MAT-2130 Midterm: MATH 2130 App State Fall2012 Test2

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turquoisegnu128

15 Feb 2019

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Answer key: 10 points) let f (x, y) = x2 + 4y2. Be sure to show your work! (a) write down a general equation for the level curves / surfaces of f (circle the correct term). Answer: x2 + 4y2 = c (where c is some constant). For c > 0 these are ellipses (centered at the origin). For c = 0 this is just the origin itself. For c < 0 the level curves are empty. (b) write down the equation of the trace of z = x2 + 4y2 in the xz-plane. The equation for the xz-plane is y = 0. So the trace is z = x2 + 4(02). Answer: z = x2 (c) make a rough sketch of z = x2 + 4y2: (10 points) consider x and y as independent variables and z as a dependent variable where 5x + xy2 + ze3x+y = 10. Using the formulas for implicit di erentiation, compute both and.

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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

MAT-2130 Midterm: MATH 2130 App State Fall2012 Test2

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