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This examination has 8 questions on 2 pages. The University of British Columbia Final ExaminationsāDecember 2002 Mathematics 226 Advanced Calculus I (Professor Loewen) Closed book examination. Time: 2 12 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. Sketch the domain of integration and evaluate:
I =
0
ā« (^) x^1 /^3 x
1 ā y^4 dy dx.
[10] 2. When x, y, u, v are related by the pair of equations x = u^3 + v^3 , y = uv ā v^2 , the symbol āu/āx has two possible interpretations. Explain what these are, and calculate both of them at the point corresponding to u = 1, v = 1.
[12] 3. Find the absolute maximum value of f (x, y) = x^2 y^2 (5 ā x ā y) in the region where x ā„ 0 and y ā„ 0. Justify the āabsolute maximumā assertion with care, including a complete statement of any theorem you apply. (If you cannot complete this justi- fication, you may earn partial credit by demonstrating that you have found a local maximum.)
[12] 4. Background Information: Given a function f : Rn^ ā R and a point x 0 in Rn, Newtonās Method for the approximate maximization of f involves two steps: (1) Find Q: Rn^ ā R, the best quadratic approximation for f near the point x 0. (2) Find a critical point for Q and call it x 1. In good cases, the critical point x 1 maximizes Q, and lies closer to a local maximizer for f than the original point x 0. (The process can be repeated.) Action Request: Using f (x, y) = x^2 y^2 (5 ā x ā y) as in Question 3, and x 0 = (1, 1), apply Newtonās Method as described above to find x 1. Is this a āgood caseā?
[12] 5. Find J =
R z dV , where R is the subset of R^3 defined by
x ā„ 0 , y ā„ 0 , x^2 + y^2 ā¤ z ā¤
12 ā x^2 ā y^2.
This examination has 8 questions on 2 pages.
[15] 6. An ant crawls on the surface of a rugby ball: the surface obeys
x^2 + y
2 2 +^ z
The temperature (in ā¦C) at each point (x, y, z) on this surface is given by T (x, y, z) = ā^82 yz sin
( (^) Ļ 2 x
As the ant passes through the point P = ( 12 , 1 , 12 ), it follows a path that makes its temperature increase most rapidly. (a) Find a vector tangent to the antās path at P. (b) If the antās speed is Ī² units/second, find its instantaneous velocity vector and its perceived rate of change of temperature at point P. Give units for your answers.
[15] 7. Let S denote the part of the sphere x^2 + y^2 + z^2 = 5r^2 where x > 0, y > 0, and z > 0. (a) Find the maximum value of 3 ln x + ln y + ln z over S. (b) Use the result in (a) to prove that for all positive real numbers a, b, c,
a^3 bc ā¤ 27
( (^) a + b + c 5
[14] 8. (a) Assuming f : R^2 ā R, give precise definitions for these statements: (i) f is continuous at (0, 0), and (ii) f is differentiable at (0, 0). Parts (b)ā(d) refer to the specific function f : R^2 ā R defined by
f (x, y) =
x^2 (1 + Ļy) + y^2 x^2 + y^2 ,^ if (x, y)^6 = (0,^ 0), 1 , if (x, y) = (0, 0). (b) Prove that f is continuous at (0, 0). (c) Find ā 1 f (0, 0) and ā 2 f (0, 0). (d) Prove that f is not differentiable at (0, 0). [Clue: Consider f (t, t).]
- BONUS QUESTION (5 MARKS): Prove: Every continuously differentiable function F : R^3 ā R such that at every point (x, y, z), āF (x, y, z) is parallel to (z, ey^ , z cos(x)), satisfies F
( (^) Ļ 2 ,^0 ,^ āa
= F
( (^) Ļ 2 ,^0 , a
for every a > 0.
