ENGM 2022 Study Guide - Final Guide: Damping Ratio, Stationary Point, University Of Manchester

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turquoisegnu128

26 Oct 2018

School

Dalhousie University

Department

Engineering Mathematics

Course

ENGM 2022

Professor

All

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Final exam sample problems, math 246, fall 2015 (1) consider the di erential equation. Identify their intervals of de nition. (a) (b) dy dt dy dx. 1 + t2 = t2 , exy + 2x. 3 + t w(0) = 3 , w(cid:48)(0) = 7. y(0) = 0 , y(cid:48)(0) = 5. (6) give an explicit real-valued general solution of the equation. Sketch a typical solution for t 0. If this equation governs a spring-mass system, is the system undamped, under damped, critically damped, or over damped? h(cid:48)(cid:48) + 2h(cid:48) + 5h = 0 . 2 (7) when a mass of 2 kilograms is hung vertically from a spring, it stretches the spring 0. 5: (gravitational acceleration is 9. 8 m/sec2. ) At t = 0 the mass is set in motion from. 0. 3 meters below its rest (equilibrium) position with a upward velocity of 2 m/sec. It is acted upon by an external force of 2 cos(5t).

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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

ENGM 2022 Study Guide - Final Guide: Damping Ratio, Stationary Point, University Of Manchester

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