fa02prelim3

# fa02prelim3 - x 2 + y 2 ≤ 1. 5. [20pts] The sphere x 2 +...

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Math 192 Prelim 3 Nov. 21, 2002 1. [15pts (5pts each)] For each of the following integrals: i) describe (in words or with a sketch) the region over which the integral is calculated and ii) evaluate the integral. (a) Z 3 0 Z y 0 x 2 + y 3 dx dy (b) Z π 0 Z y 2 0 cos( x ) dx dy (c) Z 4 0 Z 2 y ye - x 5 dx dy 2. [15pts] The height above sea level of Gorgeous State Park is given by H ( x,y ) = 4 xy - x 4 - y 2 + 350 for ± - 4 x 4 - 4 y 4 ² . Professor Pontiﬁcus goes for his daily walk in Gorgeous along a path given by x ( t ) = t y ( t ) = t 2 . Find the coordinates of all points on the path where the Professor will be neither ascending nor descending. 3. [15pts] Consider f ( x,y ) = k 2 e x - xy sin k + (6 - 5 k ) x where k is a real number. (a) Find all values of k for which (0 , 0) is a critical point of the function f . (b) Can (0 , 0) be a local extreme of f ? 4. [15pts] Find the greatest and smallest values of the function f ( x,y ) = x 2 y in the region
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Unformatted text preview: x 2 + y 2 ≤ 1. 5. [20pts] The sphere x 2 + y 2 +( z-3) 2 = 9 rests between the two planes x-2 y-2 z = c-x + 2 y-2 z = c. (a) At what points is the sphere tangent to the planes? (b) What is c ? 6. [20pts] We wish to compute the integral Z 1 . 01 . 99 Z 1 . 01 . 99 1 ln2 2-x 2-y 2 dx dy. Because there is no closed form for the antiderivative of the integrand, we must estimate. (a) Find an appropriate linear approximation, L , of the integrand and then integrate L over the region. (b) Find a bound on the error incurred by this estimation. In computing the error bounds, you may use the fact that all second partials of the integrand on the given region have absolute value less than one....
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## This note was uploaded on 09/29/2009 for the course 1920 1920 at Cornell.

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