Final 1

# D set up but do not compute the integral to compute

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Unformatted text preview: oint (0, 2, 0). (d) Set up (but do not compute) the integral to compute the length of this curve from t = −1 to t = 1. 3. Consider the function f (x, y ) = 2x3 − 3x2 + y 2 − 12x + 10. (a) Find the critical points of f . (b) Classify each critical point of f as a local minimum, local maximum or saddle point of f . 4. The base radius r and the height h of a right circular cylinder are measured as 3 centimeters and and 9 centimeters, respectively. There is a possible error of 1 millimeter in each measurement. Use a linear approximation to estimate the maximum possible error in the volume computed from these measurements. 5. UPS ground shipping limits package size by requiring the length of the package plus the girth (distance all the way around the package moving perpendicular to the length) be less than or equal to 108 inches. Find the dimensions of the package with largest volume that can be shipped by UPS ground. 6. Consider the iterated integral ￿ 2 y =0 ￿ 1 x= y 2 3 y ex dx dy . (a) Sketch the region integration; clearly label each part of its boundary with the appropriate equation. (b) Evaluate the integral. Reverse the order of integration, if necessary. 7. The gradient of a function g (x, y, z ) at the point (−3, 4, 5) is ￿−2, 1, 2￿. (a) Find the values of the partial derivatives gx , gy , and gz at the point (3, 4, −5). (b) Find the maximum rate of change of g at the point (−3, 4, 5) and the unit vector in the direction which the maximum rate of change occurs. (c) Find the rate of change of g at the point (−3, 4, 5) in the direction of the point (−1, 8, 1)....
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