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16A-WQ03

# 16A-WQ03 - MAT 16A(A002 NAME 1 2 3 4 total pinl ixm Problem...

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MAT 16A (A002) NAME: March 20, 2003 1 2 3 4 total Problem 1. (estimated time: 25mn ) (50 points) (a) Which of the following lines are parallel? Which of them are perpendicular? L 1 : 1 3 x - 1 2 y + 1 = 0 L 2 : y = - 3 2 x + 5 L 3 : x - 3 y + 3 = 0 L 4 passing by the points of coordinates (1 , 1) and (4 , 3) . (b) Find the standard form of the circle of equation 1 4 x 2 + 1 4 y 2 = y - x + 2 , and determine its center and its radius. (c) Find the standard form of the translation of the circle in question (b) two units left and one unit up . (d) Find the following limit lim x 7→ 1 x 2 + x + 3 - 5 x - 1 . Problem 2. (estimated time: 30mn ) (60 points) (i) Find the second derivatives of the following functions and simplify your results as much as possible. (a) f ( x ) = x 2 - 2 x 2 + 3 . (b) f ( x ) = x 2 cos(3 x - 1) - sin( x 2 + 1) . (ii) Using the implicit differentiation find dy/dx for the following relations. (a) - y 3 + x 3 = x 2 y 2 - 1 (b) y 2 cos x - x sin y = 0 . (iii) The radius r of a sphere is increasing at a rate of 3 inches per second. Find the rate of change of the volume when r = 5 inches. Problem 3. (estimated time: 40mn ) (70 points) Consider the function defined by f ( x ) = 2 x x 2 + 1 + 1 . (a) Find the domain of f . (b) Find the x -intercepts and

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