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Name: ___________ Date: 06-20-2016 AP Calculus Summer Assignment Due the First Week of School (Exam. 50 pts.) (Turn in on 09-09-2016) Your assignment is due the first week of school and it is graded as a test. If you have trouble with any concept, please Google, βKhan Academyβ. Make sure that you show all of your work to receive full credit for your assignment. I. Graph each pair a,b ; c, d ; e, f ; g, h ; I, j on the same piece of graph paper. Please use radians only in AP Calculus. Explain how they are related (i.e., what can you do to one function to obtain the second one?). Classify these functions into odd and even function. How can you check which trig. function is odd or even? a. π(π₯) = π ππ π₯ π’π πππ π‘βπ ππππππ [βπ, 2π] b. π(π₯) = πππ π₯ π’π πππ π‘βπ ππππππ [βπ, 2π] c. π(π₯) = βπ ππ π₯ π’π πππ π‘βπ ππππππ [βπ, 2π] d. π(π₯) = βπππ π₯ π’π πππ π‘βπ ππππππ [βπ, 2π] e. π(π₯) = π‘ππ π₯ π’π πππ π‘βπ ππππππ [βπ, 2π] f. π(π₯) = πππ‘ π₯ π’π πππ π‘βπ ππππππ [βπ, 2π] g. π(π₯) = 2 π‘ππ π₯ π’π πππ π‘βπ ππππππ [β2π, 2π] h. π(π₯) = 2 πππ‘ π₯ π’π πππ π‘βπ ππππππ [βπ, 2π] j. π(π₯) = ππ ππ π₯ π’π πππ π‘βπ ππππππ [βπ, 2π] i. 1 π(π₯) = π ππ π₯ π’π πππ π‘βπ ππππππ [βπ, 2π] II. Derive the trigonometric table values for angles that are less than or equal to special triangle, i.e. π πππ ( ) = 3 β3 . 2 π/6 π 2 from the two π/4 ? a 2a π/3 π/4 a A I need the exact values. Do not give me the decimal values of these trig. functions! Trig. Function sin x sos x tan x cot x sec x csc x 0 π/6 π/4 π/3 π/2 Observation: Write down your observation in terms of which angles and trigonometric functions have the same trig. values. How is this going to help you in working with trig. functions? III. Use the above table, the reference angle, and the unit circle to find the exact value of trigonometric function (using radian only). 2 5 a. π ππ (3 π) = ? b. πππ (6 π) = c. πππ (β 4 π) = ? 3 d. π ππ (3 π) = ? e. ππ π (β 3 π) = ? 8 f. πππ (6 π) = ? 5 9 IV. Identities and Algebraic Transformation of Expressions Use the given identities to transform each given expression from the left side to the right side. πππ 2 π₯ + π ππ2 π₯ = 1 cot π₯ = 1 tan π₯ = tan π₯ = sin π₯ cos π₯ = sec π₯ csc π₯ sec π₯ = 1 cos π₯ csc π₯ = 1 sin π₯ cos π₯ 1 + πππ‘ 2 π₯ = ππ π 2 π₯ 1 + π‘ππ2 π₯ = π ππ 2 π₯ sin π₯ 1. Prove that tan x β cot x = 1. 2. Show step-by-step how you transform each expression on the left into the one on the right. a. cos x β tan x to sin x sec x b. csc x β tan x to d. csc B β tan B β cos B to 1 e. π ππ2 π³ β sec π³ β csc π³ π‘π tan π f. cot R + tan R to g. cot D β cos D + sin D to csc D h. csc x β sin x i. (sec E β 1)(sec E + 1) to tan2E j. c. sec A β cot A β sin A to 1 csc R β sec R to cot x β cos x (1 + sin B)(1 β sin B) to cos2B k. (cos π β sin π) to 1 β 2 cos π sin π 2 m. (cos π β sin π) to 1 β 2 cos π sin π 2 o. q. s. 1 1βcos π₯ + 1 sin π₯ cos π₯ 1 sin π₯ cos π₯ β β 1 1+cos π₯ cos π₯ sin π₯ cos π₯ sin π₯ to 2 csc2 x l. n. p. ππ π 2 π₯ β1 cos π₯ 1β πππ 2 π₯ tan π₯ to cot x β csc x to sin x β cos x (1 + sin π₯)(1 β sin π₯) to cos2x = πππ§ π r. π πππ³ + π‘πππ³ = = πππ§ π t. π πππ³ + π‘πππ³ = π ππππβπππ§ π π ππππβπππ§ π x V. Write an equation for each given graph. Show how you found both a and b values in y = aβb . y Find the initial value a and the growth factor for each graph and write the equation of the exponential function. Label this graph as y1. Find the initial value a and the growth factor and write the equation of the exponential function. Label the graph as y2. y Find the initial value a and the decay factor for each graph and write the equation of the exponential function. Label this graph as y3. Find the initial value a and the decay factor and write the equation of the exponential function. Label the graph as y4. Applications of exponential functions 1. Bacteria growth at an alarming rate. In fact every hour bacteria can split effectively doubling the number of bacteria present. At the beginning of an experiment, a scientist has 20 bacteria in a dish. E-Coli bacteria reproduce every 15 minutes. (a) Write an exponential equation describing this situation. (b) If left untreated how many bacteria cells will be present in 24 hours? 2. The population of Seattle is currently around 500,000 people. If Seattle grows by 6.5% every year, (a) Write an exponential model for the scenario above. (b) What will the population be in 10 years? 3. John Adam loves snowboarding. When he graduates, he would like to purchase a new Subaru so he can go snowboarding all the time. If the Subaru that he buys costs $15,500 and depreciates at a rate of 3.6% per year, answer the following questions about this scenario. (a) Write an exponential equation to represent this situation. (b) How much will the car be worth after one year? (c) Approximately what year will Mr. Adamβs new car be worth half as much? VI. Work with Functions: A. Find domain and range of each given function. 1. π₯ 2. π₯ 2β 9 βπ₯β2 π₯ 2 β 25 2 3. βπ₯β3 4. π₯+4 π₯ 3 β16π₯ B. Find composite functions. a. (f + g)(x) b. (f β g)(x) c. 1. π(π₯) = π₯ β 5 πππ π(π₯) = 2π₯ 3 β 4π₯ 2 2. d π(π₯) = π₯+3 π₯β2 (a) (b) (c) (d) (e) (f) π₯+3 π₯β6 . Is the point (2, -5/8) on the graph of f? If x = -7, what is f(x)? What point is on the graph of f? If f(x) = 2, what is the value of x? What points are on the graph of f? What is the domain of f? What is the range of f? List the x-intercepts, if any, of the graph of f? List the y-intercept, if there is one, of the graph of f? D. Find zeros by factoring each given function f(x). 1. π(π₯) = β2π₯2 + 15π₯ β 28 2. π(π₯) = β20π₯2 β 110π₯ + 400 E. Find factors for each expression. 1. π§ 3 β 125 2. π3 + 64 4. 3π¦ 2 β 28π¦ β 96 5. 3. (π + 1)3 + 8 121π2 β 64 F. Simplify each complex fraction. 1. 2 β3 π₯ 2β3π₯ 2 2. 1 2 + πβ1 π+2 2 1 β π+2 πβ3 3. π πππ π(π₯) = C. Answer questions about a given function. 1. π(π₯) = π . ( ) (π₯) (f β g((x)) 1 1 β 2 9 π 1 1 + 3 π 4π₯ 3π₯β2 G. Graph each pair of functions on the same coordinate system. 1. π(π₯) = ββπ₯ + 3 πππ π(π₯) = 5 β βπ₯ + 3 2. π(π₯) = π π₯ and lnx. 3. π(π₯) = βπ₯ β 2 πππ π(π₯) = β2 + βπ₯ β 2 VII . Find the slope of two points and write equations of lines according to given conditions. A. Write an equation of lines going through 2 points using point-slope form, y = y1 + m(x β x1). Convert this form into the standard form ( ax + by = c). Some of the equation will be in the form (x = a or y = b due to zero slope or undefined slope). Think for a second. π= π¦2βπ¦1 π₯2βπ₯1 1. A( - 4, - 2) and B(6,-2) 2. C(4, - 7) and D(4, 8) 3. E( - 3, - 8) and F(10, -5) 4. G( - 2, - 1) and H(5, 0) B. Write an equation of a line that is parallel to the given line, 2x + 4y = 12, and goes through (-2, 6) C. Write an equation of a line that is perpendicular to the given line, 5x + 4y = - 10, and goes through (5, 0) Have a terrific summer! Get this assignment done by 09-07=2016! Remember to Google, βKhan Academyβ if you donβt know how to do a certain problem!