hw243 - MA 243 Calculus III Spring 2011 Assignments...

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MA 243 Calculus III - Spring 2011 Jacobs Assignments Assignment 1. Spheres and Other Surfaces Read 13.1 - 13.2 and 13.6 You should be able to do the following problems: Section 13.1/Problems 11 - 18, 20 - 22 Section 13.6/Problems 1 - 48 Hand in the following problems: 1. The equation x 2 + y 2 + z 2 = 2 z - 4 y + 4 describes a sphere. Find the center and the radius of this sphere? 2. A particular sphere with center ( - 3 , 2 , 2) is tangent to both the xy -plane and the xz -plane. It intersects the xy -plane at the point ( - 3 , 2 , 0). Find the equation of this sphere. 3. Suppose (0 , 0 , 0) and (0 , 0 , - 4) are the endpoints of the diameter of a sphere. Find the equation of this sphere. 4. Find the equation of the sphere centered around (0 , 0 , 4) if the sphere passes through the origin. 5. Describe the graph of the given equation in geometric terms, using plain, clear language: z = 1 - x 2 - y 2 Sketch each of the following surfaces 6 . z = - x 2 + y 2 7 . z = 1 - y 2 8 . z = 4 - x - y 9 . z = 4 - x 2 - y 2 10 . x 2 + z 2 = 16
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Assignment 2. Dot and Cross Products Read 13.3 and 13.4 You should be able to do the following problems: Section 13.3/Problems 1 - 28 Section 13.4/Problems 1 - 32 Hand in the following problems: 1. Let u = 0 , 1 2 , 3 2 and v = 2 , 3 2 , 1 2 a) Find the dot product b) Find the cross product 2. Let u = j + k and v = i + 2 j . a) Calculate the length of the projection of v in the u direction. b) Calculate the cosine of the angle between u and v 3. Consider the parallelogram with the following vertices: (0 , 0 , 0) (0 , 1 , 1) (1 , 0 , 2) (1 , 1 , 3) a) Find a vector perpendicular to this parallelogram. b) Use vector methods to find the area of this parallelogram. 4. Use the dot product to find the angle between the diagonal of a cube and one of its faces 5. Let L be the line that passes through the points (0 , 1 , 6) and (0 , 3 , 2). Find the length of the projection of k = 0 , 0 , 1 on the line L .
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Assignment 3. Lines and Planes Read 13.5 You should be able to do the following problems: Section 13.5/Problems 1 - 58 Hand in the following problems: 1a. Find the equation of the line that passes through (0 , 0 , 1) and (1 , 0 , 2). b. Find the equation of the plane that passes through (1 , 0 , 0) and is perpendicular to the line in part (a). 2. The following equation describes a straight line: r ( t ) = 1 , 1 , 0 + t 0 , 2 , 1 a. Find the angle between the given line and the vector u = 1 , - 1 , 2 . b. Find the equation of the plane that passes through the point (0 , 0 , 4) and is perpendicular to the given line. 3. The following two lines intersect at the point (1 , 4 , 4) r = 1 , 4 , 4 + t 0 , 1 , 0 r = 1 , 4 , 4 + t 3 , 5 , 4 a. Find the angle between the two lines. b. Find the equation of the plane that contains every point on both lines. 4. The following equation describes a straight line: x, y, z = - 1 , 0 , - 2 + t 1 , 2 , 2 Find the coordinates of the point where this line intersects the y -axis. 5. There is a plane that contains the y -axis as well as every point on the line described in problem 4. Find the equation of this plane.
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Assignment 4. Vector Functions and Space Curves Read 14.1 - 14.4 You should be able to do the following problems: Section 14.1/Problems 7 - 34 Section 14.2/Problems 1 - 29, 31 Section 14.3/Problems 1 - 6 Section 14.4/Problems 3 - 16, 33 - 38 Hand in the following problems: 1. Suppose the position of a particle after t
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