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0Preparation for AP Calculus AB: Summer Assignment Due: The first day of class if you are taking AP Calculus AB in the fall, or the first day of spring semester if you are taking the class in the spring. It is assumed you can complete the problems without a calculator. Please complete all of the problems. Most of the material is a review of Algebra 2, so it is expected that you can finish the problems before starting the AP Calculus class. If you experience difficulties you can find assistance online, or find examples on procedure in your Algebra 2 and Precalculus notes . I. Linear Equations There are three accepted forms for writing the equation of a linear function. Those forms are as follows: y mx b ii. Point-slope form y y1 m x x1 Ax By C where A, B, C are integers, A 0 iii. Standard form y y1 other formulas: m 2 {change in y with respect to the change in x } x2 x1 i. Slope-intercept form Problems: 1) Write the equation of the line that passes through the points point-slope form. Convert your answer to standard form. 6,3 , 3,1 in 2) Write the equation of the line that runs perpendicular to the line and passes through the point 1, 2 . Write the equation in point-slope form. 2 y x4 3 4) Sketch the graph of the linear function 3x 4 y 8 3) Sketch the graph of the linear function 4 x 3 y 17 II. Quadratic Equations There are three accepted forms for writing the equation of a quadratic function. Those forms are as follows: i. Standard form ii. Vertex form y ax 2 bx c y a x h k {where h, k is the vertex} y a x xint x xint 2 iii. x -intercept form To determine the solution to a quadratic equation, one can use factoring or the quadratic formula. Problems: 5) Solve by factoring: 3x 2 4 x 7 6) Solve using the quadratic formula: 7) Solve using the quadratic formula: 12 x 7 2 x 2 2 x2 7 4 x 8) Sketch the graph of the following parabolas. Indicate the x intercepts (if they exist), the y intercept, the axis of symmetry, and the vertex. Determine the domain and the range for each function. a) b) c) d) e) f x x2 4x 5 f x 4x x2 f x x2 6x 9 f x x 2 6 x 10 1 2 f x x 4 5 2 III. Polynomials of degree greater than two Review: For every cubic or quartic function in the form of f x ax 3 bx 2 cx d 4 3 2 or f x ax bx cx dx c ; a) the y-intercept is determined by plugging in zero for x. b) if the lead coefficient a is positive, the graph ends UP on the right c) if the lead coefficient a is negative, the graph ends DOWN on the right Examples: a>0 a<0 Problems: 9) Sketch the graph of the cubic f x x3 6 x 2 8 x . 10) Sketch the graph of the cubic 11) Sketch the graph of the quartic f x x3 x 2 9 x 9 . f x x4 5x2 4 . IV Factoring Factoring is a skill, and we use factoring daily in Calculus to solve problems. Please factor completely all of the following: Problems: i. Easy to factor: 12) 13) 14) 15) 16) 17) 18) 19) 20) x 2 14 x 48 x 2 9 x 36 x 2 14 x 24 x 2 13x 48 x 2 21x 20 x 2 13x 30 x 2 12 x 36 x 2 11x 24 x 2 5 x 36 21) x 2 x 30 ii. A little more difficult 5 x 2 33x 18 2 23) 7 x 24 x 20 2 24) 3x 13x 30 2 25) 5 x 26 x 24 2 26) 6 x 37 x 45 2 27) 8 x 2 x 1 2 28) 7 x 19 x 10 2 29) 3x 26 x 16 2 30) 7 x 13x 2 2 31) 10 x 23x 12 22) iii. Difference of squares x 2 25 2 33) x 100 2 34)1 x 2 35) 36 x 25 2 36) 100 x 1 2 2 37) x y 2 38) 49 x 64 32) iv. Factor out the GCF first and then factor again. 3x3 18x 2 48x 3 2 40) 6 x 24 x 24 x 3 41) 2 x 50 x 3 42) 3x 3x 39) 2 x 2 38 x 96 3 2 44)15 x 21x 6 x 43) v. Factor by grouping x3 2 x 2 6 x 12 3 2 46) 10 x 6 x 5 x 3 3 2 47) 2 x x 2 x 1 3 2 48) 2 x 3x 8 x 12 45) V. Trigonometry 49) Please fill in the associated values in the chart below. Angles are in radians. 0 2 3 5 6 4 3 2 3 4 6 7 5 4 3 5 7 11 6 4 3 2 3 4 6 sinx cosx tanx Trigonometric identities you need to know/ Please memorize: opposite hypotenuse adjacent cos hypotenuse opposite sin tan adjacent cos sin 2 cos 2 1 sin tan 2 1 sec 2 cot 2 1 csc 2 1 sin 1 sec cos 1 cos cot tan sin csc Problems: Using the identities on the preceding page, simplify the following: 50) sec x csc x 1 tan x 51) cot x 52) 1 1 sec x 1 sec x 1 53) cot 2 x 1 csc x 1 sin x 1 cos x 54) Sketch the graph of f graph is __________. 55) Sketch the graph of f graph is __________. 56) Sketch the graph of x cos x on the interval 0, 2 . The range for this x sin x on the interval 0, 2 . The range for this f x tan x on the interval 2 , 2 . The range for this graph is __________. 57) Solve the following trigonometric equation for on the interval 0,2 ; 2sin 2 1 0 58) Solve the following trigonometric equation for on the interval 0,2 ; 59) Solve the following trigonometric equation for on the interval 0,2 ; sec sin 2sin 0 2sin cos tan 0 VI. Exponents and Logarithms Problems: 3 2 43x 8 6 26 61) Solve: log 2 x 3 log 2 x log 2 x 2 2 3 62) Solve: log x 27 2 125 x 5 125 63) Solve: 25 x 2 64) Simplify: log 8 2 60) Solve: VII Graphing Problems: 65) Sketch the graph of 66) Sketch the graph of this function. 67) Sketch the graph of 68) Sketch the graph of function. 69) Sketch the graph of 70) Sketch the graph of 71) Sketch the graph of f x x . State the domain and range of this function. f x x 3 2 . f x x . State the domain and range of State the domain and range of this function. f x x 4 1. State the domain and range of this 1 . State the domain and range of this function. x 1 f x 2 . State the domain and range of this function. x 1 f x . State the domain and range of this function. x3 f x 72) Sketch the graph of f x x2 . 2 x x6 function. VIII Limits Problems: Determine the following limits if they exist. x4 x 2 2x 8 x 5 74) lim x 25 x 25 x 3 75) lim x 3 x 3 73) lim x 4 2x 2 x 1 76) lim x x3 1 3x 4 77) lim x 2 x x3 8 78) Let f x x2 a) lim x 2 f x = b) lim x f x = State the domain and range of this