HW12.pdf - young(toy68 HW12 villafuerte altu(53785 1 This...

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young (toy68) – HW12 – villafuerte altu – (53785) 1 This print-out should have 22 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0points If f is the function whose graph is given by 2 4 6 2 4 6 which of the following properties does f have? A. local maximum at x = 4 , B. differentiable at x = 2 . 1. neither of them correct 2. B only 3. A only 4. both of them Explanation: A. False: by inspection. B. False: the graph has a vertical tangent at x = 2 so f (2) does not exist. 002 10.0points In drawing the graph P of f the x -axis and y -axis have been omitted, but the point P = (1 , 1) on the graph has been included. Use calculus to determine which of the fol- lowing f could be. 1. f ( x ) = 8 3 + 3 x + x 2 1 3 x 3 2. f ( x ) = 2 3 + 3 x x 2 1 3 x 3 3. f ( x ) = 10 3 5 x + 3 x 2 1 3 x 3 4. f ( x ) = 4 3 + 5 x 3 x 2 + 1 3 x 3 5. f ( x ) = 14 3 3 x x 2 + 1 3 x 3 correct 6. f ( x ) = 8 3 3 x + x 2 + 1 3 x 3 Explanation: From the graph, f ( x ) + as x → ∞ . Of the six given choices, therefore, f must have the form f ( x ) = a + bx + cx 2 + 1 3 x 3 with { b, c } being one of {− 3 , 1 } , { 5 , 3 } , {− 3 , 1 } . Now f ( x ) = b + 2 cx + x 2 , f ′′ ( x ) = 2( c + x ) . Since the graph has an inflection point at x = 1, it follows that c + 1 = 0 ,
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young (toy68) – HW12 – villafuerte altu – (53785) 2 i.e. , b = 3 and c = 1. On the other hand, to determine a we use the fact that f (1) = a + b + c + 1 3 = 1 . Consequently, f ( x ) = 14 3 3 x x 2 + 1 3 x 3 . 003 10.0points Determine which one of the following could be the graph of f ( x ) = x 2 2 x 2 4 (dashed lines indicate asymptotes and solid line indicates the x -axis). 1. 2. 3. cor- rect 4. 5. Explanation: The function f ( x ) = x 2 2 x 2 4 has vertical asymptotes at x = 2 and x = 2. Thus the missing y -axis is a vertical line midway between the two dashed lines in all five graphs. On the other hand, since lim x → ∞ f ( x ) = lim x → −∞ f ( x ) = 1 , the line y = 1 is a horizontal asymptote. Now f (0) = 1 2 , while the solutions of f ( x ) = x 2 2 x 2 4 = 0 are x = ± 2. Thus the graph of f crosses the y -axis between the horizontal asymptote and the x -axis; and it crosses the x -axis to the right of x = 2 and to the left of x = 2. Consequently, the only possible graph of f is
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young (toy68) – HW12 – villafuerte altu – (53785) 3 004 10.0points In the following graph of a function f the x -axis and a vertical asymptote are shown, but the y -axis has been omitted. x Which function f could this be? 1. f ( x ) = radicalbigg x 2 x 2. f ( x ) = radicalbigg 2 x x 3. f ( x ) = radicalbigg 2 x x correct 4. f ( x ) = radicalbigg x 2 x 5. f ( x ) = 1 x 2 x 6. f ( x ) = 1 x 2 x Explanation: The graph of f shows that there exist points a, b on the x -axis with a < b so that (i) f has a vertical asymptote at x = a and an x -intercept at x = b , (ii) f ( x ) < 0 on ( a, b ], while f ( x ) is not defined for x < a .
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