23s08-Exam-Final-OLD - SAMPLE PROBLEMS OF MATH 23 SPRING...

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SAMPLE PROBLEMS OF MATH 23, SPRING 2008 (1) Find the equation of the tangent plane to the graph of z = 9 x 2 + y 2 + 6 x - 3 y + 5 at the point (1 , 2 , 18). (2) Sketch the space curve r ( t ) = h 2 sin t, t, 2 cos t i , - 2 π t 2 π and find the total length of the curve. (3) Find the parametric equations of the line through the points (1 , 0 , 7) and (1 , 2 , 3). (4) Find the intersection of the line in the above problem with the plane x + y +2 z = 6. (5) Evaluate the integral R 1 0 R 3 3 y e x 2 dxdy . (6) Find the dimensions (i.e., length, width, and height) of the rectangular box with largest volume if the total surface area is given as 64 square cm. (7) The height of a cylinder is increasing at a rate of 3 cm/sec. The radius of the cylinder is increasing at a rate of 2 cm/sec. At what rate is the volume increasing when the height is 20 cm and the radius is 15 cm? (8) Consider the surface area of the part of the sphere x 2 + y 2 + z 2 = 4 that lies above the plane z = 2. Find the area of the surface.
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