OF DEPARTMENT MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS MATH 131 FALL 2002 FINAL EXAM 1. Consider the polynomial function f (x) = x4 x3 + x2 + 2 4 3 a) [8] Determine analytically the intervals where the function is increasing and those where it is decreasing. Find also all local maxima and local minima. Show all your analytical steps. b) [8] Determine analytically the intervals where the function is concave upward and those where it is concave downward. Find also all points of inection. Show all your analytical steps. c) [4] Support your results in parts a) and b) by a graph of the function f (x) in a view window of appropriate size. Indicate clearly the inection points and the local maxima and minima. 2. [10] Consider the ellipse x2 y 2 + = 1. 8 4 Use implicit dierentiation to show, that the tangent line to the ellipse at the point (x0 , y0 ) = (2, 2) has the equation x + 2y = 4.
3. [15] Compute the following limits, using analytical steps (limit laws and/or LHospitals Rule). Justify all your steps! 1 1 a) lim ln x x x 2 x b) +x6 lim x2 cos(x 2) c) lim (1 + x) sin(x)
x0
1
4. [15] A rectangular banner has a red border and a white center. The width of the border at the top and at the bottom is 1 foot and along the sides it is 1 foot. The total area is 32 square 2 feet. Find the dimensions of the banner, which maximize the area of the white center. Justify your answer and show all your work. 5. [10] Find all the horizontal and vertical asymptotes of the function f (x) = Show all the analytical steps involved. 6. [15] A radar station on the ground is tracking an aircraft, which is ying horizontally at an altitude of 3 miles. The radar signal ind...