You are watching: Sketch the vector field f by drawing a diagram like this figure. f(x, y) = yi + xj x2 + y2
1. (45 marks) Short-Answer Questions. Placed your answer in the box detailed but present your occupational also. Each inquiry is worth 3 marks, yet not all concerns are of same difficulty. Full marks will be offered for exactly answers inserted in the box, yet at most 1 mark will be given for not correct answer. Unless otherwise stated, it is not vital to simplify your answer in this question. A) uncover the equation that the tangent airplane to the graph the the duty z = f(x,y) = offer at (0,1). Prize b) evaluate (T, 0) if f(x,y) = sin(ry). Answer c) Graph that a function z = f(x,y) is given by: i beg your pardon of the following level curves correspondes come this function. Answer d) If f(x,y) = e(3x – 2y), discover lim - Math)-f(x,y) e) find % if f(x, y) = x2 - y3 and also 2 = r sin(t), y = r cos(t). Answer f) advice the directional derivative of f(x, y) = !at (1, -1) in the direction of u =(2, 1). Prize 8) walk Simono converge? h) sketch the direction field of the differential equation = 1-2y. I) think about the differential equation ay = 15 - 31, y(0) = 0, and also use Euler"s technique with step dimension 0.1 to calculation y(0.1). Prize j) understanding that (-3,-3) is a vital point of f(x,y) = 2:3 + y2 – 4.cy -3.0, usage the second derivative test to discover out even if it is it is a local minimum, neighborhood maximum, or a saddle point. Prize k) deal with the initial value problem, dy dt 642 2y + cos(u) y(1) = . Answer 1) Is ū=(2,2, -1) perpendicular come v = (5,-4, 2)? answer m) uncover the area between the graph the y = 1 + sin(x) and also y=1-sin(x) from 0 come 7. Price n) identify the lentgh the the curve y = 8/2,0 SIS 3. Evaluate and simplify her answer. Prize o) to express the volume that the solid acquired by rotating the bounded region that lies between curves y = (x-2)2 and also y=1,150 $4 around the vertical line 2 = 5. Execute not advice this integral. Answer
2. Consider the duty f(r) = (x2 - 4x +5) e-r, for r > 0.(a) What is f(0)? (1 mark)(b) What is lim + S (2)? (No calculations necessary) (1 mark)(c) calculate the very first and 2nd derivatives: f"(o) and l" ). (4 marks)(d) using your vault answers, lay out f(x) for 2 > 0. Label the y-intercept,and any an essential or inflammation points.
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(hint: psychic to factor your answersfor f"(x) and also F"(x), this will assist you to draw sign diagrams). (4 marks)