Ch 2 Rules of Differentiation Review Assignment Answers

Ch 2 Rules of Differentiation Review Assignment Answers -...

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CHAPTER 2 REVIEW MATH 31 NAME_______________ 1. Differentiate each of the following. All answers should be left in the simplest possible factored form. (You may wish to simplify before differentiating.) a) y = 3x 7 9x 2 + 7x 2 y / = ______________________ b) y = 2 3 2 2 8 3 x y / = _______________________ c) y = 3 2 2 11 5 x x y / = _______________________ d) y = 3 ) 3 ( 6 2 x x y / = _______________________ e) y = 3x 2 (2x 3)(2x + 3) y / = _______________________ f) y = 8 2 8 6 2 x x x y / = _______________________
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2. Differentiate. Leave answers in simplest factored form. (2 marks each) a) y = (3x 1) 3/ 2 + ( 3x 1) 5/ 2 y / = _______________________ b) y = (3x + 1) 3 (2x 3) 2 y / = _______________________ c) y = 1 ) 5 2 ( 2 x x y / = _______________________ d) y = (5x 3)(2x + 5) 4/ 3 y / = _______________________ e) 3x 2 y 2y 2 + x 3y = 8 y / = _______________________
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3. Differentiate but do not factor. a) y = 3 3 2 5 x x y / = _____________________________ b) y = 3 4 3 ) 5 ( 7 x x y / = _____________________________ 4. At what value of x would the tangent line to the curve y = 2x 2 + 5x 4 be parallel to the line 3x + y 7 = 0. 5. Find the equations of the tangent lines to each of the following curves at the given points: a) at (3,5) on y = 2x 2 8x + 11 b) at (5,4) on x 2 y 2 = 9 6. Evaluate the second derivative, f // (x), at the point (2 , 27) on the curve f(x) = (2x 1) 3 . f // (2) = ______________
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7. Using first principles , the definition f / (x) = h x f h x f h ) ( ) ( lim 0 , to find f / (x) for: a) f(x) = 3x 2 2 b) f(x) = 1 2 3 x 8. By using suitable substitutions show that the second derivative of x 2 y 2 = 9 can be expressed in the form of 3 2 2 y k dx y d for some constant k. Answers: 1. a) y / = 21x 6 18x +7 b) y / = 192x(8x 2 3) 11 c) y / =(1/3)(5x 2 11x + 2) 2/3 (10x 11) d) y / = ( 4)/( x+ 3 ) 3 e) y / = 6x(8x 2 9) f) ½ 2. a) y / = (3/2)(3x 1) 1/2 (15x 2) b) y / = (2x 3)(3x + 1) 2 (30x 23) c) y / = (3/2)(x + 1) 3/2 (2x + 5)(2x + 1) d) y / = (1/3)(2x + 5) 7/3 (99 10x) e) y / = (6xy + 1) /(3 + 4y 3x 2 ) 3.a) y / = 2 1 3 2 2 1 ) 3 2 ( 5 ) 3 2 ( 5 3 1 x x x 3. b) y / = 3 1 3 2 3 1 3 4 3 2 3 ) 5 ( 3 20 ) 7 ( 3 7 ) 5 ( 3 4 7 3 1 x x or x x 4. x = 2 5. a) 4x y 7 = 0 5. b) 5x 4y 9 = 0 6. f // (2) = 72 7. See Key 8. k = 9 from y // = (x/y 2 )y / + 1/y
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CHAPTER 2 REVIEW MATH 31 NAME_______________ 1. Differentiate each of the following. All answers should be left in the simplest possible factored form. (You may wish to simplify before differentiating.) a) y = 3x 7 9x 2 + 7x 2 y / = (7)(3)x 7-1 ─ (2)(9)x 2-1 + (1)(7)x 1-1 ─ 0 y / = 21x 6 ─ 18x + 7 b) y = 2 3 2 2 3 2 12 2 2 2 8 3 8 3 8 3 x x x x x Simplify first!  power power 1 d d expression power expression expression dx dx y / = 12 1 2 2 12 8 3 8 3 d x x dx y / = 11 11 2 2 12 8 3 16 0 192 8 3 x x x x c) y = 1 3 2 2 3 5x 11x 2 5x 11x 2 1 1 / 2 2 3 1 d y 5x 11x 2 5x 11x 2 3 dx 2 / 2 3 1 y 5x 11x 2 10x 11 3 d) y = 2 3 3 2 2 x 3 2x 6 2 2 x 3 (x 3) (x 3) (x 3) Simplify first!
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