01spr-fsols

# 01spr-fsols - FINAL EXAM SOLUTIONS Math 51 Spring 2001 You...

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FINAL EXAM SOLUTIONS Math 51, Spring 2001. You have 3 hours. No notes, no books. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT Good luck! Name ID number 1. (/50 points) 2. (/50 points) 3. (/50 points) 4. (/50 points) 5. (/50 points) Bonus (/20 points) Total (/250 points) “On my honor, I have neither given nor received any aid on this examination. I have furthermore abided by all other aspects of the honor code with respect to this examination.” Signature: Circle your TA’s name: Kuan Ju Liu (2 and 6) Robert Sussland (3 and 7) Hunter Tart (4 and 8) Alex Meadows (10) Dana Rowland (11) Circle your section meeting time: 11:00am 1:15pm 7pm 1

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1. Let the function f : ± R 2 - { -→ 0 } ² R 2 have components f 1 and f 2 as described by f ³ x y ´ = ³ f 1 f 2 ´ = ³ ( x 2 ) y xy 2 ´ (a) Note that the function f is not deﬁned at the origin; this is because the component f 1 is not deﬁned there. Is this discontinuity in f 1 removable? Justify your answer. Solution: We compute limits of f 1 = ( x 2 ) y as we approach the origin from diﬀerent directions. Along the x -axis, we have y = 0, so: lim x 0 f 1 = lim x 0 ( x 2 ) 0 = lim x 0 1 = 1 Along the y -axis, we have x = 0, so: lim y 0 f 1 = lim y 0 (0 2 ) y = lim x 0 0 = 0 Since these values are diﬀerent, the limit must not exist. Therefore, the discontinuity in f 1 is not removable. (b) Find the Jacobian matrix for the function f at the point ³ x y ´ = ³ 1 3 ´ . Solution: J f = ∂f 1 ∂x ∂f 1 ∂y ∂f 2 ∂x ∂f 2 ∂y = (2 y ) x 2 y - 1 ln x 2 ( x 2 ) y y 2 2 xy J f, 0 @ 1 3 1 A = 6 0 9 6 2
(c) In what (unit vector) direction -→ u is the function f 1 increasing the fastest, at the point ± x y ² = ± 1 3 ² ? Solution: Since we know the ﬁrst row of the Jacobian matrix is the gradient vector of f 1 , we see immediately that f ± 1 3 ² = ± 6 0 ² So, the direction (unit vector) in which the function is increasing the fastest is -→ u = f k∇ f k = ± 1 0 ² (d) What is D u f 1 at the point ± x y ² = ± 1 3 ² , where -→ u is the vector determined in part (c)?

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01spr-fsols - FINAL EXAM SOLUTIONS Math 51 Spring 2001 You...

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