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The final examination for math 200 at the university of british columbia, held in april 2012. The examination covers various mathematics topics including calculus, vector calculus, and optimization. Students are required to solve problems related to finding vectors, parametric equations, tangent planes, second-order derivatives, temperature rates, lagrange multipliers, iterated integrals, and average distances. The examination is closed-book and lasts for 2 hours.
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Marks[10] 1. Let L be a line which is parallel to the plane 2x + y − z = 5 and perpendicular to the line x = 3 − t, y = 1 − 2 t and z = 3t. a) Find a vector parallel to the line L. b) Find parametric equations for the line L if L passes through a point Q(a, b, c) whereand the a < yz (^0) -plane are 2, b > 0 , c > (^) ,0, and the distances from 3 and 4 respectively. Q to the xy-plane, the xz-plane
[10] 2. Let z = f (x, y) = ln(4x^2 + y^2 ) (a) Use a linear approximation of the function f (0. 1 , 1 .2) z = f (x, y) at (0, 1) to estimate (b) Find a point P (a, b, c) on the graph of z = f (x, y) such that the tangent plane to the graph of z = f (x, y) at the point P is parallel to the plane 2x + 2y − z = 3
[10] 4. (^) is measured in centigrade andThe temperature at a point ( x, y, zx, y, z) is given by in meters. T (x, y, z) = 5e−^2 x^2 −y^2 −^3 z^2 , where T
(a) Find the rate of change of temperature at the pointtoward the point (1, 1 , 0). P (1, 2 , −1) in the direction (b) In which direction does the temperature decrease most rapidly? (c) Find the maximum rate of decrease at P.
[10] 5. Let C be the intersection of the plane x + y + z = 2 and the sphere x^2 + y^2 + z^2 = 2. (a) Use Lagrange multipliers to find the maximum value of f (x, y, z) = z on C (b) What are the coordinates of the lowest point on C?
[10] 7. The average distance of a point in a plane region D to a point (a, b) is defined by 1 A(D)
D
√(x − a) (^2) + (y − b) (^2) dxdy
where A(D) is the area of the plane region D. Let D be the unit disk 1 ≥ x^2 + y^2. Find the average distance of a point in D to the center of D.
[10] 8. (^) xLet + yE = 1 and the surface be the region in the first octant bounded by the coordinate planes, the plane z = y (^2).
Evaluate
E
zdV.
[10] 10. Evaluate I = ∫ ∫ ∫ R 3 [1 + (x^2 + y^2 + z^2 )^3 ]−^1 dV.
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The University of British Columbia Final Examination - April, 2012 Mathematics 200
Closed book examination Time: 2 12 hours
Name Signature Student Number Instructor’s Name Section Number
No information sheet allowed. No calculators allowed. Rules governing examinations 1.brary/AMS card upon request Each candidate should be. prepared to produce his or her li-
Total 100
