MATH 411 Final: MATH411 BOYLE-M SPRING2012 0101 FINAL EXAM

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turquoisegnu128

15 Feb 2019

School

University of Maryland

Department

Mathematics

Course

MATH 411

Professor

All

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Math 411 spring 2012 boyle final exam. Points in rn are column vectors (even if they are typed horizontally): (a) (15 pts. ) De ne what it means for f to be di erentiable. (b) (30 pts. ) Suppose f : rn r all partial derivatives are continuous functions. Prove that f is di erentiable. (c) (15 pts. ) A function f : rn rm is di erentiable if and only if each of its real-valued component functions f1, f2, . Suppose that k and f are nonempty closed subsets of rn, k is bounded and f k = . De ne f : u r2 by the rule f (x, y) = (xey, 1+x ln(y)), and de ne g : u r by the rule g(x, y) = x2 +y ln(x). (a) (20 pts. ) Prove there is an open set v containing the point (e, 1) and a di erentiable function h : v r such that on the set f 1(v),

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1. (45 marks) Short-Answer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for correct answers placed in the box, but at most 1 mark will be given for incorrect answer. Unless otherwise stated, it is not necessary to simplify your answers in this question. a) Find the equation of the tangent plane to the graph of the function z = f(x,y) = sell at (0,1). Answer b) Evaluate (T, 0) if f(x,y) = sin(ry). Answer c) Graph of a function z = f(x,y) is given by: Which of the following level curves correspondes to this function. Answer d) If f(x,y) = e(3x – 2y), find lim - Math)-f(x,y) e) Find % if f(x, y) = x2 - y3 and 2 = r sin(t), y = r cos(t). Answer f) Evaluate the directional derivative of f(x, y) = !at (1, -1) in the direction of u =(2, 1). Answer 8) Does Simono converge? h) Sketch the direction field of the differential equation = 1-2y. i) Consider the differential equation ay = 15 - 31, y(0) = 0, and use Euler's method with step size 0.1 to estimate y(0.1). Answer j) Knowing that (-3,-3) is a critical point of f(x,y) = 2:3 + y2 – 4.cy -3.0, use the second derivative test to find out whether it is a local minimum, local maximum, or a saddle point. Answer k) Solve the initial value problem, dy dt 642 2y + cos(u) y(1) = . Answer 1) Is ū=(2,2, -1) perpendicular to v = (5,-4, 2)? Answer m) Find the area between the graph of y = 1 + sin(x) and y=1-sin(x) from 0 to 7. Answer n) Determine the lentgh of the curve y = 8/2,0 SIS 3. Evaluate and simplify your answer. Answer o) Express the volume of the solid obtained by rotating the bounded region that lies between curves y = (x-2)2 and y=1,150 $4 about the vertical line 2 = 5. Do not evaluate this integral. Answer

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MATH 411 Final: MATH411 BOYLE-M SPRING2012 0101 FINAL EXAM

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