MATH 105 Midterm: MATH 105 2005 Winter Test 2

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turquoisegnu128

31 Jan 2019

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$MATH 105 Full Course Notes$

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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 di culty. (a) assume that z(x, y) is a linear function with slope 2 in the x-direction and slope 3 in the y-direction. If z(1, 1) = 4, nd z( 2, 1). If f (x, y) = ln(x2 + y), nd lim k 0 f (1 + k, 0) f (1, 0) k. Answer: (c) let (x0, y0) be a critical point of f (x, y) = x2 y2 + 6x + 8y 21. Find (x0, y0) and then nd the equation of the tangent plane to the surface f (x, y) at the point (cid:1)x0, y0, f (x0, y0)(cid:2). Answer: (d) suppose the marginal revenue in producing x units of a certain product is. Find the change in total revenue if production is increased from.

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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

Solve all please.