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The final exam for the egr 265: math tools for engineering problem solving course from spring 2011. The exam covers various topics in engineering mathematics, including differential equations, calculus, and vector calculus. Students are required to find explicit solutions, evaluate decay rates, and determine the work done by force fields, among other tasks.
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EGR 265, Math Tools for Engineering Problem Solving May 2, 2011, 10:45am to 1:15pm
Name (Print last name first):..........................................
Student ID Number:...........................
Final Exam
Problem 1 (8 points)
Find an explicit solution of the initial value problem
exy
dy dx
= 1, y(0) = 2.
Problem 2 (8 points)
Note: In this problem write your answers in terms of natural logarithms, which do not need to be evaluated.
Iodine-131 has a half-life of 8 days. (a) Find its decay rate k. (b) If the initial amount of Iodine-131 is 1 gram, how much of it is left after 2 days? (c) How long does it take for Iodine-131 to decay to 10 percent of its original amount?
Problem 4 (12 points)
A mass of 12 kg stretches a spring by 40 cm. 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 30 cm below the equilibrium position at a upward velocity of 2 m/s (choose the positive x-axis to be oriented downward). (c) Find the amplitude at which the mass oscillates.
Problem 5 (10 points)
(a) Find the gradient of f (x, y) = x ln(x^2 + y). (b) Evaluate the directional derivative of f (x, y) at the point P (2, −3) in the direction of the vector i − 2 j. (c) Find a unit vector in the direction of steepest decrease of f (x, y) at the point P (2, −3). Also find the rate of decrease in this direction.
Problem 8 (12 points)
(a) Verify that the force field F (x, y) = (2x cos y)i + (cos y − x^2 sin y)j is conservative. (b) Find a potential function φ(x, y) for F (x, y). (c) Find the work done by the force field F (x, y) along the curve x = t^2 + 12 , y = t, 0 ≤ t ≤ π 2.
Problem 9 (10 points)
Find the double integral of the function f (x, y) = x^3 y^2 over the triangle in the xy-plane with vertices (0, 0), (1, −1) and (1, 1).
Problem 10 (10 points)
A lamina of density ρ(x, y) = 1 + x + y occupies the half disk R that lies above the x-axis within the circle r = 2. Find the mass of the lamina.
