MATH 307 Midterm: MATH 307 UW 307 Su09 Midterm 1

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turquoisegnu128

31 Jan 2019

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Mathematics

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MATH 307

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All

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Math 307, sections a and b, summer 2009, midterm i. You have 60 minutes: you may use a calculator which does not graph and which is not programmable, show your work, please box your nal answer. 1: consider the initial value problem (a) what is the domain of the solution? ty + (1 + t)y = t, y(1) = 2. (b) find the solution to this initial value problem. 2: consider the initial value problem (a) find the solution to this initial value problem. x = tx(1 x) The net immigration out of the country is 3500 people per year. 4: given the di erential equation dy dt. = 2y(y 2)2 on the axes on the left, graph 2y(y 2)2 versus y. Then, on the right graph several possible solutions of y(t) with di erent initial values. Indicate clearly the particular solutions where the family changes behavior including concavity. Determine if these equilibrium solutions are stable, unstable or semi- stable.

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1-10 please

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 307 Midterm: MATH 307 UW 307 Su09 Midterm 1

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