0 d. lim x 4 f(x) = 6 but f(4) = 1

6 26. Determine the value of k so that lim x 1 f(x) exists. f(x) = { 2kx2 + 3x x < 1 4x 9 x 1 E. Short Answer. (you may need calculator assistance here; ALWAYS round to three decimal places) 27. When an airplane leaves the runway, its angle of climb is 17 and its speed is 250 feet per second. Find the plane s altitude after 2 minutes. 28. A buoy oscillates in simple harmonic motion as waves go past. At a given time it is noted that the buoy moves a total of 2.5 feet from its low point to its high point. It returns to its high point every 5 seconds. Write an equation that describes the motion of the buoy if at its high point t = The population of City A in 1990 is 14 thousand and increases exponentially at a rate of 6% annually. The population of City B in 1990 is 26 thousand and decays continuously with a rate of 3%. a. Write the model for each population. b. Determine the year in the populations are equal. c. Determine the half life of the population of City B.

7 30. Determine the tripling time of a population that grows continuously with a rate of 7.5%. 31. Let f be a linear function where f(2) = 5 and f( 3) = 1. Determine f x. 32. Use the table to calculate the average rate of change from t = 1 to t = 4. t xt Find the equation of the line that contains the points (-1, 4) and (6, -2). 34. Find the equation of the line that is perpendicular to the line 3x + 2y = 4 and contains the point (3, 1). 35. The number of elk after t years in a state park is modeled by the function P(t) = e 0.03t. a) What was the initial population? b) When will the number of elk be 750? c) What is the maximum number of elk possible in the park?

8 F. Sigma Notation 36. Write out the terms indicated by the following summations, and then evaluate the sums. a. 5 ( 1 2 i) i i=0 = b. 10 (4)(0.1) k = i=0 37. Write the following sums in sigma ( ) notation. a b [( 1 10 )2 + ( 2 10 )2 + ( 3 10 )2 + + ( )2 ]

9 G. Solve each equation. Give exact answers. 38. a. 9 2x x 2 = 4 + 2x b. 2 5 y y = 1 c. xe 2x + 3ex 2 e 2x = 0 d. log 4 ( x 6 ) = 5 2 e. log 2x (81) = 4 3 f. 5(3.2) x 1 = 145 g. 2xe x 3x = 0 h. 3x 4 x 5 = x + 1 i. 1 x 2 1 x = 6 j. x 10 x + 9 = 0

10 H. Solve the following inequalities. Support your answer with a graph or sign chart. Express your answers in in interval notation. 39. x 2 16 > x 2 + 6x 16 > x 2 3x x 2 + 5x x 3 + 4x 2 x sin 2 sin x I. Simplify using only positive exponents x ( 3 ) (4 2 9x) 1 2 ( 9) ( 2 ) [ 2 x 2 (2 x) 2] 48. (16x 2 y) x1 2 sin x 50. 4x (x 4) (2x+5) ( x 2 x 1 y 1 y 2) 1

11 J. Functions 53. Given f(x) = 5x 1 and g(x) = 2x 2, a) Evaluate g ( f ( 2)) b) Evaluate f ( f (3)) c) Solve f ( x ) = 0 d) Is g x even, odd or neither? e) What is the inverse of f x? 54. For the function y = - ½ (x + 3) 2 + 1, explain the effects of the negative, the one-half, the three, and the one. 55. If f(x) = x 2, g(x) = 2x 1, and h(x) = 2 x, evaluate the following. (a) f(g(2)) (b) g(f(2)) (c) h 1 (x) (d) f(h( 1)) (e) g (f (h ( 1 2 ))) 56. Let g(x) = x 5 + x 1 a) If (2, 33) is a point on the graph of g, what is a point on the graph of g -1? b) Without graphing, can you determine if gxcrosses ( ) the x-axis? Explain.

12 57. Let f(x) = x Sketch graphs of f, f 1, 1, and f using different colors for each graph. f Also, determine an equation for f -1 (x). 58. If f(x) = 3x 2 + 2x and g(x) = 3 x (a) f(x+h) f(x) h, then find (b) f(x + 2) f(2) (c) g(x+h) g(x) h

13 K. Exponents and Logarithms 59. Simplify. (a) e ln(5x +2) (b) log 7 (7 x 49 y ) (c) log (d) 3 2 log 9 5 (e) log (f) log Solve. (a) log x 3 64 = 2 3 (b) log(7250) = log(7.25) + x (c) 5e 9x 7 = 17 (d) log 2 (2x + 5) = 3

14 L. Limits 61. Determine each limit: If a limit is d.n.e. because it is, then state whether the limit is or. ex 5x ( ) x x a. lim c. lim 1001x4 +2x 9 x (1.001) x 11x b. lim 2 +9 x 3+2x d. lim 2ex 3e x x 5e x +7e x 62. Determine each limit. If a limit is d.n.e. because it is, then state whether the limit is or. 7 2x+8x a. lim 3 x x 2 +6 c. lim x ln(x 2 +1) 5 x 3 +9 b. lim 6 x +5 x e x 2 d. lim 2e x 3e x x 5e x +7e x 63. Determine each limit. x 3 a. lim x 3 x b. lim x 0 x 2 c. lim x 2 1 x 2