Exam 2 - . Express the area of a triangle in terms of . 10....

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MATH 2010 EXAM 2, March 11, 2011 Write your solutions in a bluebook. 1. (8 points) Find f 0 ( x ) if f ( x ) = sin( x 4 + e x 2 ). 2. (8 points) Express f 0 ( x ) in terms of g , g 0 and x if f ( x ) = g ( x 4 ) + ( g ( x )) 4 . 3. (10 points) Let y = f ( x ) be defined implicitly by the equation xy + x 4 y 2 - 6 = 0 . a) Evaluate f (1), assuming f (1) < 0. b) Express y 0 in terms of x and y . 4. (8 points) Find the derivative of f ( x ) = ln(1 + x 6 ) + tan - 1 ( x 2 ). 5. (8 points) Find f 0 ( x )) if f ( x ) = (7 + x 2 ) 7+ x 2 . 6. (8 points) Suppose the position of an object moving on a line at time t is given by f ( t ) = - 2 t 3 + 3 t 2 , where -∞ < t < . Find the intervals where the object is speeding up. 7. (10 points) a) Find dy if y = x 1 / 5 . b) Use the differential approximation to estimate (32 . 08) 1 / 5 - (32) 1 / 5 . 8. (10 points) Suppose that x and y are functions of time t and are related by the equation 2 x 3 + y 3 = 29. Find the rate of change of x with respect to t when x = 1, if y is decreasing at the rate of 3 units per second when x = 1. 9. (5 points) Two sides of a triangle have length 3 and 5 and the angle between them is
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Unformatted text preview: . Express the area of a triangle in terms of . 10. (10 points) Find the absolute maximum and minimum of the function f ( x ) = 32 x 1 3-x 4 3 on the closed interval [-1 , 27]. (For evaluating outputs, note that f ( x ) = x 1 3 (32-x ) . ) 11. (15 points) Sketch the graph of the function f based on the following information; (Show limits, intervals of increase and decrease, concavity, x coordinates for local maxima, minima, and points of inection.) f is continuous on (0 , ) . lim x - f ( x ) = . lim x f ( x ) = . f ( x ) = 4 x +12 x 2 +16 . f 00 ( x ) =-4 ( x +8)( x-2) x 2 +16 . f (0) = 0 ....
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