AP CALCULUS summer2009 workshopII

AP CALCULUS summer2009 workshopII - AP CALCULUS SUMMER...

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AP CALCULUS SUMMER WORKSHOP II Problems : 1) Solve for x : a) x x 42 17 16 0 −+ = b) x x x 32 57 3 0 ++ + = c) () xx x 14 1 2 3 22 = d) +− = 31 0 0 4 e) 21 8 23 2 0 2 = f) ()() 2 2 2 26 120 0 −− += g) 2 2 12 8 + = h) 27 3 0 2 cos cos x x = , [] x 02 , π i) 4 8 2 6 22 26 0 2 sin sin +− − + = , x , 2) Let fx a x x x b =− + . When fx is divided by x 1 the remainder is 6 and when is divided by x + 2 , the remainder is 9. Find the values of a and b . 3) If x =+ 1 2 12 , then what is the value of 44 2 x x ? 4) Solve for x and y : 16 32 8 83 2 4 2 4 xy x y = = 5) Prove the following trigonometric identities: a) 1 1 2 2 2 + + = tan cot tan θ b) csc cot csc 2 x x x −= c) sin csc cot cos x x x x 1 d) sin cos cot 2 = e) 1 1 1 2 2 = tan sin sin cos sin x x x f) 1 11 + + = sin cot cos tan sin cos α αα 6) a) Show that ( ) ( ) cos cos cos sin sin xxx x x 24 1 8 8 = b) Find the exact values of i) cos cos cos πππ 7 2 7 4 7 ii) cos cos cos 9 2 9 4 9 iii) sin cos cos 16 16 8 iv) ( )( )( ) sin sin sin 10 50 70 ooo Hint : If is acute, then sin cos θθ 90 o and sin sin 180 o v) ( )( )( )( ) sin sin sin sin 64 26 67 8 oooo 7) Consider the function fx x x 1 , x 0. a) Show that ( ) can also be written as x x + 1 2 2 , x b) Let gx x x 2 2 1 , x What is the minimum value of and when does it occur ? c) Let hx x x x + sin sin sin 13 , ( ) x 0, . What is the minimum value of and when does it occur ? 8) What is the domain and range of = 1 2 ? Hint : First try sketching the function yx x =−+ 2 and then make some observations from your sketch.
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Answers : 1) a) Using the substitution u x = 2 , x x 42 17 16 0 −+ = is transformed into uu 2 17 16 0 = which factors to () ( ) −− = 16 1 0 . Thus, u x == 16 2 or u x 1 2 which implies that x 4 or x = ± 1 .
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This note was uploaded on 07/27/2010 for the course MATH MATH 1025 taught by Professor Tanny during the Spring '09 term at York University.

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AP CALCULUS summer2009 workshopII - AP CALCULUS SUMMER...

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