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The final exam for egr 265-6d: math tools for engineering problem solving, fall 2009. The exam covers various mathematical topics including differential equations, calculus, and vector calculus. Students are required to find explicit solutions, decay rates, unit vectors, and perform integrations.
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EGR 265-6D, Math Tools for Engineering Problem Solving December 7, 2009, 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
(1 + x^2 )yy′^ = x, y(0) = 2.
Problem 2 (8 points)
A radioactive isotope has a half-life of 10 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 5 years? (c) How long does it take for the isotope 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 mass of 10 kg stretches a spring by 50 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 20 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) =
x^2 + y^3. (b) Evaluate the directional derivative of f (x, y) at the point P (1, 2) in the direction from P to the point Q(3, 3). (c) Find a unit vector in the direction of steepest decrease of f (x, y) at the point (1, 2). Also find the rate of increase in this direction.
Problem 7 (8 points)
Find the line integral (^) ∫
C
x^2 y 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 8 (12 points)
(a) Show that the force field F (x, y) = (4ey^ − 2 yex)i + (4xey^ − 2 ex)j is conservative and find a potential function φ(x, y) for it. (b) Find the work done by the force field F from part (a) along the curve x(t) = t^2 , y = t^3 , 0 ≤ t ≤ 1.
Problem 10 (10 points)
Find the double integral of the function f (x, y) = e
x^2 +y^2 over the region in the first
quadrant which is bounded by the circles r = 1 and r = 2.
