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The final exam for the egr 265, spring 2012 course on 'math tools for engineering problem solving'. The exam consists of 10 problems covering various topics in engineering mathematics, including differential equations, vector calculus, and integral calculus.
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EGR 265, Math Tools for Engineering Problem Solving May 7, 2012, 10:45am to 1:15pm
Name (Print last name first):..........................................
Student ID Number:...........................
Final Exam
Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7 Problem 8 Problem 9 Problem 10 Total
Problem 1 (8 points)
Find an explicit solution of the initial value problem
y′^ − 2 x = 2xy, y(1) = 0.
Problem 3 (12 points)
Consider the second order differential equation
y′′^ − 4 y′^ + 4y = 2e^3 x. (1)
(a) Find the general solution of the homogeneous equation corresponding to (1). (b) Find a particular solution of the inhomogeneous equation (1). (c) Solve the initial value problem given by (1) and initial conditions y(0) = 0, y′(0) = −1.
Problem 4 (12 points)
A mass of 100 kg stretches an undamped spring by 10 cm. Assume that g = 10 m/s^2. Include the correct units in all your answers below. (a) Find the spring constant k and its correct unit. (b) Set up the second order differential equation which governs the motion of the spring-mass system, choosing the x-axis to be oriented downwards. Find the general solution of this equation. (c) Find the particular solution of the equation if the mass is released 50 cm below the equilibrium position from rest. (d) What is the first positive time at which the mass returns to the equilibrium position?
Problem 6 (10 points)
(a) Determine the equation of the tangent plane to the graph of z = (^) xx+y through the point (2, − 1 , 2). (b) Also find parametric equations for the normal line to the graph from (a) at (2, − 1 , 2).
Problem 7 (8 points)
Find the line integral (^) ∫
C
xy^2 ds,
where C is a quarter of a unit circle centered at the origin and contained in the first quadrant, starting at (1, 0) and ending at (0, 1).
Problem 9 (11 points)
A lamina of constant density ρ(x, y) = 1 is bounded by the curves y = x^2 and y = 1. (a) Find the lamina’s mass. (b) Find the lamina’s centroid. Use geometric considerations to simplify your work.
Problem 10 (8 points)
Rewrite the function f (x, y) = x + y using polar coordinates and find its integral over the quarter disk of radius 1 in the first quadrant.
