EGR 265-6D Fall 2011 Final Exam: Math Tools for Engineering Problem Solving, Exams of Mathematics

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EGR 265-6D, Math Tools for Engineering Problem Solving

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Number Score Max Score 1 8 2 8 3 14 4 12 5 10 6 8 7 8 8 12 9 12 10 8 Total 100

• For graders only

EGR 265-6D, Math Tools for Engineering Problem Solving December 9, 2011, 1:30pm to 4:00pm

Name (Print last name first):..........................................

Student ID Number:...........................

Final Exam

Problem 1 (8 points)

Find an explicit solution of the initial value problem

3 y^2 y′^ = 2x(y^3 + 1), y(0) = 0.

Problem 3 (14 points)

Consider the second order differential equation

y′′^ − 3 y′^ + 2y = 2 cos(x) + sin(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 1 m spring measures

m long after a mass of 10 kg is attached to it. The medium

through which the mass moves offers a damping force with damping coefficient β = 80 kg/s. Include the correct units in all your answers below. (a) Find the spring constant k, assuming that g = 10 m/s^2. (b) Find the equation of motion of the mass if it is released from a position 10 cm above the equilibrium position with a downward velocity of 40 cm/s (Choose the positive x- axis to be oriented downward). (c) Will the mass return to the equilibrium position? If yes, when is the first time? If no, why not?

Problem 6 (8 points)

Determine the equation of the normal line to the graph of z =

ln x x + y

• 1 through the

point (1, 0 , 1).

Problem 7 (8 points)

Find the work done by the force field

F(x, y) = y^2 i − exj

along the curve given by the graph of y = x

2 2 −^ 1, 0^ ≤^ x^ ≤^ 1.

Problem 9 (12 points)

A lamina of density ρ(x, y) = x is bounded by the triangle with vertices (0, 0), (1, 0) and (1, 1). (a) Find the lamina’s moment of inertia Iy with respect to the y-axis. (b) Find the lamina’s moment of inertia Ix with respect to the x-axis.

Problem 10 (8 points)

Let R be the region in the second quadrant which lies between the circles of radius r =

2 and r =

5. Find ∫

R

1 + x^2 + y^2 ) dA.

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