CALCULUS AB/BC SUMMER REVIEW PACKET (Answers)

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1 Name CALCULUS AB/BC SUMMER REVIEW PACKET (Answers) I. Simplify. Identify the zeros, vertical asymptotes, horizontal asymptotes, holes and sketch each rational function. Show the work that leads to your graph. ) f() = 3 ) f() = 4 3 3) f() = Zeros: = ± VA: = ± HA: y = Holes: (0, ) 8 Zeros: = 8 VA: = 4 HA: y = Holes: ( 4, 3 ) Zeros: none VA: = HA: y = 0 Holes: (4, ) 5 (see calculator for graphs) II. Complete the following identities: ) sin + cos = ) cos = sin (= 3) cot + = csc 4) sin = cos (= cos + cos 5) sin() = sin cos 6) cos() = cos sin 7) tan + = sec cos ) ) III. Simplify each epression: sin ) 3) +h = h +3 3 = 3(+3) 5) ) = = (+h) ) 4) 0 5 = = ( 3) (+) + 6) = ) +h = h +h+

2 IV. Solve for z: ) 4 + 0yz = 0 5y ) y + 3yz 8z 4 = 0 4 y 3y 8 V. Epand and Simplify: ) 4 n n=0 = = 5 ) 3 n= = n = VI. Determine each of the following given that f() = {(3,5), (,4), (,7)} g() = 3 h() = {(3,), (4,3), (,6)} k() = + 5 ) (f g)(3) = f(0) = und ) (g k)(7) = g(54) = 5 3) f() ) (kg)() = ( + 5) 3 5) (f + h)() = = 3 6) (k g)(5) = 5 3 7) f () = {(5,3), (4,), (7,)} 8) k () = ± 5 VII. Miscellaneous: Follow directions for each problem. ) Evaluate f(+h) f() and simplify if f() =. + h h ) Epand ( y) 5 using Pascal s Triangle y y 40 y 3 + 0y 4 y 5 3) Simplify 3 ( + 5 ) ) Eliminate the parameter and write a rectangular equation for = t and y = t. 4 = y VIII. Simplify: ) ) e ln 3 = 3 3) e +ln = e, > 0 4) ln = 0 5) ln e 7 = 7 6) log 3 ( 3 ) = 7) log 8 = 3 9) e 3 ln = 3 8) (6a 5 5 3) = 64a 4 0) 4y = 3y 3 3y 5 3 3

3 ) 7 3 = 9 ) (5a 3) (4a 3 ) = 0a 3 6 3) a 3 a a = 4 a+3a 4) ( + ) + ( + ) = 4a 3 ++ (+) IX. Using the point-slope form y y = m( ), write an equation for the line: ) with slope -, containing the point (3,4) y 4 = ( 3) ) containing the points (,-3) and (-5,) y + 3 = 5 ( ) or y = 5 ( + 5) 6 6 3) with slope 0, containing the point (4,) y = 4) perpendicular to the line 3y + =6, containing the point (3,4) y 4 = 3 ( 3) 5) parallel to the line y 3 = 7, containing the point (3,4) y 4 = 3( 3) X. Without a calculator, determine the eact value of each epression: ) sin π = 0 ) sin π = 3) sin 3π 4 = 4) cos 0 = 5) cos 3π = 4 6) cos π = 3 7) tan 7π = 4 8) tan π = ) cot 5π = 3 6 0) sec 4π = 3 ) csc π = 3 3 ) cos (sin ) = 3 3) arcsin = π 4) arctan = π 4 5) sin (sin 7π 6 ) = π 6 XI. Determine all points of intersection: ) Line + y = 8 and line 4 y = 7 (3, 5) ) Parabola y = and line y = 5 + (5, 36) and (-3, -4) 3) Parabola y = 4 and parabola y = + (-, 0) and (, 3) XII. For #-6, determine the function s domain and range. For #7-0, evaluate. f() = 4 g() = 4 h() = ln Function Domain Range ) ln(f()) = ln( 4) (4, ) (, ) ) g() = 4 (, ] [, ) [0, ) 3) f(g()) = 8 (, ) [ 8, ) 4) g(f()) = 8 + (, ) [ 4, ) 5) h() = ln (0, ) (, ) 6) h () = e (, ) (0, )

4 7) g(f(7)) = g(3) = 5 8) f (g()) = f (0) = 4 9) h() = 0 0) h () = e XIII. Solve for, where is a real number. Show the work that leads to your solution. ) = 4 = 6,3 ) 4 3 = 0 = ± XIV. Given the vectors v = -i + 5j and w = 3i + 4j, determine: ) v = i + 5 j ) w v = 5i j 3) length of w = 5 XV. Epress each of these fractions as the sum of two or more fractions with simpler denominators. (This is called partial fraction decomposition.) ) = ) + = ( ) ( ) ) = 3+ 4) = (+) XVI. Sketch a graph of each. ) r = ) r = 3 sec θ 3) r = + sin θ 4) r = cos(3θ) (see calculator for graphs) XVII. Graph each function. Give its domain and range. ) y = sin ) y = e Domain: (, ) Range: [,] Domain: (, ) Range: (0, ) 3 3) y = 4) y = Domain: [0, ) Range: [0, ) Domain: (, ) Range: (, ) 5) y = ln 6) y = + 3 Domain: (0, ) Range: (, ) Domain: (, ) Range: [, ) 7) y = < 0 8) y = { Domain: (, 0) (0, ) 4 > 3 Range: (, 0) (0, ) Domain: (, ) Range: (0, )

5 9) y = 0) y = csc Domain: πn, n Z Domain: (, 0) (0, ) Range: (0, ) Range: (, ] [, ) ) y = ( + ) 3 4 ) y = tan Domain: (, ) Range: [, ) Domain: π + n, n Z Range: (, ) XVIII. Answer the following questions with a graphing calculator. All answers should be accurate to 3 decimal places. ) In your calculator, graph f() = using a window size as follows: min = -0, ma = 0, y min = -00, y ma = 60. Sketch the result below. (see calculator for graph) ) Find all roots (zeros) of f(). =.5, 0,, 5 3) Find all local maima of f(). (.067, 0.0) 5) Find the intervals over which f() is positive. 4) Find all local minima of f(). ( 0.890, 8.483) (3.948, 88.55) 6) Find the intervals over which f() is increasing. 7) Use the table function in your calculator to complete the table. X f () ) In your calculator, graph g() = and h() = using a window size: min = -8, ma = 4, y min = -0, y ma = 50. Sketch the result below. (see calculator for graph) 9) Find all points of intersection of g() and h(). ( 5.773, 6.664), ( 0.787, 0.4), (.760, 0.60) XIX. Function f, defined on the closed interval [-3,3], is graphed to the lower left: Graph of y = f() ) Sketch y = f().

6 ) Sketch y = f( ). 3) Sketch y = f ( ). 4) Sketch y = f( ) + 5) Sketch y = f( ).