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The final exam for the egr/ma 265 course, math tools for engineering problem solving. The exam covers various topics in mathematics, including differential equations, calculus, and vector calculus. Students are required to find explicit solutions, evaluate decay rates, and determine parametric equations, among other tasks.
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EGR/MA 265, Math Tools for Engineering Problem Solving May 03, 2010, 10:45 AM - 1:15 PM
Name (Print Last Name First):..........................................
Student Signature:......................................................
Problem 1 (8 points)
Find an explicit solution of the initial value problem
2(1 + x)yy′^ = 1, y(0) = − 1.
Problem 2 (8 points)
A radioactive isotope has a half-life of 20 years. (a) Find its decay rate k (which should be a negative number). (b) If the initial amount of the isotope is 1 gram, how much of it is left after 10 years? (c) How long does it take for the isotope to decay to 20 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 mass of 10 kg stretches a spring by 100 cm. Include the correct units in all your answers below. (a) Find the spring constant k, assuming that g = 10 m/s^2. (b) What is the frequency at which the mass oscillates? (c) Find the equation of motion of the mass if it is released from rest at a position 50 cm below the equilibrium position (choose the positive x-axis to be oriented downward). (d) Find the first positive time at which the mass passes through the equilibrium position.
Problem 5 (10 points)
(a) Find the gradient of f (x, y) = ln
x^2 + y^2
(b) Evaluate the directional derivative of f (x, y) at the point P (1, 0) in the direction from P to the point Q(4, 4). (c) Find a unit vector in the direction of steepest decrease of f (x, y) at the point (1, 0). Also find the rate of increase in this direction.
Problem 7 (8 points)
Find the line integral (^) ∫
C
xy^4 ds,
where C is the right half of the unit circle centered at the origin, starting at (0, −1) and ending at (0, 1).
Problem 8 (12 points)
(a) Is the force field F (x, y) = 3x^2 y^4 i + 4x^3 y^3 j conservative? If your answer is yes, then find a potential function Φ(x, y) for it. (b) Find the work done by the force field F from part (a) along a portion of the parabola given by the parametric equations x(t) = t, y = t^2 , 0 ≤ t ≤ 1.
Problem 10 (10 points)
Find the double integral of the function f (x, y) = x^2 + y^2 over the region in the second quadrant which is bounded by the circles r = 2 and r = 4.
