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The University of British Columbia Final Examinations—December 2010 Mathematics 226 Advanced Calculus I (Professor Loewen) Closed book examination. Time: 2.5 hours
Notes, books, and calculators are not allowed. Write your answers in the booklet provided. Start each solution on a separate page. SHOW ALL YOUR WORK!
[10] 1. The plane P passes through the points A(1, 1 , 0), B(0, 1 , 1), and C(1, 3 , 1).
(a) Find an equation for P. (b) Show how to split the given vector F = (− 4 , 5 , −7) as
F = u + w, where u is normal to the plane P and w is perpendicular to u.
[8] 2. Riverboat Slim earns his living as a calculus shark on the paddlewheelers of the Mississippi. An undercover cop once learned Slim’s trick for finding the tangent plane for any surface of the form
Ax^2 + By^2 + Cz^2 = K. (∗) “If (x 0 , y 0 , z 0 ) is on that surface and I need the tangent plane right there”, Slim confided, “I just rewrite (∗) but change x^2 to x 0 x, change y^2 to y 0 y, and so on.” The cop got the hint: Ax 0 x + By 0 y + Cz 0 z = K. (∗∗)
(a) Justify Slim’s method. (b) Generalize Slim’s method to produce the tangent hyperplane at a = (a 1 , a 2 ,... , an) for the (n − 1)-dimensional hypersurface in Rn^ defined by c 1 xp 1 + c 2 xp 2 + · · · + cnxpn = K. (†)
(Here K ∈ R, c ∈ Rn, and p > 1 are given constants; point a lies on the surface.) Your goal is an equation named (‡) that is related to (†) in the same way that Slim’s equation (∗∗) is related to (∗).
[12] 3. (a) Use Lagrange Multipliers to minimize f (x, y, z) = x^2 + y^2 subject to the simul- taneous constraints
x + 2y + 3z = 6 and x − y − 3 z = − 1. [A correct solution that does not use Lagrange Multipliers will earn some partial credit.] (b) Discuss the problem produced by changing “Minimize” to “Maximize” in part (a).
[10] 4. Evaluate I =
0
y
x^2 sin(πxy) dx dy.
[15] 5. Near Saddleback Pass, the elevation z is related to the map coordinates (x, y) by
z = f (x, y) def = z 0 −
x^2 4
y^2 8
We have volunteered to build a trail from the point A, where x = 1 and y = 0, to the mountain pass at B, where x = 0 and y = 0. Our instructions specify that
(i) the trail must start out in the direction of increasing y, and
(ii) hikers on the trail must not face a slope larger than 1/4.
(a) Sketch a contour map showing level curves of f near the mountain pass. Make a sketch that is large enough and neat enough to support the additions described below.
(b) If we choose a slope of exactly 1/4 to meet rule (ii), in what direction does the trail leave point A? Show this direction on your map from (a).
(c) If an extreme runner sprints uphill along our trail, maintaining a speed of ‖v‖ = 2, what will be his/her three-dimensional velocity vector v at the point A? [Assume all coordinates are measured in meters and the given speed is in meters per second.]
(d) Near the pass at B, the ground is almost level, so rule (ii) is satisfied for motion in any direction at all. Find all points where this is true, and show them on your map from (a).
[12] 6. (a) Write the precise logical definition of the statement below:
lim x→a f (x) = L. (∗)
(b) If the following limit exists, evaluate it and prove in detail that the definition in part (a) holds. Otherwise, prove in detail that existence fails.
lim (x,y)→(0,0)
2 x^2 + x^2 y − y^2 x + 2y^2 x^2 + y^2
(c) Apply the instructions from part (b) to
lim (x,y)→(0,1)
x^2 y^2 − 2 x^2 y + x^2 (x^2 + y^2 − 2 y + 1)^2
