Welcome to AP Calculus!

AP Calculus has a rigorous curriculum and a pace that must be maintained (regardless of the weather or changes in the school schedule) in order to be ready for the AP test in May of 2012. Your commitment to a minimum of one hour of study per night is essential and by taking the class, you are agreeing to give your math class a high priority in terms of your busy schedule.

Attached is the summer assignment, it will be taken for a grade and used as a guide for review of foundational topics. We will move quickly through the review, so it is imperative that a majority of the work be done before school starts. There will be a test over this material at the end of the 1st full week of school. Please show all your work in a neat, numbered and orderly fashion so it can be easily checked. Failure to prepare the summer assignment will impact your grade for the 1st 9-week term, as the pre-requisite topics will be roughly 15% of the first 9-weeks grade. Check the school website this summer for tutoring opportunities in June for any help you may need with the packet.

Please take time this summer to become familiar with the functions on the attached list. (You have encountered these throughout Algebra I, II, Trigonometry and Pre-Calculus.) You need to be able to graph each by hand and state the domain and range of each of these as well as other key characteristics. Also, the trigonometry and logarithm worksheets are very important as each covers material you should already know and will be required to use at sometime during the Calculus course.

A graphing calculator is essential for this course. A Barron’s AP Calculus review book (available at local bookstores) is extremely helpful, as well. You will also be required to read “A Tour of the Calculus” by David Berlinski before the 2011-2012 school year begins. You will be given an assignment related to the book on the first day of school. Remember that AP Calculus is a demanding course. Please be prepared to work hard.

Have a wonderful summer and be ready to start work in August!

Brooke Taylor

AP Calculus Summer Assignment 2011 Name_______________________

SHORT ANSWER. Write the work or phrase that best completes each statement or answers the question.

Graph the equation by plotting points.

1) y = –x – 2 2) y = – x2 + 5

Solve the problem.3) If (a, 3) is a point on the graph of y = 2x – 5, what is a?

4) If (3, b) is a point on the graph of 3x – 2y = 17, what is b?

5) Sketch the graph of the equation by hand: y = | –x – 3|

List the intercepts for the equation.6) y = x2 + 1

7) Find the intercepts of the graph of y = (x – 2)2 – 1

8) Find the x and y-intercepts of the equation. Write answer as ordered pairs: 3x2 – 10x – 8 – y = 0

Determine whether the function is symmetric with respect to the y-axis, symmetric with respect to the x-axis, symmetric with respect to the origin, or none of these.9) y = 3x2 – 5 10) y = –5x3 + 2x 11) y = –2x4 + 4x + 1

12) y2 – 1 = x 13) y = −x5

x2−7

14) a) Calculate and b) interpret the slope of the line containing the points (1, –3) and (7, 8).Write the equation of the line satisfying the given conditions.15)Find an equation of the vertical line passing through the point (–3, –5).

16) The horizontal line through the point (−34,2).

17) Give the equation of the line that contains (–2, 4) and (–6, 4).

Find an equation for the line with the given properties. Write the answer in slope-intercept form.

18) A line containing the point (2, –5) and having the same x-intercept as the line given by 3x – 6y = –9.

Write an equation in general form for a line satisfying the given conditions.19) Write the general form of the equation of the line with slope 4 passing through the point ( 3, –5).

20) Through (9, 4) and (–8, 2).

Write an equation for the line.21) Write the slope-intercept from of the equation of the line passing through the point (6, 0) and parallel to the line y = –6x + 1.

22) Find an equation of the line perpendicular to 6x – 3y = 8 and containing the point (0, 8).

Solve the problem.23) Each week a soft drink machine sells x cans of soda for $0.75/soda. The cost to the owner of the soda machine for each soda is $0.10. The weekly fixed cost for maintaining the soda machine is $25/week. Write an equation that relates the weekly profit, P, in dollars to the number of cans sold each week. Then use the equation to find the weekly profit when 92 cans of soda are sold in week.

24) Each day the commuter train transports x passengers to or from the city at $1.75/passebger. The daily fixed cost for running the train is $1200. Write an equation that relates the daily profit, P, in dollars to the number of passengers each day. Then use the equation to find the daily profit when the train has 920 passengers in a day.

25) Each month a beauty salon gives x manicures for $12.00/manicure. The cost to the owner of the beauty salon for each manicure is $7.35. The monthly fixed cost to maintain a manicure station is $120.00 Write an equation that relates the monthly profit, in dollars, to the number of manicures given each month. Then use the equation to find the monthly profit when 200 manicures are given in a month.

26) Find the value of the function g(x) = –4x2 + 3x + 15, when x = –2

27) Evaluate the function. Express the answer in simplified form:

f(x) = 3x2 – 5x +2. Evaluate f ( x+h )−f ( x )

h , where h ≠ 0.

Determine the domain of each function

28) h(x) =

4 xx( x2−4 ) 29)

x√x−3

30) For the given function, find (a) the domain of f, (b) the x-intercepts, if any, of the graph of f, (c) the y-intercept, if there is one, of the graph of f.

f(x) = x2±49x+9

Solve the problem.31) Malin sells stereo equipment and earns a weekly base pay of $250 plus $30 per stereo set sold. Express his salary S as a function of the number x of stereo sets sold.

32) Express the area A of a parallelogram as a function of the height h if the length of the base of the parallelogram is 4.5 times the height of the parallelogram.

Plot and interpret the appropriate scatter diagram.33) The table shows the study times and test scores for a number of students. Draw a scatter plot of score versus time treating time as the independent variable.

34) Use a graphing utility to find the equation of the line of best fit for: x 1 2 3 4 5 6 y 17 20 19 22 21 24

35) Solve the problem. Use a graphing calculator to graph the data.A marina owner wishes to estimate a linear function that relates boat length in feet and its draft (depth of boat below the water line) in feet. He collects the following data. Let boat length represent the independent variable and draft represent the dependent variable. Use a graphing utility to draw a scatter diagram and to find the line of best fit. What is the draft for a boat 60 ft in length (to the nearest tenth)?Boat Length (ft) Draft (ft)

25 2.525 230 3

30 3.545 645 750 750 8

Identify where the function is changing as requested.

40) The graph of a function f is given. Use the graph to find (a) the numbers, if any, at which f has a local maximum. What are these local maxima?, and (b) the numbers, if any, at which f has a local minimum. What are these local minima?

41) Using a graphing utility, determine where the function is increasing and decreasing. Round answers to 3 decimal places: f(x) = 4x3 – 5x2 – 7x +3

The graph of a function is given. Decide whether it is even, odd or neither.

42) 43)

44) Determine if the graph is even, odd or neither.Determine if the function is even, odd or neither:45) f(x) = x3 – 4x

46) f(x) = 2x2 + 4x4

47) f(x) = x3 – x2

Graph the piece-wise defined function. The graph of a piece-wise defined function is given. Write a definition for the function.

48) f(x) = ¿¿¿¿

49)

Graph the function.50) f(x) = x3 + 2 51) f(x) = √ x−4+7

52) f(x) = |x + 2| + 2

Solve the problem.53) The graph of y = x2 is shifted to the right by 3 units. Write the resulting equation.

Using transformations,sketch the graph of the function.54) The graph of y = f(x) is as shown. Sketch the graph of y = f(x + 2) – 1.

55) Graph the function whose graph is that of y = x3 – x2 – 6x but is reflected about the y-axis.

56) Find the function that is finally graphed after the following transformations are applied to the graph of y = | x| . The graph is shifted right 3 units, stretched by a factor of 3, shifted vertically down 2 units and finally reflected across the x-axis.

Find the indicated composite for the pair of functions.

57) (f º g)(x): f(x) = 7x + 8, g(x) = 5x – 1

58) (f º g)(x): f(x) = 7x−6 , g(x) =

87 x

59) (g º f)(x): f(x) = 4x2 + 5x + 3, g(x) = 5x – 5

Find the domain of the composite function f º g.

60) f(x) = 6x + 54, g(x) = x + 1

61) f(x) = 8x+8 , g(x) = x + 5

62) f(x) = √ x−1 , g(x) = 1x−6

Solve the problem.

63) An oil well off the Gulf Coast is leaking, with the leak spreading oil over the surface of the gulf as a circle. At any time t, in minutes, after the beginning of the leak, the radius of the oil slick on the surface is r(t) = 6t ft. Find the area A of the oil slick as a function of time.

64) Let f(x) = √2−x and g(x) = | 2x – 1|. Find the domain of (f º g) (x). Express answer in interval notation.

65) Let g(x) = x−1x+2 and h(x) = 4x – 3. Find (h º g)(x).

Express answer as a single fraction in reduced form.

Find functions f and g so that the composition of f and g is H.

66) H(x) = 3√ x+1

Solve the problem.

67) A wire of length 6x is bent into the shape of a square. Express the area of the square as a function of x.

68) Two boats leave a dock at the same time. One boat is headed directly east at a constant speed of 35 knots (nautical miles per hour), and the other is headed directly south at a constant speed of 22 knots. Express that distance d between the boats as a function of the time t.

69) An open box with a square base is required to have a volume of 27 cubic feet. Express the amount A of material used to make such a box as a function of the length x of a side of the base.

70) Let P(x, y) be a point on the graph of y2= 4x + 4. Express the distance, d, of the point P from the origin. Express your answer in a simplified form.

In the problem, t, is a real number and P=(x, y) is the point on the unit circle that corresponds to t. Find the exact value of the given trigonometric function.

71) ( 56 , √116 )

; find sin t 72) ( 56 , √116 )

; find tan t

73) (√558 , 3

8 ); find sec t 74) (−√33

7, 47 )

; find cos t

75) (−√65

9, 49 )

; find cot t

A point on the terminal side of angle q is given. Find the exact value of the given trigonometric function.

76) (12, 16); Find sin q .

77) (6, 8); Find cos q .

78) (–15, 36); Find sin q

79) (21, 28); Find csc q .

Find the value of the expression.

80) sin–1 √22 81) cos–1

√22 82) cos–1

(−√22 )

83) tan–1 –1 84) sin–1 –0.5 85) tan–1(√33 )

Use a calculator to find the value of the expression in radian measure rounded to 2 decimal places.

86) sin–1 (0.4) 87) cos–1 (16 ) 88) tan–1 (1.5) 89) sin–1(√53 )

Solve the problem.

90) A building 150 feet tall casts a 80 foot shadow. If a person looks down from the top of the building, what is the measure of the angle between the end of the shadow and the vertical side of the building (to the nearest degree)? (Assume the person’s eyes are level with the top of the building.)

91) Two surveyors 180 meters apart on the same side of a river measure their respective angles to a point on the other side of the river and obtain 54º and 68º. How far from the point (line-of-sight distance) is each surveyor? Round your answer to the nearest 0.1 meter.

92) A hill slopes at an angle of 15° with the horizontal. From the base of the hill, the angle of elevation of a 600 ft tower at the top of the hill is 40°. How much rope would be required to reach from the top of the tower to the bottom of the hill? Round answer to the nearest foot.

93) A room in the shape of a triangle has sides of length 7 yd, 10 yd, and 15 yd. If carpeting costs $18.50 a square yard and padding costs $4.25 a square yard, how much to the nearest dollar will it cost to carpet the room, assuming that there is no waste?