Are You Ready For AP Calculus BC? Show all work on a separate sheet of paper. 1.) Simplify: (a) ! ! !!! ! ! !!!!!" (b) ! ! !!!!! ! ! !! ! !!! ! ! ! ! ! ! ! !" ! (c) !! (d) !!! !! !!! !! 2.) Rationalize the denominator: (a) ! !! ! (b) ! !! ! (c) ! !! !! ! 3.) Write each of the following expression in the form π π ! π ! where c, p, and q are numbers: (a) (!! ! )! ! 9ππ ! (b) (c) ! ! ! ! ! (d) !"!! ! ! !! (e) ! !! (! !! ) ! ! (f) ! !! ! !! ! !! ! !! 4.) Solve for x (do not use a calculator): ! (a) 5 !!! = 25 (b) = 3!!!! (c) log ! π₯ = 3 (d) log ! π₯ ! = 2 log ! 4 β 4 log ! 5 ! 5.) Simplify: (a) log ! 5 + log ! (π₯ ! β 1) β log ! (π₯ β 1) ! 6.) Simplify: (a) log 10! (b) log ! !"! (c) 3! !"#! ! (b) 2 log ! 9 β log ! 3 ! (c) 2 log π₯ + 3 log π₯ ! 7.) Solve the following equations for the indicated variables: ! ! ! ! ! ! ! (a) + + = 1, πππ π (b) π = 2 ππ + ππ + ππ , πππ π (c) π΄ = 2ππ + 2ππβ, πππ πππ ππ‘ππ£π π (d) π΄ = π + πππ, πππ π (e) 2π₯ β 2π¦π = π¦ + π₯π, πππ π (f) !! !! + !!! ! = 0, πππ π₯ 8.) For the equations (a) π¦ = π₯ ! + 4π₯ + 3 (b) 3π₯ ! + 3π₯ + 2π¦ = 0 (c) 9π¦ ! β 6π¦ β 9 β π₯ = 0 complete the square and reduce to one of the standard forms π¦ = π(π₯ β β)! + π or π₯ = π(π¦ β π)! + β 9.) Factor completely: (a) π₯ ! β 16π₯ ! (b) 4π₯ ! β 8π₯ ! β 25π₯ + 50 (c) 8π₯ ! + 27 (d) π₯ ! β 1 10.) Find all real solutions to: (a) π₯ ! β 16π₯ ! = 0 (b) 4π₯ ! β 8π₯ ! β 25π₯ + 50 = 0 (c) 8π₯ ! + 27 = 0 11.) Solve for x: (a) 3 sin! π₯ = cos ! π₯ ; 0 β€ π₯ < 2π (c)tan π₯ + sec π₯ = 2 cos π₯ ; ββ < π₯ < β (b) cos ! π₯ β sin! π₯ = sin π₯ , βπ < π₯ β€ π 12.) Without using a calculator, evaluate the following: (a) cos 210° (e) cos !! ! (b) sin !! ! (f) sin!! (c) tan!! (β1) ! ! (g) tan !! ! (d) sin!! (β1) (h) cos !! β1 13.) Solve the equations: (a) 4π₯ ! + 12π₯ + 3 = 0 (b) 2π₯ + 1 = ! !!! !!! ! (c) β€ 1 (c) π₯ ! + π₯ + 1 > 0 ! β !!! = 0 14.) Solve the inequalities: (a) π₯ ! + 2π₯ β 3 β€ 0 (b) !!!! !!!! 15.) Solve for x: (a) βπ₯ + 4 β€ 1 (b) 5π₯ β 2 = 8 (c) 2π₯ + 1 = π₯ + 3 16.) Determine the equations of the following lines: (a) the line through (β1, 3) and (2, β4); (b) the line through (β1, 2) and perpendicular to the line 2π₯ β 3π¦ + 5 = 0; (c) the line through (2, 3) and the midpoint of the line segment from β1, 4 to (3, 2) 17.) Find the point of intersection of the lines: 3π₯ β π¦ β 7 = 0 and π₯ + 5π¦ + 3 = 0 !!!! 18.) (a) Find the domain of the function π π₯ = (b) Find the domain and range of the functions: iii) β π₯ = π !! β 4 iv) π π₯ = log ! (2π₯ β 3) + 1 v) π¦ = β2 3 β π₯ + 6 19.) Simplify ! !!! !! ! (a) π π₯ = 2π₯ + 3 (b) π π₯ = (a) π π₯ = 2π₯ + 3 (b) π π₯ = ! , where ! ! !!!! . i) π π₯ = 7 ii) π π₯ = ! !!! !!!! !!!! (c) π π₯ = π₯ ! 20.) Find the inverse of the functions: ! !!! !!!! (c) π π₯ = π₯ + 2π₯ β 1, π₯ > 0 21.) You should know the following trigonometric identities. (A) sin(βπ₯) = βsin π₯ (B) cos(βπ₯) = cos π₯ (C) cos(π₯ + π¦) = cos π₯ cos π¦ β sin π₯ sin π¦ (D) sin(π₯ + π¦) = sin π₯ cos π¦ + cos π₯ sin π¦ (a) sin! π₯ + cos ! π₯ = 1 (b) sin 2π₯ = 2 sin π₯ cos π₯ (c) cos 2π₯ = cos ! π₯ β sin! π₯ (d) cos 2π₯ = 2cos ! π₯ β 1 (e) cos 2π₯ = 1 β 2 sin! π₯ (f) cos ! π₯ = !!!"# !! ! (g) sin! π₯ = !!!"# !! ! 22.) Find the derivative of each function. ! (a) π π₯ = 4π₯ ! β 7π₯ + 3 (b) π π₯ = 4π₯ ! + 1 (c) π π₯ = 3 cos ! (2π₯) (d) π π₯ = π !! (e) π π₯ = 3π₯ ! tan π₯ (f) π π₯ = sin!! (3π₯) 23.) Find the integral of each function. ! (a) 3π₯ ! + 7π₯ β 4 ππ₯ (b) 4π₯ 3π₯ ! β 2 ππ₯ (c) sec ! (5π₯) ππ₯ (d) 7π₯π !! ππ₯ 24.) Find the derivative of the equation with respect to x. (a) 2π₯π¦ + π₯ ! β 4 = 0 (b) cos π₯π¦ = 5π₯ β 3π¦ + 9 25.) Solve the differential equations. (a) !" !" = 2π₯π¦ (b) !" !" = !! ! ! (c) !" !" = 5π₯ β π₯π¦ 26.) Graph the parametric or polar equation. (a) π₯ = 4π‘ ! β 1, π¦ = β2π‘ + 1 (b) π = 3 cos 2π (c) π = 2 β 4 sin π (d) π = 2 cos π (e) π = 3
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