MATH 139 Midterm: MATH139 Fall 2001 Exam

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turquoisegnu128

31 Jan 2019

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McGill University

Department

Mathematics and Statistics

Course

MATH 139

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All

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$MATH 139 Full Course Notes$

MATH 139 Full Course Notes

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Mathematics 189-139a: (9 marks) find the following limits. You may use l"hospital"s rule, when appropri- ate, if you wish. (a) lim x 1 x2 + 10x + 1 x2 + 1 (b) lim x . 3 (c) f(x) = x cos(x: (10 marks) find the equation of the line tangent to the curve. 10x + yx2 + y3 + 12 = 0 at the point (x, y) = ( 1, 1). 5. (i) (3 marks) find all the critical points of the function f(x) = 2 sin(x) cos(2x) in the interval 0 < x < What is the largest possible volume of such a box: a function f is de ned on the whole real line by f(x) = x if < x < 0, If not, what kind of discontinuity does f. If not, what kind of discontinuity does f (i) (2 marks) have at 0? (ii) (2 marks) have at ? (iii) (2 marks)

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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

MATH 139 Midterm: MATH139 Fall 2001 Exam

MATH 139 Full Course Notes

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