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27 Sep 2018

3. (8 marks) A function y(x) obeys this relationship: y' = xy2 + ax2, where a is a constant. (a) Determine the value of a such that the tangent line to the graph of y(x) at (1,1) passes through (2,5).

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Casey Durgan
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29 Sep 2018

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3. (8 marks) A function y(1) obeys this relationship: y' = ry² + ar?, where a is a constant. (a) Determine the value of a such that the tangent line to the graph of y(«) at (1,1) passes through (2,5). (b) For this a value, determine the concavity of the function at r= 1.

1. (16 marks, 2 for each part.) The problem consists of 8 short-answer questions: V22 +12 - 1 (a) Evaluate lim - or determine that it does not exist. In the latter case, write DNE as answer. 1+2 -1 Answer (b) Find values of a that would make the following function continuous everywhere, or determine that no such value exists. In the latter case, write DNE as answer. g(x) = ( 22+2 2+1 ar if I 1. Answer (c) Find the derivative of y = r" lnr. Answ (d) Find the equation for the tangent line to the curve z-y + xy2 = 2 at (1,1). Answer (e) Use linear approximation to estimate (0.998) Answer if any exists. Write DNE if none r2 - 4 (f) Find the vertical, horizontal or slant asymptotes of f(x) = exists. Answer (g) Find any local maxima and minima of h(x) = r- in the interval (1,4). Give both r and h(2) values and state whether it is a local minimum or maximum. Answer (h) On which part of the interval [0,1] is f(x) = et sin x concave upward? Answer

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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