extramidterm2 - x : dy dx cos y. (c) Compute the derivative...

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Math 1a – Practice Midterm 2 GSI: David Freeman November 11, 2005 Part I Each question is worth 2 points. 1. Suppose f and g are increasing functions on an interval I . Which one of the following must be true? (a) f ( x ) > 0 for all x in I . (b) g 00 ( x ) > 0 for all x in I . (c) fg is increasing on I . (d) f - g is decreasing on I . (e) 1 /f is increasing on I . (f) g 3 is not decreasing on I . 2. Suppose f ( x ) is defined and twice differentiable for all real x . Which one of the following could be a possible description of f and its first and second derivatives? (a) f ( x ) < 0, f 0 ( x ) < 0, and f 00 ( x ) > 0 for all x . (b) f ( x ) has two local minima and f 00 ( x ) has three zeroes. (c) f (1) = - 2, f (3) = 0, and f 0 ( x ) > 0 for all x . (d) f ( x ) has two local maxima and f 00 ( x ) has one zero. (e) f ( - 1) = 0, f (0) = - 1, f 00 ( x ) < 0, and f 0 ( x ) ≤ - 1 for all x . (f) f ( x ) has three zeroes and no local maxima. 1
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Part II Each question is worth 3 points. 1. Answer all of the following. (a) Find a function f ( x ) that is defined for all x > 0, has f (1) = 1, and has f 0 ( x ) = 1 /x . (b) Differentiate the following with respect to
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Unformatted text preview: x : dy dx cos y. (c) Compute the derivative of f ( x ) = x e x . 2. Compute the following limits. (a) lim x cosh x-1 x 2 (b) lim x 1 + 3 x x (c) lim x + 1 x + ln x 3. Let f ( x ) be the following function, dened for all positive real numbers: f ( x ) = 3 + ln x &lt; x &lt; 1 3 cos( x/ 2) 1 x 5 tan-1 x x &gt; 5 . . If x is your answer to this question, then your score is f ( x ), rounded down to the nearest integer if necessary. If f ( x ) is negative you will get a score of zero. 4. Sand falls from a conveyor belt at a rate of 10 m 3 /min onto the top of a conical pile. The height of the pile is always equal to the diameter of the base. How fast is the height changing when the pile is 4 m high? 5. Sketch a graph of the function y = e-x 2 . 2...
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extramidterm2 - x : dy dx cos y. (c) Compute the derivative...

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