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16Amathfinal - Math 16a Problem List Spring 2008 H Woodin 1...

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Math 16a Problem List Spring 2008 H. Woodin 1. For each of the following, determine if the limit exists and compute the limit if it does exist. (a) lim x 0 x q 1 + (1 /x 2 ) (b) lim x 0 x 2 q 1 + (1 /x 4 ) 2. Find the points on the graph of y = x 3 + 1001 where the tangent is parallel to the line y = 3 x . 3. Find the derivative of y = ( x 3 + x 2 + 2001) 16 at x = 1. 4. Find the equation of the line which is tangent to the curve y = 4 x 1 / 4 at the point where x = 16. 5. Suppose f ( x ) = | 2 x + 1 | 3 . Find f 0 ( x ). 1
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6. Suppose f ( x ) is a function with domain (0 , ) and that f ( x ) = x 2 - 4 x + 3 x 2 - 1 for x 6 = 1 and that f (1) = a . Suppose f ( x ) is continuous at x = 1. Find a . 7. Using the derivative, find an approximate value of ln 3 in terms of e by using the fact that ln e = 1. 8. Suppose g ( x ) = 1 3 x 3 - x . (a) Identify the inflection points of g ( x ) or explain why there are none. (b) Find the maximum value of g ( x ) for 0 x 2. (c) Does g ( x ) have a mimimum value for x > 2? Why? 9. Consider the curve defined by the equation x 3 + y 2 x = y . (a) Find dy/dx by implicit differentiation. (b) Find the point ( a, b ) on the curve where a > 0 and the line tangent to the curve is vertical. 10. Find the minimum possible value of 3 a + 5 b given that a > 0, b > 0 and that a · b = 75.
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