AP Calculus Summer Assignment
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1 AP Calculus Summer Assignment Welcome to AP Calculus. This summer assignment will help you review some Algebraic and Trigonometric topics that you will need for some problems in Calculus. This summer assignment is designed to help you review/relearn topics that I feel you need refreshed. It is important that you understand the algebraic, eponential, trigonometric topics you may have forgotten. Take you time doing the problems. Actually do the problems. This packet is due the first day of school. You will be tested on this material the first week of school. I will also have a copy of your function notebook on Blackboard. Please make sure it is up to date. If not please copy the pages you do not have. You will also have a functions quiz the first week of school. You should be able to recognize the basic graphs as well as the trig, eponential and log graphs. I am looking forward to working with you in September. I you have any questions, feel free to contact me at Mbrumberg@wtps.org. Helpful Websites: Algebra: Trigonometry: General Math including Calculus If you do not have a graphing calculator, please obtain one of the following recommended calculators: TI 84+, TI-nspire (CAS or not)*. If you already have a calculator, make sure to update the OS. * CAS is not permitted for the ACT
2 AP CALCULUS SUMMER ASSIGNMENT 07 Mrs. Brumberg I. Basic Algebraic Rules. Are the following statements true? If not, change them to make them true. k k + y y = b. = + c. = + k + 4 k + 4 p+ q p q d. a a = b b e. a a = b b f. a + b a + b = c c g.! " #$!%$ ='+ II. Comple Fractions & Rational Epressions. Simplify. 4 b. ' + h h c. h 5 +. Write as a single fraction with the denominator in factored form b c. ( ) + ( ) d. ( ) ( ) + e.
3 4. Evaluate without a calculator: *64 -// 0 b. *6 / 0 c. *7 -/ 0 d. * 6/- 0 III. Negative and Fractional Eponents 5. Simplify using only positive eponents. Do not rationalize the denominators ( 4) b y y c. y IV. Solving Equations and Factoring 6. Solve for y in simplest form. y + y = + y b. y y + yy = 5y + c. yy + y = yy 7. Solve the quadratic equation. Use any means from algebra: factoring, quadratic formula, graphing. Be sure answers are simplified. If you use graphing, state what you did on the calculator = b. 0 + = c = Factor completely (There should be negative eponents.) + 9 b. 9 c. 6
4 d. sin + tan e. e e + e f V. Equations of lines 9. Find the equation of the line that passes through the point (, 4) and is parallel to the line + y 8 = Find the equation of the line that is perpendicular to the line + y 8 = 0 at the point (, ).. The line with slope 5 that passes through the point (-, ) intersects the -ais at a point. What are the coordinates of this point?. What are the coordinates of the point at which the line passing through the points (, -) and (-, 4) intersects the y-ais?. Graph the equation Is the point (, ) on the graph? y = and answer the following questions. b. Is the point (, 6) on the graph?
5 c. Is the function odd, even or neither? d. Find the and y intercept(s). VI. Asymptotes and Intercepts 4. Find all intercepts and asymptotes. (NO slant asymptotes.) + y = 4 b. y = ( + ) c. y = 4 d. y = + 6 VII. Domain 5. Use interval notation to identify the domain for each of the following functions. h ( ) = b. k( ) = c. 6 0 d. d( ) = ln( ) VIII. Graphing Functions 6. Graph the following functions. 0 f() = > 0 b. f() = (, ) [,) + [, ) c. f() = 6
6 d. y = > e. 4 < y = + + > f. y = + + g. y= + IX. Functions 7. Find f() f(5) given f() = Find f( + ) f() given f() = Find f( + h) for f() =. 0. Find f( + h) f( ) h if f() = Given f() = and g() =, complete the following. f(g()) = b. g(f()) = c. f(f()) =. Given f( ) = 5 and g() = 5, complete the following. f(g(7)) = b. g(f(v)) = c. g(g()) = X. Binomial Theorem. Use the Binomial Theorem to epand and simplify the following epressions. ( + 5) 5 b. (a 4b) 4 c. (w + ) 6
7 XI. Factor Theorem 4. Use the p over q method and synthetic division to factor the polynomial P(). Then solve P()=0. P ( ) = b. P 4 ( ) = XII. Logarithms 5. Condense the epression: ln( ) + ln( + ) 6ln 6. Epand the epression: ln!" 9:% ; < =. 7. Epress y in terms of. ln y = + b. ln y = ln + ln0 c. ln y = 4ln + d. = e ln 4 y 8. Solve for. ln e = b. ln e = 4 c. ln + ln = 0 d. ln 5 e = e. ln lne = f. ln6 + ln ln = g. ln( + 5) = ln( ) ln( +) XIII. Trigonometry 9. Evaluate (without a calculator!!). NO decimals. cos 0 b. sin 0 c. tan π d. cos 4 π e. sin π f. sin π
8 g. sin h. tan i. cos j. sec %$ k. tan k. If 5 cosθ = and θ is in Quadrant II, Find the all the remaining trig functions. 0. Which of the following epressions are identical? cos b. ( cos ) c. cos. Which of the following epressions are identical? ( sin ) b. BCDEFG ' c. sin d. sin. Solve the following for the indicated variable on the interval[ 0, π ). cos = b. sinklm =0 c. tan = 0 d. + = sin sin. Complete the following trig identities $%Ksin P#cos sin + cos = b. tan PM" + = c. sin P = XIV. Word Problems 4. Find the surface area of a bo of height h whose base dimensions are p and q and satisfies the following conditions: The bo is closed.
9 b. The bo has an open top. c. The bo has an open top and a square base with side length p. 5. A seven foot ladder, leaning against a wall, touches the wall feet above the ground. Write an epression in terms of for the distance from the foot of the ladder to the base of the wall. 6. A piece of wire 5 inches long is to be cut into two pieces. One piece is inches long and is to be bent into the shape of a square. The other piece is to be bent into the shape of a circle. Find an epression for the total area made up by the square and the circle as a function of. 7. A police car receives a radio call to catch a vehicle which is speeding down the highway at 80 mph. The police car, which is miles away, drives after it at 08 mph. How long will it take for the police car to catch up? 8. The base of a triangle is 6 cm more than the height. If the area of the triangle is 40 square cm, what is the length of the base?
10 9. Two trains, the Epress and the Commuter, leave the same station at the same time. The Epress, which heads north, travels 0 km per hour faster than the Commuter, which goes east. If the trains are 00 km apart after hours, find the speed of each train. 40. The depth, d, of a buoyant object t seconds after plunging into water can be found using the equation d = 6t + rt, where r is the velocity at which the object strikes the water. If the object strikes the water at a velocity of 40 feet per second, find the maimum depth the rocket will reach and at what time. When will the rocket surface again? 4. Find the average rate of change (i.e. slope) for the following functions on the indicated intervals. f( ) = ;[0,4] b. f( ) = ;[4,5] c. A car travels 40 miles over a period of 0 minutes. Find the average velocity of the car in miles per hour over this time period. d. On January st 00, the value of a stock was $5 per share. By December st 00, the value of the stock had fallen to $8 per share. What is the average rate of change in the value of the stock in dollars per month? e. In 984, the Fizzy Cola Company sold million gallons of sod By 00, the company was selling 7 million gallons of sod What is the average rate of change in the number of gallons of soda sold per year? f. During a recent trip to the store, a car s velocity went from 0 to 60 mph in 0 seconds. What is the average acceleration of the car in miles per hour per hour?
11 () Basic Algebraic Rules ) Are the following statements true? If not, change them to make them true.
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