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The final exam for egr 265-6d, a university course titled 'math tools for engineering problem solving', which was offered in fall 2010. The exam covers various mathematical topics including differential equations, calculus, and vector calculus. Students were required to solve problems related to these topics and provide answers in terms of natural logarithms.
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EGR 265-6D, Math Tools for Engineering Problem Solving December 10, 2010, 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
xyy′^ = 1 + y^2 , y(1) = 1.
Problem 2 (8 points)
A radioactive substance has a half-life of 10 days. The initial amount of the substance is 100 milligrams. (a) Determine the decay rate of the substance. (b) How much of the substance is left after 5 days? (c) How long does it take for the substance to decay to 10 percent of its original amount? Note: Write your answers in terms of natural logarithms, which do not need to be evaluated.
Problem 4 (12 points)
A 100 cm spring measures 140 cm 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 = 100 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 below the equilibrium position with an upward velocity of 50 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 5 (10 points)
(a) Find the gradient of f (x, y) =
2 x^2 + 2xy + y^2. (b) Evaluate the directional derivative of f (x, y) at the point P (1, 2) in the direction from P (1, 2) to Q(2, 3). (c) Find a unit vector in the direction of steepest decrease of f (x, y) at the point P (1, 2). Also find the rate of decrease in this direction.
Problem 7 (8 points)
Find the work done by the force field
F(x, y) = y^2 i +
x^2 y
j
along the curve given by the graph of y = ex, 0 ≤ x ≤ 1.
Problem 8 (12 points)
(a) Verify that the force field F (x, y) = (cos x − cos y)i + x sin yj is conservative. (b) Find a potential function φ(x, y) for F (x, y). (b) Find the work done by the force field F (x, y) along the curve x = t/2, y = (π −t)/2, 0 ≤ t ≤ π.
Problem 10 (8 points)
Let R be the region in the first quadrant which lies between the circles of radius r =
and r =
R
1 + x^2 + y^2 dA.
