University of Saskatchewan Department of Mathematics & Statistics MATH. Final Instructors: (0) P. J. Browne (0) B. Friberg (0) H. Teismann December 9, 000 Time: :00-:00 pm This is an open book exam. Students may use Stewart: Calculus. Calculators, notes, formula sheets are not permitted. Only a soft lead pencil should be used on the opscan sheet. Your student ID number should be encoded and written, with your name, in the upper left hand corner of the opscan sheet. ALL QUESTIONS AE OF EQUAL VALUE Section A: Answer these questions on the opscan sheet. Find the point of intersection (if it exists) of the lines x = y = z and x = y, z =. (,, ) (B) (,, ) (, 0, 0) (,, ) (E) (, 0, ) (0,, ) (0,, 0) Lines do not intersect None of the above.. Find the equation of the plane passing through the point (,, ) which is parallel to the plane containing the points (0,, 0), (,, ) and (,, ). x y + z + = 0 (B) x y + z = 0 x + y + z = 0 x y + z + = 0 (E) x y + z = 0 x + y + z = 0 x + y z = 0 x + y z = 0 None of the above. Find the length of the curve given by r (t) =< t, t, t >, 0 t. [/ 8] (B) 6 [/ 8] 8 [/ 8] 8 [/ + ] (E) [/ + 8] 8 [/ ] [9/ ] [/ 8] None of the above.../
Math. Final December 9, 000 Page. The set of all points P (x, y, z) whose distance to the plane z = is twice the distance from P to (,, 0) forms an ellipsoid with equation (x x 0 ) a + (y y 0) b + (z z 0) Find (x 0, y 0, z 0 ) and a, b, c. (x 0, y 0, z 0 ) = (,, ), a =, b =, c = (B) (x 0, y 0, z 0 ) = (,, ), a =, b =, c = (x 0, y 0, z 0 ) = (,, ), a =, b =, c = (x 0, y 0, z 0 ) = (,, ), a =, b =, c = (E) (x 0, y 0, z 0 ) = (,, ), a =, b =, c = c =, a > 0, b > 0, c > 0. (x 0, y 0, z 0 ) = (,, ), a =, b =, c = (x 0, y 0, z 0 ) = (,, ), a =, b =, c = (x 0, y 0, z 0 ) = (,, ), a =, b =, c = None of the above.. Given that the equation z xz y = 0 implicitly defines z as a function of x and y, find z z x and y, and use these to find z x y in terms of x, y, z. (z + x)/(z x) (B) /(z x) /(z x) (z + x)/(z x) (E) (z + x)/(z x) (z + x)/(z x) (6z x)/(z x) (6z + x)/(z + x) None of the above 6. A particle starts at time t = 0 from position (, ) with speed m/sec in the negative Y -direction. It is subject to an acceleration given by Find its position at t =. a (t) = i t + + 6t j. (/, ) (B) (/, ) (, ) (/, ) (E) (/, 0) (/, ) (/, ) (6/, ) None of the above.../
Math. Final December 9, 000 Page. Find the equation of the tangent plane to the surface xyz + yz = at the point (,, ). x + y + z 8 = 0 (B) x + y + z = 0 x + y + z = 0 x + y + z = 0 (E) x + y + 0z = 0 x + y + 0z = 0 x y + z 6 = 0 x + y + 0z = 0 None of the above 8. Find the directional derivative of f(x, y) = arctan(xy) at the point (, ) in the direction of the line y = x for increasing x. 9. Evaluate (B) 0 (E) None of the above (x+y)da where is the region bounded by y = 0 and y = x. / (B) 6 6/ 0 (E) 6/ 8 6 8/ None of the above 0. Find the maximum value of f(x, y, z) = x + y + z subject to x + y + z = 6. 6 (B) 6 (E) 6 None of the above. A lamina occupies the region inside the circle x + y = in the first quadrant. Its density at any point is proportional to the square of the distance from (0, 0). Find the co-ordinates of its centre of mass. ( ) 8, 8 ( (B) 0 0), ( 8, ) 8 ( 0, 0 ) ( ), (E) ( ), ( ), (, ) None of the above.../
Math. Final December 9, 000 Page. Evaluate x 0 0 dy dx y (B) (E) + + + None of the above. Find the surface area of the surface z = x + y lying above the triangle with vertices at (0, 0, 0), (,, 0) and (0,, 0). /9 (B) / 6/9 (E) /9 /9 6/ / None of the above. Calculate z dv where is the region in the first octant bounded by y + z =, y = x, z = 0, x = 0. [Hint: set up the integral in the form the integrations.]. Calculate x + y = y. (B) 8 z dz dx dy with appropriate terminals for 6 (E) 8 None of the above x x + y da where is the region outside x + y = and inside /8 (B) / / 0 (E) /8 / / None of the above 6. Use spherical co-ordinates to calculate (x + y + z )dv where is the region whose lower boundary is the cone z = x + y x + y + z =. ( ) (B) ( ) and whose upper boundary is ( ) (E) None of the above.../
Math. Final December 9, 000 Page. Find the work done by the force F (x, y, z) = x + (y + ) <,, > + (z + ) acting on a particle which moves in a straight line from (,, ) to (,, ). (B) (E) 0 None of the above 8. Find the line integral to (,, ). C x ds where C is the curve y = x, z = x from (0, 0, 0) / (B) 9/ 9/ / (E) 9/8 / /6 9/6 None of the above 9. Find the integral C arctan y dx y x dy where C is the boundary of the square + y with vertices (0, 0), (0, ), (, ), (, 0) described in an counterclockwise fashion. (B) 0 / (E) / None of the above 0. Calculate F dr where F (x, y) =< e y + ye x, xe y + e x > and C is the curve C r (t) =< sin ( ) t, ln t >, t. [Use the Fundamental Theorem for Line Integrals] ln (B) ln + ln (E) ln + ln ln None of the above.../6
Math. Final December 9, 000 Page 6 Section B: Bonus Questions. Write complete solutions to these problems in the answer booklet provided. B. A solid has lower boundary given by the cone z = x + y and upper boundary the plane z =. Its density is given by f(x, y, z) = z. Find its mass. [Use cylindrical co-ordinates] B. Find the point on the plane x + y + z = that is closest to (,, 0). ** THE END **