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Calculus IV [2443002] Midterm I
Q1]. Find an equation for the tangent plane to the graph of f (x, y ) = x2 + 2xy y 2 at the point (2, 1, 7). Ans: Equation is given by (z z0 ) = fx (2, 1)(x x0 ) + fy (2, 1)(y y0 ). We have fx = 2x + 2
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Calculus IV [2443002] Midterm II
Q1].[10 points] Consider the double integral
1 0 22y 1y
f (x, y ) dx dy
Sketch the region of integration. Soln. The limits x = 2 2y and x = 1 y tell us that the region is bounded on the right by the l
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Calculus IV [2443002] Midterm III 1
1.1
Q1 [15 points]
Part 1
Write down the change of variables formula for triple integrals.
1.2
Answer to part 1
Suppose the change of variables (x(u, v, w), y (u, v, w), z (u, v, w) takes a region
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Calculus IV [2443002] Quiz I
Q1]. Which one of the four functions listed below has the following level curves?
1. 2. 3. 4.
f (x, y ) = (x + 1)(y 2). g (x, y ) = (x 1)(y + 2). h(x, y ) = (x + 1)2 (y 2)2 . k (x, y ) = (x 1)2 (y + 2)2 .
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Calculus IV [2443002] Quiz II
Q1]. State the second derivative test for functions of two variables. Ans: Let (a, b) satisfy fx (a, b) = 0 and fy (a, b) = 0. Dene D(x, y ) = (fxx )(fyy ) (fxy )2 If D(a, b) > 0 and fxx (a, b) > 0, then
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Calculus IV [2443002] Quiz III
Tuesday, April 4, 2000
Q1]. Write the following triple integral out as a spherical coordinates triple integral.
3 9x2 9x2 y 2
z (x2 + y 2 + z 2 )dzdydx
3 0
0
Soln: The region is precisely one quarter o
Calculus IV [2443004] Midterm I
For full credit, give reasons for all your answers. Q1].[15 points] Draw the level curves f = 0, f = 1, f = 4, and f = 1 for the function f (x, y ) below. Also, sketch the graph of f in a neighborhood of the origin. f (x, y
Calculus IV [2443004] Midterm II
For full credit, give reasons for all your answers. Q1].[15 points] For the double integral below, rst sketch the region of integration, and then convert it to a polar coordinares integral.
2 0
2y y 2
2y y 2
f (x, y ) dx
Calculus IV [2443004] Midterm III
For full credit, give reasons for all your answers. Q1].[15 points] Evaluate the following triple integral by rst sketching the region of integration, and then converting it to a spherical coordinates integral.
1 0 1
1y