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A university exam for mathematics 226, advanced calculus i, held in december 2005 at the university of british columbia. The exam consists of 9 questions covering various topics such as vector calculus, directional derivatives, optimization, and integrals in multiple dimensions. Students are expected to demonstrate their understanding of calculus concepts and problem-solving skills.
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Marks
[12] 1. (6 marks for each part) (a) Prove that the line given by the parametric equations x = 1 + 4t, y = 2 − t, z = − 3 t, is parallel to the plane 2x + 5y − z = 4.
(b) Find the distance between the plane and the line in (a).
[10] 2. Find all points on the surface 3x^2 − y^2 + 2z^2 = 1 where the tangent plane is parallel to both of the vectors (2, 2 , 1) and (4, 1 , −5).
(a) (10 marks) Find the local maximum and minimum values and saddle points of the function f (x, y) = x^4 + y^4 − 4 xy + 6.
(b) (3 marks) Does the function in (a) have a global maximum or minimum? Explain why or why not.
[10] 5. The plane x + 2y + z = 2 intersects the paraboloid z = x^2 + y^2 in an ellipse. Find the points on this ellipse which are nearest to and farthest from the origin.
(c) Is f differentiable at (0, 0)? Explain why or why not.
[5] 7. Let f : Rn^ → R be a function of class C^1 such that f (tx) = taf (x) for all x ∈ Rn, t > 0 for some fixed a ∈ R (such functions are called homogeneous of degree a). Prove that x · ∇f (x) = af (x). (Hint: for fixed x, differentiate f (tx) with respect to t.)
[12] 8. Evaluate the following integrals. (6 marks for each part) (a)
D^ xdA, if^ D^ is the region bounded by the parabola^ y
(^2) − x − 5 = 0 and the line x + 2y = 3. (Hint: pay attention to the choice of the order of integration.)
(a)
0
x^2 x
(^3) sin(y (^3) )dydx. (Hint: reverse the order of integration.)
Be sure that this examination has 10 pages including this cover
The University of British Columbia Sessional Examinations - December 2005 Mathematics 226 Advanced Calculus I Closed book examination Time: 2.5 hours
Print Name Signature Student Number Instructor’s Name Section Number
No calculators, notes, or books of any kind are allowed. Show all calculations for your solutions. If you need more space than is provided, use the back of the previous page.
Rules governing examinations
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