MAT-2130 Midterm: MATH 2130 App State Spring2013 Test2

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turquoisegnu128

15 Feb 2019

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If f (x, y) = c, then dy dx. If f (x, y, z) = c, then. Fz: (12 points) three level curve plots are shown below. The level curves 9 x2 y2 = c can be rewritten as x2 + y2 = 9 c. these are circles of radius 9 c centered at the origin (assuming 9 c > 0). The level curves p3x2 + y2 = c can be rewritten as y2 c2 = 1 (when c 6= 0). [note: we could have matched these by choosing particular values of c and matching up with the graph. For graph. example: c = 0 for 9 x2 y2 yields x2 + y2 = 9 (a circle of radius 3). 3x2 + y2 = c2 so that x2 c2/3 (b) sketch a gradient vector at the points a, b, and c. if the vector is 0, draw an x on the point.

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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 Spring2013 Test2

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