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Unformatted text preview: rabbani (tar547) – Homework12 – Fouli – (58395) This print-out should have 22 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points 003 10.0 points 1 Find the value of f (0) when f ′ (t) = 3 sin 2t , f π 2 = −1 . Find all functions g such that g ′ ( x) = √ 4 x2 + x + 3 √ . x 1. f (0) = −4 2. f (0) = −5 3. f (0) = −3 4. f (0) = −6 5. f (0) = −7 004 10.0 points cos 2x , 4 cos2 x , 2 1. g (x) = x 42 1 x + x+3 +C 5 3 √ 2. g (x) = 2 x 4x2 + x − 3 + C √ 3. g (x) = 2 x 42 1 x + x+3 +C 5 3 √ 4. g (x) = 2 x 4x2 + x + 3 + C 5. g (x) = √ x 4 x2 + x + 3 + C 42 1 x + x−3 +C 5 3 10.0 points Consider the following functions: (A) (B ) (C ) F1 (x) = √ 6. g (x) = 2 x 002 F2 (x) = − F3 (x) = sin2 x . Which are anti-derivatives of f (x) = sin x cos x ? Determine f (t) when f ′′ (t) = 4(3t + 1) and f (1) = 6, ′ 1. F2 only f (1) = 1 . 2. F3 only 3. none of them 4. F1 and F3 only 5. F1 and F2 only 6. F2 and F3 only 7. F1 only 8. all of them 005 10.0 points 1. f (t) = 2t3 − 4t2 + 4t − 1 2. f (t) = 2t3 + 2t2 − 4t + 1 3. f (t) = 6t3 + 4t2 − 4t − 5 4. f (t) = 6t + 2t − 4t − 3 5. f (t) = 2t3 − 2t2 + 4t − 3 6. f (t) = 6t3 − 4t2 + 4t − 5 3 2 rabbani (tar547) – Homework12 – Fouli – (58395) Find f (π/2) when 1 2 f ′ (t) = 3 cos t + 4 sin t 3 3 and f (0) = −1. 1. f (π/2) = 2. f (π/2) = 3. f (π/2) = 4. f (π/2) = 5. f (π/2) = 17 2 11 2 13 2 19 2 15 2 10.0 points 1. F (3) = 2. F (3) = 101 4 103 4 2 3. F (3) = 25 4. F (3) = 51 2 5. F (3) = 26 008 10.0 points 006 Let f be a twice-differentiable function and let g be its inverse. Consider the following equations: A. f ′′ (g (x))(g ′(x))2 = f ′ (g (x))g ′′(x) ; g ′ ( x) = 1 f ′ (g (x)) 1 . f ( x) ; A particle moves along the x-axis so that its acceleration at time t is a ( t) = 3 − 4 t in units of feet and seconds. If the velocity of the particle at t = 0 is 2 ft/sec, how many seconds will it take for the particle to reach its furthest point to the right? 1. 6 seconds 2. 3 seconds 3. 4 seconds 4. 5 seconds 5. 2 seconds 007 10.0 points B. C. g ( x) = Which ones do f, g satisfy? 1. all of them 2. A and B only 3. C only 4. A and C only 5. B and C only 6. A only 7. none of them 8. B only 009 10.0 points If F = F (x) is the unique anti-derivative of f (x) = (4 − x)2 + 6 (4 − x)2 Find the value of g ′ (1) when g is the inverse of the function f (x) = 2 sin x, −π/2 ≤ x ≤ π/2 . which satisfies F (0) = 0, find F (3). rabbani (tar547) – Homework12 – Fouli – (58395) 1 1. g ′ (1) = − √ 3 1 2. g ′ (1) = √ 3 3. g ′ (1) = 1 4. g ′ (1) = −1 1 5. g ′ (1) = − √ 2 1 6. g ′ (1) = √ 2 010 10.0 points Suppose g is the inverse function of a differ1 . entiable function f and G(x) = g ( x) 1 If f (5) = 6 and f ′ (5) = , find G′ (6). 25 1. G′ (6) = 6 2. G′ (6) = −1 3. G′ (6) = 8 4. G′ (6) = −2 5. G′ (6) = −3 011 10.0 points 3. g ′ (3) = 4. g ′ (3) = 5. g ′ (3) = 3 π 2 π π 2 012 10.0 points 3 Find the value of g ′ (−1) when g is the inverse of the function f defined by f ( x) = 3 x 3 − (Hint: f (1) = −1.) 1. g ′ (−1) = 3 2. g ′ (−1) = 1 13 1 13 4 . x 3. g ′ (−1) = − 4. g ′ (−1) = 13 5. g ′ (−1) = −13 6. g ′ (−1) = −3 013 10.0 points 1 + 6x . 7 − 5x Find the inverse of f ( x) = πx 2 1. f −1(x) = 2. f −1(x) = 3. f −1(x) = 4. f −1(x) = 7x − 1 6x + 5 7x + 1 6x + 5 7x + 1 5x + 6 7x − 1 5x + 6 On (−1, 1) the function f (x) = 3 + x2 + tan has an inverse g . Find the value of g ′ (3). (Hint: find the value of f (0)). 1. g ′ (3) = 1 2. g ′ (3) = π 3 rabbani (tar547) – Homework12 – Fouli – (58395) 5. f −1 (x) = 6x − 1 5x + 7 10.0 points 016 10.0 points 4 014 Find the inverse function, f −1, for f when f ( x) = 9 − x2 , 0 ≤ x ≤ 3. Find the inverse function, f −1 , of f ( x) = √ 1. f −1 (x) = 2. f −1 2x . 3 x2 + 4 √ √ 1. inverse doesn’t exist 2. f −1(x) = 3. f −1(x) = 4. f −1(x) = √ 5. f −1(x) = 017 9 − x2 , x2 − 9 , 1 , 9 − x2 9 − x2 , 0≤x≤3 −3 ≤ x ≤ 3 0≤x<3 −3 ≤ x ≤ 3 4 − 4 x2 3 4 + 3 x2 2x 3 4 − 4 x2 2x 4 − 3 x2 2x 4 + 3 x2 4 − 3 x2 3x 10.0 points ( x) = 3. f −1 (x) = √ 4. f −1 (x) = √ 5. f −1 (x) = √ 6. f −1 (x) = √ 10.0 points 015 Which of the following functions will fail to have an inverse? Find the inverse function, f −1 , of f when f is defined by f ( x) = 1. f −1 (x) = 2. f −1 (x) = 3. f −1 √ 4x − 7 , 7 x≥ . 4 4 7 12 ( x + 7), x ≥ 4 12 ( x + 7), x ≥ 0 4 7 4 4 7 4 1.3 2 1 0 -1 -2 -3 -4 -5 4 2.3 2 1 0 -1 -2 -3 -4 -5 4 2 −4 −2 −2 −4 2 4 1 (x) = ( x2 − 4), x ≥ 7 1 7 x2 + 4 , x ≥ x2 x2 -5 -4 -3 -2 -1 0 1 2 3 4 4 2 −4 −2 −2 −4 2 4 4. f −1 (x) = 5. f 6. f −1 1 ( x) = 4 1 ( x) = 7 − 7, x ≥ 0 − 4, x ≥ 0 −1 -5 -4 -3 -2 -1 0 1 2 3 4 rabbani (tar547) – Homework12 – Fouli – (58395) 5 4 3.3 2 1 0 -1 -2 -3 -4 -5 4 5 4 2 −4 −2 −2 −4 2 4 -5 -4 -3 -2 -1 0 1 2 3 4 10 9 8 7 1. 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 10 9 8 7 2. 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 10 9 8 7 3. 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 8 4 −8 −4 4 −4 −8 -10-8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8 9 -9 8 4.3 2 1 0 -1 -2 -3 -4 -5 4 5.3 2 1 0 -1 -2 -3 -4 -5 4 2 −4 −2 −2 −4 2 4 8 4 −8 −4 4 −4 −8 -10-8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8 9 -9 -5 -4 -3 -2 -1 0 1 2 3 4 8 4 2 −4 −2 −2 −4 2 4 8 4 −8 −4 4 −4 −8 -10-8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8 9 -9 -5 -4 -3 -2 -1 0 1 2 3 4 018 10.0 points 8 If the graph of f is 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 8 4 −8 −4 −4 −8 -10 -8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8 9 -9 9 8 7 4. 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 8 4 −8 −4 4 −4 −8 -10-8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8 9 -9 4 8 8 which of the following is the graph of f −1 (x)? rabbani (tar547) – Homework12 – Fouli – (58395) 10 9 8 7 5. 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 6 8 4 −8 −4 4 −4 −8 -10-8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8 9 -9 Consider the following properties that a function f might have: A. graph of f passes horizontal line test; 8 C. f is 1 − 1. B. f ′ (x) ≥ 0 for all x; Which properties always ensure that f has an inverse f −1 ? 10.0 points 1. all of them 2. B only 3. A only 4. A and B only 5. B and C only 6. none of them 7. A and C only 019 Consider the following functions: F1 (x) = sin x, F2 (x) = x2 + 2x, F3 (x) = |x + 1|, 1. none of them 2. F2 only 3. F1 and F2 only 4. F2 and F3 only 5. all of them 6. F3 only 7. F1 only 8. F1 and F3 only 020 ′ 0 ≤ x ≤ 2π/3 , −1 ≤ x ≤ 1 , −1 ≤ x ≤ 1 . Which have an inverse on the given domain? 8. C only 022 10.0 points If the graph of f is 10.0 points Use f (x) to determine whether x4 − 2 x2 4 has an inverse on (−∞, ∞). f ( x) = 1. f does not have inverse 2. f has inverse 021 10.0 points which one of the following contains only graphs of anti-derivatives of f ? rabbani (tar547) – Homework12 – Fouli – (58395) 7 1. 6. 2. 3. 4. 5. ...
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This note was uploaded on 04/13/2010 for the course M 408 K taught by Professor Jouve during the Fall '08 term at University of Texas.

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