MTH 162 Midterm: MTH 162 University of Rochester Fall 15Exam1 Solutions2

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turquoisegnu128

31 Jan 2019

School

University of Rochester

Department

Mathematics

Course

MTH 162

Professor

All

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October 15, 2015: (20 points) (a) use integration by parts to express in =r /2. Answer: cosn x dx in terms of in . Let u = cosn v = sin x. Use sin2 x = 1 cos2 x to get. 1x and dv = cos x dx, so du = (n 1) cosn . We have and therefore cosn x dx = uv(cid:12)(cid:12)(cid:12) = (n 1)z /2 cosn x dx = (n 1)z /2 cosn . 2 x dx (n 1)z /2. 2 x sin2 x dx cosn x dx. nz /2. 0 (b) use the formula of part (a) repeatedly to nd i6. You will not get partial credit here if the formula you are using is incorrect. A tank has the shape of an inverted pyramid (point down) with a square base. 8 ft tall and the area of its base is 16 ft2. The tank is lled with uid to a height of 2 ft.

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Please show detailed steps/work thank u! really appriciate it !

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

MTH 162 Midterm: MTH 162 University of Rochester Fall 15Exam1 Solutions2

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