AP Calculus AB Summer Assignment Mrs. Berkson

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1 AP Calculus AB Summer Assignment Mrs. Berkson The purpose of the summer assignment is to prepare ou with the necessar Pre- Calculus skills required for AP Calculus AB. Net ear we will be starting off the ear with Calculus and will not be reviewing an Pre-Calculus topics at the beginning of the ear. Instead we will review the Pre-Calculus topics as needed. However, to make sure ou have the pre-requisites required to handle Calculus ou will be given a 50 question multiple choice Pre-Calculus test during the first week of school. The date of the test will be announced on the first da of school. To prepare for this test, ou should complete the review provided. The answers are included for ou to check to see if ou are answering the questions correctl. The review will not be collected. We will spend some time of the first da of school going over an questions ou have on the review. In additon to the review, complete the following: Memorize the unit circle (radians onl) and complete the table. You will have a speed quiz on the first quadrant of the unit circle during the nd week of school. You will have a speed quiz on the entire unit circle during the 3 rd week of school. All quiz and tests dates will be posted on m website site in August. Purchase a TI-nspire CX CAS graphing calculator and bring it to first da of school. If ou are taking and or plan to take AP Statistics, ou will need this calculator in that class as well. You ma elect to use another calculator; however ONLY the TI-nspire CX CAS will be taught in class. Learning to use another calculator will be our own responsibilit. A limited number of TInspire calculators will be available to rent from the school. Starting Tuesda August 15 th. Please me to arrange pick up. If ou have an questions please me at lberkson@dadeschools.net or visit

2 Unit Circle sint cost tant csct sect cot t

3 AP Calculus AB Summer Assignment Stud Guide Name Period: Date: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Evaluate the eponential epression. 1) (-)- 1) ) -- ) Simplif the eponential epression. 3) ) ) () ) 5) -0 5) ) ) 7) - 7) ) 3 z3 3 ) 9) 3-9) Rationalize the denominator. 9 10) 10) 11) ) 1) ) 13) ) 1

4 Find the product. 1) ( - )( + + 3) 1) 15) ( + 9)( ) 15) 1) (3-5)( - 5) 1) Solve the problem. 17) Write a polnomial in standard form that represents the volume of the open bo. 17) ) Write a polnomial in standard form that represents the area of the shaded region. 1) Factor out the greatest common factor. 19) ) 0) ( + ) + 13( +) 0) 1) ( - 3) - ( - 3) 1) Factor b grouping. Assume an variable eponents represent whole numbers. ) ) 3) ) Factor the trinomial, or state that the trinomial is prime. ) )

5 5) + - 5) ) ) 7) ) ) ) 9) ) Factor the difference of two squares. 30) 1-30) 31) ) 3) (1-1) 3) Factor the perfect square trinomial. 33) ) Factor using the formula for the sum or difference of two cubes. 3) 3-3) 35) ) 3) ) Factor completel, or state that the polnomial is prime. 37) ) 3) - 3 3) Solve the problem. 39) Write an epression for the area of the shaded region and epress it in factored form. 39)

6 Factor and simplif the algebraic epression. 0) /5-1/5 0) 1) ( + )1/ + ( + )3/ 1) ) ( + 5)/5 - ( + 5)1/5 ) 3) ( + 9)-1/5 + ( + 9)-/5 3) Simplif the rational epression. Find all numbers that must be ecluded from the domain of the simplified rational epression. + ) ) ) ) Add or subtract as indicated. ) ) 7) ) ) ) Simplif the comple rational epression ) - 9) 50) ) 51) )

7 Simplif the epression. 1-5) 5) First, write the value or values of the variable that make a denominator zero. Then solve the equation. 53) 7 = ) 5) = ) Solve the formula for the specified variable. 55) V = 1 Bh for h 55) 3 5) S = πrh + πr for h 5) Solve the quadratic equation b the method of our choice. 57) ( + 7) = 3 57) 5) = ) 59) + = 7 59) 5

8 Use the graph to find the indicated function value. 0) = f(). Find f(). Find the domain. Find the range. Intervals in which the function is increasing. Intervals in which the funciton is decreasing. Intervals in which the function is constant. The minimum value. The maimum value. 0) Evaluate the piecewise function at the given value of the independent variable. 1) g() = if if = -3 ; g(-) 1)

9 Graph the function. + if -9 < ) f() = -9 if = if > 10 ) Find and simplif the difference quotient f( + h) - f(), h 0 for the given function. h 3) f() = ) ) f() = 5 ) Use the given conditions to write an equation for the line in point-slope form. 5) Slope = 5, passing through (, ) 5) ) Passing through (-3, -7) and (-5, -) ) 7) Passing through (1, -5) with -intercept = -1 7) Begin b graphing the standard quadratic function f() =. Then use transformations of this graph to graph the given function. ) h() = ( + ) - 3 )

10 Begin b graphing the standard square root function f() = function. 9) g() = Then use transformations of this graph to graph the given 9) Begin b graphing the standard absolute value function f() =. Then use transformations of this graph to graph the given function. 70) g() = ) Find the domain of the function. 71) f() = ) - 7) h() = 3-1 7) 73) f() = 3-73) 7) - 7)

11 For the given functions f and g, find the indicated composition. 3 75) f() = - 7, g() = 5 7 (f g)() 75) 7) f() = - 5, g() = + 5 (g f)() 7) Find functions f and g so that h() = (f g)(). 77) h() = (3 + 1)9 77) Find the inverse of the one-to-one function. 7) f() = ( - )3 7) 79) f() = ) Determine whether the given quadratic function has a minimum value or maimum value. Then find the coordinates of the minimum or maimum point. 0) f() = - - 0) 1) f() = ) Solve the problem. ) A developer wants to enclose a rectangular grass lot that borders a cit street for parking. If the developer has 3 feet of fencing and does not fence the side along the street, what is the largest area that can be enclosed? ) Find the -intercepts of the polnomial function. State whether the graph crosses the -ais, or touches the -ais and turns around, at each intercept. 3) f() = ( + 1)( - )( - 1) 3) ) = 0 ) Find the -intercept of the polnomial function. 5) f() = ( + 1)( - )( - 1) 5) Use the Leading Coefficient Test to determine the end behavior of the polnomial function. ) f() = ) 7) f() = ( - 3)( - )( - 1) 7) Find the zeros of the polnomial function. ) f() = ) 9

12 9) f() = ) Use the Intermediate Value Theorem to determine whether the polnomial function has a real zero between the given integers. 90) f() = ; between 1 and 90) Divide using snthetic division. 91) ) 9) ) Solve the polnomial equation. In order to obtain the first root, use snthetic division to test the possible rational roots. 93) = 0 93) 9) = 0 9) Solve the problem. 95) A bo with an open top is formed b cutting squares out of the corners of a rectangular piece of cardboard and then folding up the sides. If represents the length of the side of the square cut from each corner, and if the original piece of cardboard is 1 inches b 1 inches, what size square must be cut if the volume of the bo is to be cubic inches? 95) Use the graph of the rational function shown to complete the statement. 9) 10 9) As -+, f()? 10

13 Find the vertical asmptotes, if an, of the graph of the rational function. 97) f() = ) 9) h() = - ( - 1) 9) Find the horizontal asmptote, if an, of the graph of the rational function. 99) f() = ) 0 100) g() = ) ) h() = ) Graph the function b making a table of coordinates. 10) f() = 5 10) Graph the function. 103) Use the graph of f() = to obtain the graph of g() = )

14 Write the equation in its equivalent eponential form. 10) log 5 5 = 10) Write the equation in its equivalent logarithmic form. 105) 3 = 105) Evaluate the epression without using a calculator. 10) log ) 107) log 1 107) Find the domain of the logarithmic function. 10) f() = log ( + ) 10) 109) f() = log ( - ) 109) Use properties of logarithms to epand the logarithmic epression as much as possible. Where possible, evaluate logarithmic epressions without using a calculator ) log 110) 3 111) log b 7 z3 111) Use properties of logarithms to condense the logarithmic epression. Write the epression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic epressions. 11) 5 log b q - log b r 11) 113) 1 3 [5ln ( + 10) - ln - ln ( - 7)] 113) Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places 11) log ) Solve the eponential equation. Epress the solution set in terms of natural logarithms. 115) 3 =. 115) 11) e + 5 = 11) Solve the eponential equation. Use a calculator to obtain a decimal approimation, correct to two decimal places, for the solution. 117) e = ) 1

15 Solve the logarithmic equation. Be sure to reject an value that is not in the domain of the original logarithmic epressions. Give the eact answer. 11) log ( + 3) = -1 11) 119) ln + = 9 119) 10) log ( + ) - log = 10) Use the Pthagorean Theorem to find the length of the missing side.then find the indicated trigonometric function of the given angle. Give an eact answer with a rational denominator. 11) Find sin θ. 11) 9 1) Find sec θ. 1) 5 13) Find tan θ. 9 13) A point on the terminal side of angle θ is given. Find the eact value of the indicated trigonometric function of θ. 1) (9, 1) Find cos θ. 1) 15) (1, ) Find csc θ. 15) 1) (, -9) Find cot θ. 1) 13

16 The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not eist. 17) a. lim f() b. f(0) 17) ) a. lim 0 f() b. f(0) 1) 19) a. lim f() b. f() 19)

17 130) a. lim f() 3 b. f(3) 130) ) a. lim f() 3- b. f(3) 131) Use properties of limits to find the indicated limit. It ma be necessar to rewrite an epression before limit properties can be applied. 13) lim 0 ( - 5) 13) 133) lim 1 ( - ) 3 133) ) lim ) A piecewise function is given. Use the properties of limits to find the indicated limits, or state that the limit does not eist. 135) f() = - + if < 1 135) - + if > 1 a. lim f() b. lim f() c. lim f()

18 Answer Ke Testname: OVERVIEWTEST 1) 1 1 ) ) 5 ) 31 5) - ) -5 7) ) 9 z9 9) 10 10) 7 11) ) ) 5-3 1) ) ) ) ) ) (7 - + ) 0) ( +)( + 13) 1) ( - 3)( - 1) ) ( + )( + 5) 3) ( - 5)( + 7) ) ( - 9)( + 3) 5) ( + )( - ) ) (3 + )( + 3) 7) ( - 3)(5 - ) ) ( + 3)( - 9) 9) ( - 3)(3 - ) 30) (9 + )(9 - ) 31) (13 + 3)(13-3) 3) ( + 1)( + 1)( - 1) 33) ( - 1) 3) ( - )( + + 1) 1

19 Answer Ke Testname: OVERVIEWTEST 35) (5 + 1)( ) 3) ( - 5)( ) 37) ( + )( - ) 3) ( + )( + )( - ) 39) (11 + )(11 - ) 0) 1/5(3/5-1) 1) ( + )1/ (1 + ( + )1/) ) ( + 5)/5 ( ) ( + 10) 3) (+ 9)/5 ) 1 5 +, - 5, - 1 5) +, -7, ) 7) 3-5 ( + 5)( - 5) ( - 1)( + 1)( - ) ) ) 1 50) ) - 5) ) 0; 1 5) 5; {9} 55) h = 3V B 5) h = S - πr πr 57) - 13, - 1 5) 5, -1 59) {- - 3, - + 3} 0) )

20 Answer Ke Testname: OVERVIEWTEST ) 10 (, ) 5 (, 1) (-9, -5) (, -9) 3) 3 ) 5(+h) 5) - = 5 ( - ) ) + 7 = - 3 ( + 3) or + = - 3 ( + 5) 7) + 5 = - 5 ( - 1) or = - 5 ( + 1) ) 10 9)

21 Answer Ke Testname: OVERVIEWTEST 70) ) (-, ) 7) (-, -9) (-9, 0) (0, 9) (9, ) 73) (-, 3] 7) (, ) 1 75) 5-9 7) 77) f() = 9, g() = ) f-1() = ) f-1() = + 5 0) minimum; 1, - 3 1) minimum; - 1, 0 ) 7 ft 3) -1, crosses the -ais;, crosses the -ais; 1, touches the -ais and turns around ) 0, touches the -ais and turns around; -, crosses the -ais;, crosses the -ais 5) - ) falls to the left and falls to the right 7) rises to the left and rises to the right ) = 0, = -, = 3 9) = -1, = 1, = - 90) f(1) = -3 and f() = ; es 91) ) ) {1, -1, -} 9) {1, 3 +, 3 - } 95) 3 in. b 3 in. square 9) + 97) no vertical asmptote 9) = 0 and = 1 99) = 0 100) = 19

22 Answer Ke Testname: OVERVIEWTEST 101) no horizontal asmptote 10) ) ) 5 = 5 105) log = ) 1 107) -1 10) (-, ) 109) (-, ) or (, ) 110) log 3 ( + 5) - log 3 111) log b + 7log b - 3log b z 11) log b q5 r 113) ln 3 ( + 10) 5 ( - 7) 11) 1.31 ln. 115) 3 ln 11) {ln - 5} 117). 0

23 Answer Ke Testname: OVERVIEWTEST 11) ) {e1 - } 10) { 15 } 11) 5 5 1) 1 13) 9 1) ) 5 1) ) a. lim f() = 1 0 b. f(0) = 1 1) a. lim 0 f() = 1 b. f(0) does not eist 19) a. lim f() = b. f() = 1 130) a. lim 3 f() does not eist b. f(3) = 5 131) a. lim f() = 3 3- b. f(3) = 5 13) ) -1 13) 0 135) a. b. c. 1

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