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Unformatted text preview: rabbani (tar547) – Homework10 – Fouli – (58395) This print-out should have 17 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Let f be the function defined by f (x) = x − cos 2x, −π ≤ x ≤ π . 1 Determine all interval(s) on which f is decreasing. π 1. [− π , − 12 ], [ π , 6 6 11π 12 ] 11π 12 ] Determine the increasing and decreasing properties of the function f (x) = (x − 1) (x + 2) on its natural domain. 7 7 1. inc: [− 5 , 1], dec: [−2, − 5 ] ∪ [1, ∞) 7 2. inc: [−2, − 5 ], dec: [− 7 , ∞) 5 7 7 3. inc: (−∞, − 5 ] ∪ [1, ∞), dec: [− 5 , 1] 4 5 1 5 2. [− 5π , − π ], [ π , 12 6 6 3. [−π, − 5π ], [ 7π , π ] 12 12 π 4. [− 5π , − 12 ], [ 7π , 12 12 π 5. [− 5π , − π ], [ 38 , 12 8 11π 12 ] 11π 12 ] 004 (part 1 of 2) 10.0 points Let f be the function defined by 1 1 f ( x) = x 2 + x2 − x4 . 3 5 (i) Determine the derivative of f . 1. f ′ (x) = (1 − x2 )(2 − x2 ) 2. f ′ (x) = (1 + x2 )(3 − x2 ) 3. f ′ (x) = (1 + x2 )(2 − x2 ) 4. f ′ (x) = (1 − x2 )(2 + x2 ) 5. f ′ (x) = (1 + x2 )(3 + x2 ) 6. f ′ (x) = (1 − x2 )(3 + x2 ) 005 (part 2 of 2) 10.0 points (ii) Find the interval(s) on which f is decreasing. 4. inc: (−∞, −2] ∪ [1, ∞), dec: [−2, 1] 7 7 5. inc: [−2, − 5 ] ∪ [1, ∞), dec: [− 5 , 1] 002 10.0 points Find all values of x at which the graph of y = x2 − 4 cos x changes concavity on (−π/2, π/2). ππ 1. x = − , 33 π 2. x = 6 ππ 3. x = − , 66 π 3 π 6 003 10.0 points 4. there are no values of x 5. x = − 6. x = π 3 7. x = − 1. ( − ∞, −1 ], [ 1, ∞) rabbani (tar547) – Homework10 – Fouli – (58395) √ √ 2. (−∞, − 3 ], [ 3, ∞) √√ 3. [− 3, 3 ] √√ 4. [− 2, 2 ] 5. [ − 1, 1 ] √ √ 6. (−∞, − 2 ], [ 2, ∞) 006 10.0 points 4. local maximum at x = −4, 2 5. local maximum at x = 4 008 10.0 points 2 Let f be the function defined by f (x) = 1 + x2/3 . Consider the following properties: A. has local maximum at x = 0 Which does f have? 1. A only 2. B only 3. both of them 4. neither of them 009 10.0 points B. concave down on (−∞, 0) ∪ (0, ∞) Find all points x0 at which 4+x f ( x) = (x + 2)2 has a local minimum. 1. x0 = −4 2. x0 = −4 , −2 3. x0 = 6 4. no such x0 exist 5. x0 = 6 , −6 6. x0 = −2 7. x0 = −6 007 10.0 points The derivative of a function f is given for all x by f ′ (x) = (2x2 + 4x − 16) 1 + g (x)2 where g is some unspecified function. At which point(s) will f have a local maximum? 1. local maximum at x = −4 2. local maximum at x = 2 3. local maximum at x = −2 If f is a continuous function on (−5, 3) whose graph is 4 2 −4 −2 2 which of the following properties are satisfied? A. B. C. f has exactly 4 critical points, f has exactly 2 local extrema, f ′′ (x) > 0 on (−5, −3). rabbani (tar547) – Homework10 – Fouli – (58395) 1. A and B only 2. none of them 3. B and C only 4. C only 5. A only 6. all of them 7. B only 8. A and C only 010 10.0 points 8 7 6 3. 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 3 6 4 2 −6 −4 −2 −2 −4 −6 1234567 2 46 Which of the following is the graph of f ( x) = 8 7 6 1. 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 7 6 2. 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 x−2 ? x+1 6 4 2 −6 −4 −2 −2 −4 −6 6 4 2 −6 −4 −2 −2 −4 −6 24 6 24 6 -8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 -8-7-6-5-4-3-2-10 7 6 4. 6 5 4 4 3 2 2 1 0 -1 -2 −6 −4 −2 −2 -3 -4 −4 -5 -6 −6 -7 -8 8 -8-7-6-5-4-3-2-10 7 6 5. 6 5 4 4 3 2 2 1 0 -1 -2 −6 −4 −2 −2 -3 -4 −4 -5 -6 −6 -7 -8 -8-7-6-5-4-3-2-10 7 6 6. 6 5 4 4 3 2 2 1 0 -1 -2 −6 −4 −2 −2 -3 -4 −4 -5 -6 −6 -7 -8 -8-7-6-5-4-3-2-10 2 46 1234567 2 46 1234567 2 46 1234567 011 10.0 points Which of the following statements about the absolute maximum and absolute mini- -8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 rabbani (tar547) – Homework10 – Fouli – (58395) mum values of f ( x) = x3 − x2 − 3 x − 1 x+1 1. 4 on the interval [0, ∞) are correct? 1. abs. max. = 7, abs. min. = −2 2. no abs. max., abs. min. = −2 3. abs. max. = 2, no abs. min. 2. 4. no abs. max., abs. min. = −1 5. abs. max. = 7, 012 Find the value of x→∞ no abs. min. 10.0 points lim f (x) 3. when 4 x2 − 3 2 x3 + 3 x + 1 ≤ f ( x) < . 2 x2 x3 1. limit = 4 2. not enough information given 4. 3. limit = 1 4. limit = −∞ 5. limit = 2 6. limit = ∞ 013 10.0 points 5. The following graphs have similar horizontal asymptotes, as indicated by the dashed lines, and each graph passes through the origin. Decide which one of them is the graph of 4x . f ( x) = √ x2 + 1 rabbani (tar547) – Homework10 – Fouli – (58395) 6. 5. limit does not exist 016 Determine if 5 x3 − 3 x lim x → ∞ 2 x3 + 4 x 2 + 5 exists, and if it does, find its value. 1. limit = 3 2 10.0 points 5 014 10.0 points Find all asymptotes of the graph of y= 3x2 − 5x − 12 . 3x2 − 11x + 6 1. vert: x = 3, horiz: y = 1 2 2. vert: x = − , horiz: y = 1 3 2 3. vert: x = , horiz: y = 1 3 2 4. vert: x = , horiz: y = −1 3 5. vert: x = −3, horiz: y = −1 015 10.0 points 2. limit = 3 3. limit = 5 2 4. limit = 1 5. limit = 2 6. limit does not exist 017 10.0 points A certain function f is known to have the properties x → −∞ Which of the following is the graph of f ( x) = x3 ? x2 − 9 lim f ( x) = 2 , x→∞ lim f (x) = 8 . Determine if x → 0+ lim 5 + 4x 1 1+f x exists, and if it does, compute its value. 1. limit = 3 2. limit = 3. limit = 5 9 4 9 5 4. limit = 3 11 10 9 1. 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 8 4 −8 −4 −4 −8 -12 -10 -8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8 9 10 -11 -9 11 4 8 rabbani (tar547) – Homework10 – Fouli – (58395) 11 10 9 2. 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 6 8 4 −8 −4 −4 −8 4 8 -11 -9 11 11 -12 -10 -8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8 9 10 10 9 3. 8 8 7 6 5 4 4 3 2 1 0 -1 -2 −8 −4 4 8 -3 -4 −4 -5 -6 -7 -8 −8 -9 -10 -11 -12 12 -12 -10 -8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8 9 10 11 11 -11 -9 10 9 4. 8 8 7 6 5 4 4 3 2 1 0 -1 -2 −8 −4 4 8 -3 -4 −4 -5 -6 -7 -8 −8 -9 -10 -11 -12 -12 -10 -8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8 9 10 -11 -9 11 ...
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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 at Austin.

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