AP CALCULUS BC ~ (Σer) ( Force Distance) and ( L1,L2,...) of Topical Understandings ~

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1 Name: Previous Math Teacher: AP CALCULUS BC ~ (Σer) ( Force Distance) and ( L1,L,...) of Topical Understandings ~ As instructors of AP Calculus, we have extremely high expectations of students taking our courses. As stated in the district program planning guide, we expect a certain level of independence to be demonstrated by anyone taking AP Calculus. Your first opportunity to demonstrate your capabilities and resourcefulness to us is through this summer work packet. We expect you stay current on your skills as well as improve upon them. Therefore, this packet is a requirement Calculus BC. It will be your first major grade of the new school year. SHOW US YOUR BEST WORK. It needs to be completed when handed in on the first day of class. Requirements The following are guidelines for completing the summer work packet There are 66 questions you must complete. Please do all your work in this packet. You must show us all of your quality work. Please include all the steps used to answer each question. Your work should lead to the correct answer. Please circle or box each answer!! Be sure all problems neatly organized, and all writing is legible. If you have trouble answering any question, be resourceful. This might include looking back to your notes from previous years or even finding a book or website with formulas or explanations. In the event that you are unsure how to perform functions on your calculator, you may need to read through your calculator manual to understand the necessary syntax or keystrokes. You must be familiar with certain built-in calculator functions such as finding maximum and minimum values, intersection points, and zeros of a function. You will also need to be able to do regression analysis on your calculators. We expect you to come in with certain understandings that are prerequisite to Calculus. A list of these topical understandings can be found below. Please be familiar with all of these and ready to apply them to a higher level. Topical understandings within summer work Factoring Conics Limits of Functions Prove Trig Identities Graphing Piecewise Functions Regression Analysis Graphing, simplifying expressions, and solving equations of the following types: Trigonometric, rational, logarithmic, exponential, Polynomial/Power, Radical, Polar, and Parametric.

2 Finally, we suggest not waiting until the last two weeks of summer to begin on this packet. If you spread it out, you will most likely retain the information much better. Once again this is due, completed with quality, on the first day of class, and it counts as a grade. It is your ticket into the class. Best of luck and if you have any questions, feel free to contact us. Mr. Yanisch Calc BC (60) bill_yanisch@ipsd.org Mrs. Liveris Calc BC (60) vanessa_liveris@ipsd.org Ms. Sheila Roth Calc BC (708) sheila_roth@ipsd.org First impressions are lasting impressions impress us!! "If I have seen farther than others it is because I have stood on the shoulders of giants"--newton "In most sciences one generation tears down what another has built. In mathematics, each generation builds a new story to the old structure"

3 AP CALCULUS SUMMER PACKET Simplify 5 x 6 x. Express answer using a single radical.. Factor completely. ac + cd ab bd 5x 6. Solve the double inequality, 1 < 7. Express your answer in interval notation. 4. Solve x x 10. Use factoring and a number line sign chart. 5. Determine the range of: where they occur. 4 = 1 0x x. Also, find the max and min values of f ( x) f ( x) x, and state 6. Write the equation of the line through: (-, 7) and (, 5) in point -slope and slope-intercept form.

4 7. Solve the equation 4 x = 5 x + 4 graphically and algebraically: 8. The table shows average salaries for employees of Mediocre Tools, Inc. Find a linear regression equation for the data and graph the equation with a scatter plot of the data. (Use x = 0 for 1975, and express values to the nearest hundredth.) Then estimate the average salary in 008. Year Average Salary ($) , , , , , , Three sides of a fence and an existing wall form a rectangular enclosure. The total length of a fence used for the three sides is 40 feet. Let x be the length of the two sides perpendicular to the wall as shown. Write the area A of the enclosure as a function of the length x of the rectangular area as shown in the figure. Then find the value(s) of x for which the area is 5500 ft. X? Existing wall X 10. Rewrite the expression log 5 ( x + ) into an equivalent expression using only natural logarithms.

5 11. Simplify the following expressions. ( ) ( 1+ ln ) 1 ln x a e ( b) ( c) e ( d ) log 8 ( e) ( n 1 )! 5n! x + log 1. Factor completely. x + 1x + 6 9y + 1. Let f ( x) = x, and g( x) = x 1. Compute ( go f )( x), and state its domain in interval notation. 14. Graph, and show direction for, the relation defined by the parametric equations. x = t 5, y = t 1, t 15. The following three transformations are applied (in the order given) to the graph of y = x. I. A vertical stretch by a factor of II. A horizontal shift right 5 units III. A vertical shift down 6 units Graph it. Which of the following is an equation for the graph produced as a result of applying these transformations? A. y = 5x 1 B. y = ( x 5) 6 C. y = ( x + 5) 6 D. y = x 1 + E. y = ( x 6) 5

6 16. Let y = x + 7 f ( x) =. Find a rule for x 1 f. 17. Find an equation for the parabola whose vertex is (, -5) and passes through (4, 7). Express your answer in the standard form for a quadratic function. 18. Which of the following could represent a complete graph of f ( x) = ax x, where a is a real number? 19. Find a degree polynomial with leading coefficient 4 and zeros -, 1, and Let g ( x) be a sinusoidal function with a min at ( π,5) and the next max at (,8) Write an equation for g ( x). State the amplitude and the period of g ( x). 5π. 1. Write the absolute value function as a piecewise function. y = 4x x

7 . Determine if the function is even, odd, or neither. f ( x) Show the work. 1 = x x. The graph of x+ y = a for a > 1 is best represented by which graph? 4. Simplify. 1 x 4 x 9x 5. Describe the transformations that can be used to transform the graph of log x to a graph of f ( x) = 4log( x + ). 6. The number of elk after t years in a state park is 116 modeled by the function P( t) =. 0.0t e Graph it. a) What was the initial population of elk? b) When will the number of elk be 750? c) What is the maximum number of elk possible in the park?

8 7. Arturo invests $700 in a savings account that pays 9% interest, compounded quarterly. If there are no other transactions, when will his balance reach $4550? 8. If the point P, ( 5, ) P, is on the terminal side of θ, find all six trigonometric functions of θ. 9. Simplify (csc x tan x)sin x cos x. A. sin x cos x B. cos x sin x C. sin x + cos x D. cos x sin x 0. π Which transformation was not performed on y = sin x to obtain y = sin(x + )? A. π Horizontal shift left by units 9 B. Horizontal stretch by a factor of C. Vertical stretch by a factor of D. Reflection through the x-axis 1. Solve the equation on the interval [ 0,π ]. sin ( x ) cos ( x) = 0

9 . Use the double angle identities to rewrite sin ( x), in terms of linear powers of common trig functions. o. An airplane is flying on a bearing east of north at 650 mph. Express the velocity of the airplane as a vector. 4. Without using a calculator, find the exact value of cos 1 ( cos 17 π ) 5. Justify your answer. 5. Determine whether the vectors, 1 and, 5 are orthogonal. 6. Find the angle between vector u =, 5 and vector v = 1,. 7. Factor and solve the inequality. Use a number line analysis and write the answer using interval notation. ( + 1) ( x ) + ( x + 1)( x ) > 0 x.

10 8. Eliminate the parameter and identify the graph of the parametric curve given by x = t, y =. Graph it! t 9. Evaluate and simplify f ( x + h) f ( x) h if ( x) x x f =. 40. Determine sin x lim, if possible. x 0 x x 41. Sketch the functions. Shade the region bounded by the graphs and the y-axis. Find the area of the shaded region. y = x 1 y = x Given the force vector of magnitude 55N, acting in the direction 75. Write the vector in r r i, j notation. 4. students are 180 feet apart on opposite sides of a telephone pole. The angles of elevation from the o o students to the top of the pole are 5 and. Find the height of the pole.

11 44. Graph the piecewise function. x x < 1 f ( x) = x = 1 x < x 45. Using f ( x) from #44 above, at what points c in the domain of ( x) f does Lim f ( x) x c exist? 46. Decompose the function y = sin ( ln( x + ) ) into 4 basic functions, ( x), g( x), h( x), and j( x), State the functions, and give the order of compositions to get y. f. 47. Sketch the graphs of the polar equations = 9sin( ) and r = r ϑ. Find the 4 points of intersection. Show work. 48. The graph of the polar equation = 5sin( ϑ ) r is a(n) (a) petal rose, starting at the polar axes (b) petal rose, starting in the 1 st quadrant (c) 6 petal rose, starting at the polar axis (d) 6 petal rose, starting in the 1 st quadrant 49. Solve the equation sin x cos x cos x 0. = by factoring. Give the solutions on [,π ]

12 50. Find all the exact solutions of the equation on the interval [ 0, π ). sin x + sin x = Using partial fraction decomposition, decompose the fraction x + x ( x + )( x 1). 5. Find the equation that describes the set of points (x, y, z) that are 5 units from the point (, -1, 6). 5. Determine the sum, if it exists, of the infinite geometric series below Write the equation of a quadratic function whose x-intercepts are -1 and, and a range consisting of all numbers less than or equal to The graph of y = f ( x) is given. Sketch the following. y = f ( x) f ( x) f ( x) f ( x)

13 56. Find a formula for the Perimeter and Area of the shape. r r 57. Find the points of intersection of x + y = 4 and x + y 4x 4y = 4, graphically and algebraically. 58. For the function f (x) graphed, evaluate lim f ( x). x A. lim f ( x) = 10. x C. lim f ( x) =. x B. lim f ( x) =. x D. lim f ( x) does not exist x 59. Solve for x: e x = x. 60. Solve (sin 1 x)(cos x) + (cos x)(cos x) = for x over [ 0, π ). x Find the domain of the function f ( x) =. Express your answer in interval notation. Explain how x + you got your answer.

14 x + 6x Find all asymptotes for the function y =. State your answers as equations. x Graph the function y = x x + x 1. Find the local maximum/minimum values, and all the x-intercepts. Sketch the graph and state the window dimensions. 64. Use a graphing calculator to approximate all of the function s real zeros. Round to decimal places. 6 5 f ( x) = x 5x 4x + x + x Find all (if any) points of discontinuity of h ( x). Write your answer(s) as a coordinate point. h ( x) x 4x 5 = x Eliminate the parameter and name the endpoints. x = t, y = t +, t.