Summer Review for Students Entering AP Calculus AB

Transcription

1 Summer Review for Students Entering AP Calculus AB

2 Class: Date: AP Calculus AB Summer Packet Please show all work in the spaces provided The answers are provided at the end of the packet Algebraic Manipulation 1 Evaluate x y (x y) if x = 5 and y = 6 Expand (x 4 7y ) Simplify: x y ˆ x y ˆ x y + ˆ 7 Multiply x ˆ + x x x + ˆ 5x y x 5 y 1 z 0 (xy ) xy 8 Expand using Pascal s Triangle ( x ) 5 4 Simplify 18xy 7a b 1x y 5a b 5 Subtract ( g + g 9) from (g 4g 6) 1

3 9 Factor completely a) a 6a Factor completely a) 5x 4 y + 0xy + 15xy 4 b) 6y + 1y 5 b) x + x 15x 5 c) 1m n 75mn c) 0x 15y d) 49x 100y d) 4x 1xy + 9y e) 6xp + 4x 5yp 5y e) x 6x + 9 4y

4 11 Factor the expression completely 15a 8b 6 14 Divide using synthetic division (x 5 + 5x 4 + x + 5) (x + ) 1 Factor the expression completely 4 x ˆ + 5 4(x)(x 1) 4 + x ˆ (9)(x 1) 15 Divide: x 5 + x 4 8x ˆ + x + x + x 7 ˆ 1 Factor the expression completely 1 x ˆ x ˆ + 4 1

5 16 Simplify a x a a x a 0 Simplify the compound fractional expression 5 x 1 4 x + 1 x x x Simplify (4w wy)(w + y) (y 4w)(5w y ) 1 Simplify 5 6x 18 x 1 4x 14x Simplify x 64 x + 64 x 16 x 4x + 16 Perform the subtraction and simplify x x x 0 1 x + 4 x 5 19 Simplify x 1 x y 4

6 Simplify a) Evaluate without a calculator: a) 1 ˆ 4 ˆ b) b) 5 ˆ c) + 5x 5x ˆ d) m 4m 4m 5 c) 7 d) 0049 e) p ˆ 5 Simplify a) 5 4a 0 b 5 x f) 1 y yx 7 4 ˆ 4 g) x 1 y x y ˆ b) 18a b 1ab 5 5

7 6 Rationalize the denominator and simplify a) a b 8 Simplify and rewrite using only positive exponents 4x 16 4 (x 4) b) c) Simplify the expression 4 x 1 ˆ È (x + 1) (x 1) x + 1 ÎÍ (x + 1) d) y x 5 7 Rationalize the numerator and simplify a) x 40 5 ˆ + x 1 1 y y 1 b) x + 5 x 66 6

8 1 Simplify each expression and write the result in the form a + bi a) 6 ( + 9i) + ( 1 + 4i) Use the Laws of Logarithms to expand the expression ln x ˆ 4 x 1 x + b) (7 i)( + i) Use the Laws of Logarithms to combine the expression c) ( 5 7i) 5logx 1 log x ˆ log(x 1) d) i + i 4 Evaluate the expression 10 log π 5 Evaluate the expression e) 5 i 4i log Evaluate the expression f) i 1795 ln e 7 ˆ 7 Evaluate the expression log( ) 7

9 Equations and Inequalities 8 Solve (x + ) = 1 (1x + 4) 5x 4 41 Solve the equation x 5 = 7 9 Solve the equation 4 Solve the equation for the indicated variable: 5 y + 1 y + 1 (y 4) = 4 a) ax + b cx + d = 11, solve for x b) x a = c x, solve for x b 40 Solve 4 x = 18 c) x a = c x, solve for b b 8

10 4 Solve each equation: a) n(9 n)(n + 5) = 0 44 Solve by completing the square x + 8x + = 0 b) x + 8x 0 = 0 45 Solve the equation by completing the square x 1x 7 = 0 c) 0x(x 1) = 4 9x d) 8x + 46x 0 = 0 46 Solve by using the Quadratic Formula x 6x = 10 e) x 6x = 10 9

11 47 Find all solutions of the equation and express them in the form a + bi 1 50 Solve (6x ) 4 = 0 5x + 16 = 0 51 Solve 1 x 1 x = 6 48 Find all real solutions of the equation x 6 9x 10 = 0 5 Solve a) x = 7 49 Solve 4 4x 7x + 9 = 0 b) = b 10 b 10

12 5 Solve for x 4 x = x 1 57 Solve for x e x = 54 Solve for x 4 x + = 5x 4 58 Solve the equation for x e 4x + e x 8 = 0 55 Use the definition of the logarithmic function to find x a) log x 4 = 1 59 Solve for x ln(7 x) = 0 b) log x 6 = 1 56 Solve for x: log (x 6x) = 4 60 Solve for x ln(1) 4x = ln 1 ˆ e 11

13 61 Solve the inequality Graph the solution on a number line x Solve the nonlinear inequality Express the solution using interval notation and graph the solution set x + x > 10 6 Solve and graph your answer on a real number line: x < 6 64 Solve the nonlinear inequality Express the solution using interval notation and graph the solution set x 1 x x

14 Lines and Coordinate Geometry 65 Graph the line of each equation: a) x + y = 6 66 Find the slope of: a) a line passing through (-4, 4) and (, -5) b) a line parallel to y = x + 7 c) the line whose equation is x y = 1 b) y = x + 4 d) the line whose equation is y = 5 e) a line perpendicular to 6x + 5y = 9 c) x = 67 Determine whether the lines are parallel, perpendicular, or neither Explain your reasoning x + y = 1 x + y = 4 1

15 68 Find an equation, in point-slope form, of the line that satisfies the given conditions Through ( 1, 11); perpendicular to the line passing through (, 1) and (7, 1) 70 Solve the system of equations by graphing x + y = 7 11x y = 1 Then, rewrite the equation in standard form 69 Solve the system of equations by substitution 8x y = 10 x y = 9 71 Solve the system of equations by elimination x y = 6 9y 6x = 9 14

16 7 Solve they system of equations by any method 06x + 16y = 8 005x + 008y = Find the solution set for the following system of inequalities x + y 4 x y > 4 7 Solve the system of equations by any method x y = x 1 10 y = A pair of points is graphed (a) Find the distance between them (b) Find the coordinates of the midpoint 15

17 Name: 78 Find the center and radius of the circle 76 Prove whether or not the points A(4, ), B(, 6), C(7, 5), and D(5, 8) form a square x + y 6x + 4y = 0 (a) The center is (b) The radius is (c) Graph and label important points (d) Domain: Range: 77 Find all points of intersection of the graphs of x + x y = and x + y = 79 If the point 1,1 ˆ lies on the graph of the equation kx xy + y = 5, find the value of k 16

18 Functions 80 If f(x) = x + x +, evaluate each of the following: 81 The graph of a function g is given a) f( ) b) f(m) (a) Find g( 5) c) f(p 5 ) (b) Find g( ) (c) Find g( 1) (d) Find g(1) (e) Find g() d) f(x + h) (f) Find the domain of g (g) Find the range of g (h) Estimate the values of x for which is g(x) = 1 17

19 Name: 8 Consider the following function: f(x) = x 1x Consider the following function: f(x) = x 1x 14 a) Write the quadratic function in vertex form a) Write the quadratic function in vertex form b) State the vertex and whether the graph has a minimum or maximum b) State the vertex and whether the graph has a minimum or maximum c) State the equation of the axis of symmetry c) State the equation of the axis of symmetry d) Find the x intercepts and approximate the values d) Find the x intercepts and approximate the values e) Find the y intercept e) Find the y intercept f) Graph using all the points f) Graph using all the points 18

20 84 Find the domain of the function h(x) = 8x 7 88 Explain how the graph of g is obtained from the graph of f g(x) = 1 f( x 5) 85 Find the domain of the function f(x) = x + 9 x 4 89 Sketch the graph of the function using transformations f(x) = x Find the domain of the function g(x) = x 4x State the domain and range 87 Find the domain of the function g(x) = ln( x) 7x + 90 Determine whether f(x) = neither x x + 4 is even, odd, or 19

21 91 Evaluate the piecewise defined function at the indicated values Ï x + 4x if x f(x) = Ô Ìx if < x 1 9 ÓÔ if x > 1 9 Sketch the graph of the piecewise defined function Ï 1 if x < 1 f(x) = Ô Ìx if 1 x 1 x + if x > 1 ÓÔ (a) Evaluate f( 4) (b) Evaluate f 7 ˆ (c) Evaluate f( ) 9 Write the equation of the piecewise defined function shown below: (d) Evaluate f(0) (e) Evaluate f(5) 0

22 94 Write the following absolute value functions as piecewise functions without absolute value a) f(x) = x 4 95 Write the following absolute value functions as piecewise functions without absolute value a) f(x) = x 9 b) g(x) = 6 x b) g(x) = x + x 1 c) h(x) = 4x x c) h(x) = x + 4x + 4 1

23 96 The graph of a function is sketched below 98 Determine the average rate of change of the function between the indicated values of the variable f(x) = 10 6x + x a) x = 1, x = a) Determine the intervals on which the function is decreasing and which it is increasing a) x = a, x = a + h b) State the relative minimum and maximum values 97 The graph of a function is given Determine the average rate of change of the function between the indicated values of the variable 99 Explain what the difference quotient represents in terms of the graph of the function

24 100 Evaluate the difference quotient for each function a) f(x) = x Given f(x) = 4 9x and g(x) = x 5 6, evaluate (f û g)(x) and (gû f)(x) and simplify as much as possible b) g(x) = x 4x 10 Find f û gû h f(x) = 5 x, g(x) = x, h(x) = x + 4 c) g(x) = x + 5

25 10 Use the given graphs of f and g to evaluate the expression 106 Factor the function completely and then find all zeros of f(x) = x x 8x Find a polynomial with integer coefficients that satisfies the given conditions (gû f)() = Q has degree 4, and zeros, 0, and 7i 104 Find the inverse function of f f(x) = 8 x ˆ Find the inverse function of f 108 Find all vertical and horizontal asymptotes of f(x) = 6x 4x x 6x 8x f(x) = 5x x + 7 4

26 109 Sketch the graph of the function using transformations f(x) = ln( x + 4) Sketch the graph of the function using transformations f(x) = e x Domain: Range: Equation of Asymptote: Calculate the x and y intercepts Domain: Range: Equation of Asymptote: Calculate the x and y intercepts 110 Let f(x) = x + and g(x) = x Which of the following are true? a) g(x) = f 1 (x) for all real values of x b) f û gˆ(x) = 1 for all real values of x c) f(x) is one to one 11 Which of the following are true? a) ln x + yˆ = ln(x) + ln(y) b) ln x y ˆ = y ln(x) c) ln( x) y = y ln(x) d) lnx lny = log y x e) lnx lny = ln(x y) 5

27 Trigonometry 11 Which of the following expressions are equivalent? a) cos x b) cos x c) ( cos x) 114 Which of the following expressions are equivalent? a) ( sinx) 1 b) arcsinx c) sinx 1 d) 1 sinx e)csc x f)sin 1 x 118 Find an angle between 0 and π that is coterminal with the given angle 19π Find the reference angle for the given angle a) Find the terminal point P(x,y) on the unit circle determined by the given value of t = 14π b) 5 c) Convert 150 to radians d) 5π 117 Convert π 4 radians to degrees d) 8π 5 6

28 10 Find the exact value for each trigonometric function (a) sec π 6 1 Find the values of the trigonometric functions of θ from the information given tanθ = 6, sinθ > 0 (a) sinθ (b) csc 7π 6 (b) cosθ (c) cot π ˆ (c) cscθ (d) tan 9π (e) sec( 4π) ˆ (d) secθ (e) cotθ 1 Simplify completely the trigonometric expression cos x sec x + tanx 11 Evaluate the expression without using a calculator sin 45 cos 60 +sin 60 cos 45 7

29 14 Use the addition and subtraction formulas to simplify the expression cos(x + y) cos x cos y 17 Evaluate in radians a) arccos ˆ b) arctan( 1) c) sin 1 1 ˆ 15 Use an appropriate half-angle formula to find the exact value of the expression sin15 d) sec 1 1 ˆ e) arcsin( 1) 18 Evaluate the expression by sketching a triangle sec sin 1 1ˆ 1 16 Use the double angle identity to rewrite the expression sin6x 19 Rewrite as an algebraic expression by sketching a triangle cot sin 1 x ˆ 7 8

30 10 Find all solutions of the equation 4cos x 4cos x + 1 = 0 1 Solve the inequality sin x sinx over 0 x < π 11 Find all solutions of the equation 1 sinx = cos x 14 Given that tanθ =, and θ is in the third quadrant, 5 find the values of: a) sec θ b) sin(θ) 1 Find all solutions of the equation tan ( 5x) 1 = 0 c) tan θ ˆ 9

31 Name: 15 Graph the function and state the amplitude, period, domain, and range Label the values on the x-axis clearly g(x) = sin(x) Graph one period of the function and state the period, domain, and range π π ˆ g(x) = tan x 4 0

32 Problem Solving You must use an ALGEBRAIC process for all of the word problems Correct answers obtained by guess-and-check will not obtain full credit 17 Find four consecutive odd integers whose sum is An orange has 15 calories more than a grapefruit Twenty oranges and ten grapefruits have 1800 calories together Jim ate three oranges and half a grapefruit How many calories did Jim eat? 18 The product of and a number, decreased by 8, is the same as twice the number, increased by 15 Find the number 140 A sports club publishes a monthly newsletter Expenses are $090 for printing and mailing a copy, plus $600 total for research and writing Write a function that represents the total monthly cost of publishing the newsletter and use it to determine the cost of sending 1000 copies 1

33 141 A company manufactures and sells small weather radios If the cost of producting the radio can be expressed as C(x) = 10, x and the revenue produced from the sales can be expressed as R(x) = 50x, how many radios must be produced and sold for the company to make a profit? (If you don t know the meaning of a term, look it up!) 144 Find the area and perimeter of a right triangle with a leg of length 8 and hypotenuse The area of a triangle is 7 in and the base is 8 in Find the height 14 Five times the supplement of an angle is 60 more than its complement Find the angle, its supplement, and its complement 146 In the figure below, find a polynomial expression that represents the area of the shaded region 14 Find the value of x Then, find the perimeter of the triangle

34 147 Phyllis invested $1,000, a portion earning a simple interest rate of 4 1 % per year and the rest earning a rate of 4% per year After one year the total interest earned on these investments was $555 How much money did she invest at each rate? 149 A wire 0 in long is cut into two pieces One piece is formed into a square and the other into a circle If the two figures have the same area, what are the lengths of the two pieces of wire (to the nearest tenth of an inch)? 148 Solve the right triangle in (a) Find b Please give the answer to two decimal places (b) Find r Please give the answer to two decimal places (c) Find m A

35 150 A cylindrical can has a volume of 90π cm and is 10 cm tall 15 The volume of a cone is 80 in Find a function that models the height h of the cone in terms of its radius r a) What is its diameter? b) If the can has a bottom, but no top, what is its sufrace area? 15 A rancher with 650 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle (see the figure) 151 A poster is 1 inches longer than it is wide Find a function that models its area A in terms of its width w (a) Find a function that models the total area of the four pens (b) Find the largest possible total area of the four pens 4

36 Answer Section AP Calculus AB Summer Packet NUMERIC RESPONSE x 4y x 11 y 4 15y bx 5 4g 6g + 6 9x 8 4x 4 y + 49y 6 7 x 4 + x 7x + 6x 9 8 4x 5 810x x 70x + 40x 9 a) ( a 10) ( a + 4) b) y 1 ˆ y + 5 ˆ c) mn( m 5) ( m + 5) d) 7x 10yˆ 7x + 10y ˆ e) 6x 5yˆ p + 7 ˆ 10 a) 5x y x + 4y + y ˆ b) (x + 1) x 5 ˆ c) 5(x 5y) (x + 5y) d) x yˆ e) x yˆ x + yˆ 11 5a b ˆ 5a + 10a b + 4b 4 ˆ 1 x ˆ (x 1) 7x ˆ 4x + 15 x x x 4 x + x 4x x + 15 x x x + 9 x + x 7 a + x 16 a w 17 5 w yˆ x + 4x + 16 ( x + 4) y( x + 1) x y xˆ 1

37 0 1 x + 9 x + x 1 7x (x 1)(x ) x 7 (x 5) (x + 4) a) 1 d) m e) 4 a) 16 b) - 8 c) 5 a) a 4 b 5 b) 6ab 6 a) a b b 7 a) x 4 16(x + 1) ( x 1) 1 4x 5y 7 b) 4 b) 15 1 b) c) 5x 1 9p f) x 9 y g) ˆ x d) x 8 y 4 1,000,000 = c) 15 d) x y 1 a) 5i b) i c) 4 70i d) i e) 4 5 i f) i 4 4ln(x) + 1 ln(x 1) ln(x + ) log x 5 (x 1) 4 ˆ x π y COMPLETION 8 x = 1 9 y = 45 1

38 40 x = 6 41 x = 4,x = 4 a) x = 11d b a 11c 4 a) n = 0,, or 5 44 x = 4 ± i 6 45 x = ± 5 46 x = ± i 47 x = 4 5 i, 4 5 i 48 x = 10,x = 1 b) x = cab a + b c) b = ax x ac or b) x = 10, c) x = 6 5,x = 7 4 ax ac x d) x = 5 or 4 e) x = ± 9 49 x = 1 8 or 7 50 x = x = 1 or 1 5 a) x = 7 b) b = 6 (reject b = 1) 5 x = 5 ln4 + 4ln 54 x = 5ln ln ; x = 8 or 57 ln ˆ 5 58 x = ln 59 x = 7 e 60 x = ± 1 61 x 5 or x < x < 11

39 6 (, 5) (, ) 64 (, 1] (,6] SHORT ANSWER 65 a) 66 a) b) b) c) d) 0 e) c) 5 6, m = Since these slopes are neither equal nor opposite reciprocals, the lines are neither parallel nor perpendicular 68 y + 11 = (x + 1) 67 m1 = x y = 9 69 (-4, -1) 70, 7ˆ No solution (, 1) 4, 7ˆ 4

40 ; (1, ) 76 You need to find the slopes and lengths of all the sides If the consecutive sides are all congruent and perpendicular, then the shape is a square 77 5,7 ˆ and 1,1 ˆ 78 (, ); 4 79 k = ESSAY 80 a) 7 b) 18m + m + c) p 10 + p ; ; ; 1; 0; [ 5, ]; [, ] d) x 4xh h + x + h + 5

41 8 a) f(x) = (x ) 5 b) Vertex (, -5) minimum c) x= d) x intercepts x = ± 5 ± 1 07 and e) f(0) = 7 f) 8 a) f(x) = (x + ) + 4 b) Vertex (-, 4) maximum c) x = - d) x intercepts x = ± ± and 44 e) f(0) = 14 f) ÈÍ Í 7 ˆ 84 ÍÍÍÍ, ÍÎ 8 6

42 85 (, ) (,) (, ) 86 (, 4] [8, ) È 87 7, ˆ ÎÍ 88 reflect over the x axis ; vertical shrink by a factor of ; shift 5 units right 89 D [, ) R (,] 90 odd 91 0; 7 ; ; 0; Ï x if x < Ô Ì 4 x if x if x > ÓÔ 94 a 95 a È 96 a) Dec : ÎÍ, 1 Inc: (, ] [ 1, ) b) rel min value 08 rel max value = , a + h 6 7

43 99 The DQ represents the slope between any two points on the function whose x values are h units apart x + h + + x + ; x + xh + h 4 ; (x + h + 5)(x + 5) x 5 ; 9x 10 f g h(x) ˆ ˆ 5 = x f 1 5 (x) = 8 x 105 f 1 (x) = 7x + x ( x 1) ( x + ) ( x 1) x = 1, x =, x = 1 ˆ 107 Q(x) = x 4 x + 49x 147x 108 y =, x = 4, x = = 5 x 6 + 1x x + 64 Domain ( 4, ) Range (, ) Asy: x = 4 Intercepts: x = 1 e 4, y = ln(4) a and c only 8

44 111 Domain (, ) Range (, ) Asy: y = Intercepts: x = ln, y = 1 11 b and d only CASE 11 a and c 114 a, d, e and b,f ˆ, π π ; 45; 0, π ; π ; ; ; undefined; ˆ 6 7 ; 1 7 ; 7 6 ; 7; sin(x) 14 1 tan(x) tan(y) sin(x) cos(x) 9

45 17 a) 5π b) π 4 (NOT π 4 or 7π 4 ) c) π 6 (NOT 7π 6 or 11π 6 ) d) undefined e) π x x 10 x = π + k π, 5π + k π where k is an integer 11 x = π + π k, where k Z where k is an integer 1 x = π 0 + π k, where k is an integer È π 1 6, 5π ÎÍ 6 [π,π) ; 0 9 ; \ Amplitude = Period = π Domain, ˆ Range È ÎÍ 1,7 10

46 16 Period = 4 Ï Domain Ô Ìx x 4k, where k Z Ô ÓÔ Ô Range, ˆ PROBLEM 17 11, 115, 117, n = 19 0 calories 140 The cost is $1500 for 1000 copies 141 They must sell at least 500 radios 14 angle=45, comp=45, supp = x = 1; P = A = 16 5 P = h = 18in 146 6x $7,000; $6, ; 4175; , d = 6, SA= 69π 151 A(w) = w + 1w 15 h(r) = 40 π r 15 A(w) = 5 w (10 w); 10,565 11