OLD MIDTERM EXAMS AND CLASS TESTS

Transcription

1 189-65A: Advanced alculus OLD MIDTERM EXAMS AND LASS TESTS Midterm Eam October 1996 Answer all questions in Part A. Answer two () question from Part B and one (1) question from Part for a total of si (6) questions. PART A: Answer each of the following three (3) questions A1 ompute (a) ( + y)d ( y)dy + ( + y + z)dz if is parametrized by = t, y = t, z = t, where t goes from 1 to 1. (b) {y y sin(y)}d + {y sin(y)}dy, where is the line segment from (3, 0) to (0, 3). A alculate the line integral ( 1 3 y3 5 )d + ( y 4 )dy, where denotes the unit circle + y = 1 traversed once counterclockwise. A3 ompute ( y)ddy, where D is the region bounded by y = 3, y = 1, + y = 5, and D + y = 4. PART B: Answer two () of the following questions B1 A wire along the 1st quadrant part of + y = a has density δ(, y) given by δ(, y) = y. Find the position of the center of mass. B ompute +y d + +y dy, where is (a) the circle + y = a traversed once counterclockwise, and (b) the perimeter of the triangle with vertices (1, 0), ( 1, 1), (0, 1). Justify your calculation. B3 Use a line integral to compute the area of the triangle with vertices at (0, 0), (, 3), and (3, 4). Then calculate the area of the same triangle using a double integral (or integrals). B4 If F (, y) = (y +, ) compute the flu of F (a) across the line segment from (1, 1) to (3, 1) (observe the standard orientation convention of the assignments), and (b) across the boundary of the triangle with vertices at (1, 0), (0, 1), and (0, 0) (in the outward sense).

2 PART : Answer one (1) of the following questions 1 Find the maimum value of y + yz if + y + z = 1. If f(, y, u, v) = e u cos v and g(, y, u, v) = y e u sin v show that near = 1, y = 1, u = 0, v = π 4, one can solve the equations f(, y, u, v) = 0 g(, y, u, v) = 0 for and u as functions of y and v. Determine for = 1, y = 1, u = 0, v = π 4. [ ] ************************************** Midterm Eam October 1997 Answer all questions in Part A. Answer two () question from Part B and one (1) question from Part for a total of si (6) questions. PART A: Answer each of the following three (3) questions A1 ompute (a) ( + z)d ( z)dy ( + y z)dz if is parametrized by = t, y = t, z = t, where t goes from 0 to 1. (b) {y ye y }d + { y e y }dy, where is the line segment from (0, 0) to (1, 0) followed by the arc of the circle + y = 1 from (1, 0) to (0, 1) that lies in the first quadrant. A alculate the line integral ( 1 y 5 )d + ( 1 + y 4 )dy, where denotes the boundary of the rectangle determined by (0, 0), (4, 0), (4, ), and (0, ) traversed once clockwise. A3 ompute ( y)ddy, where D is the triangle with vertices at (1, 1), (3, ) and (4, 6). D

3 Part B: Answer any two () of the following questions. B1 A wire in the shape of a coil of a heli is parametrized by (t) = (cos t, sin t, t), 0 t π. Assume the mass density of the wire is δ(, y, z) = z + 1. Find the z-coordinate of the center of mass of the wire. B ompute +y d + +y dy, where is (a) the circle + y = 4 traversed once counterclockwise, and (b) the perimeter of the square determined by (3, 3), ( 3, 3), ( 3, 3), and (3, 3). Justify your calculation. B3 Use a line integral to compute the area of the triangle with vertices at (0, 0), (3, 4), and (6, ). Then calculate the area of the same triangle using a double integral (or integrals). B4 If F (, y) = ( + y, y) compute the flu of F (a) across the line segment from (1, 1) to (3, 1) (observe the standard orientation convention of the assignments), and (b) across the boundary of the triangle with vertices at (1, 0), (0, 4), and (0, 0) (in the outward sense). PART : Answer one (1) of the following questions 1 Find the maimum value of yz if + y + z = 9. If f(, y, u, v) = yu + v and g(, y, u, v) = y + v u show that near = 1, y = 1, u = 0, v = 1, one can solve the equations f(, y, u, v) = g(, y, u, v) = for and u as functions of y and v. Determine [ ] when = 1, y = 1, u = 0, v = 1. ********************************** 65 Midterm class test November 1998 Instructions: Answer all questions. Note that in question 4 there is a choice. 1. ompute the following line integrals: (a) yd + ydy, where is the line segment from (, 0) to (6, 5).

4 (b) (y + z + yz cos(yz))d + ( + z + z cos(yz))dy + ( + y + y cos(yz))dz, where is the line segment from (0,, 0) to (1, 1, 1) followed by the line segment from (1, 1, 1) to (3, 6, 1) followed by the line segment from (3, 6, 1) to (1, π/, 1).. alculate (y )d+3ydy, where is the curve ( 1) +y = 1 traversed once counterclockwise. 3. ompute the outward flu of F (, y) = (y, ) across the boundary of the triangle with vertices at (1, 1), (5, 1), and (3, 1). 4. Either compute the line integral +y d + +y dy where is (a) the boundary of the square with corners at (1, 1), (3, 1), (3, 1) and (1, 1) traversed counterclockwise. (b) the boundary of the square with corners at ( 1, 1), (1, 1), (1, 1) and ( 1, 1) traversed counterclockwise. (c) the boundary of the rectangle with corners at ( 1, 1), (3, 1), (3, 1), and ( 1, 1). (d) the line segment from (0, 1) to ( 1, 1). Eplain your calculations. Or compute the outward flu of F (, y) = ( +y, y +y ) across (a) the boundary of the triangle T with vertices at (, ), (, ) and (0, 4). (b) the boundary of the square S with vertices at (, ), (, ), (, ) and (, ). (c) the boundary of the union of the square S and the triangle T. (d) the line segment from (, ) to (, ) using the convention that the normal is to the right of the curve. Eplain your calculations. 5. Find the maimum value of 3 5y + z on the sphere ( + ) + (y 1) + (z + 4) = 4. ************************************ Midterm lass test October ompute the line integral ( y sin(y) + y4 + y ) ( ) d + + y sin(y) dy where is the boundary of the triangle with vertices at (1, 1), ( 1, ) and ( 1, 1), taken counterclockwise.. Use Green s formula to compute the area enclosed by the curve (t) = ( + cos t) cos t, y(t) = ( + cos t) sin t, 0 t π.

5 3. Eplain why (u, v) can be solved for as functions of (, y) near the point ( 0, y 0, u 0, v 0 ) = (1, 1, 1, 1, 1) from the relations ompute,,, at (1, 1, 1, 1). u + yvu = u 3 + y v 4 =