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The final exam for egr 265, a university course titled 'math tools for engineering problem solving', held in fall 2008. The exam covers various topics in engineering mathematics, including differential equations, calculus, and vector calculus. Students are required to solve problems related to these topics, such as finding explicit solutions of initial value problems, determining charges and currents in rc-series circuits, and calculating moment of inertia and double integrals.
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EGR 265, Math Tools for Engineering Problem Solving December 8, 2008, 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
dy dx
= ey+2x, y(0) = 0.
Problem 2 (12 points)
(a) Write down the differential equation for the charge q(t) in coulombs at the capacitor in an RC-series circuit including a resistor of R ohms, a capacitor of capacitance C farads and an impressed voltage E(t). (b) By solving an initial value problem for the differential equation from part (a) determine the charge q(t) in an RC-series circuit if R = 50 ohms, C = 2 ร 10 โ^3 farads and a constant voltage of E = 200 volts is impressed. Assume that after one second of charging the charge on the capacitor is 2 coulombs. (c) Determine the current i(t) in amperes for the circuit in part (b).
Problem 4 (11 points)
A 100-kilogram mass stretches a spring by 10cm. The spring/mass system has no damping and no exterior forcing. (a) Find a second order differential equation for the position x(t) of the mass relative to its equilibrium position. Use the approximate value g = 10 m/s^2 for the gravitation constant and assume that the positive x-direction is measured downward from the equilibrium. (b) Assuming that the mass is released 20cm below the equilibrium position from rest, determine its position x(t).
Problem 5 (11 points)
(a) Find the directional derivative of f (x, y) = xexy^ at the point (1, 0) in the direction of the vector 4i โ 3 j. (b) Find a unit vector in the direction of steepest decrease of f (x, y) = xexy^ at the point (1, 0).
Problem 6 (8 points)
Determine the equation of the tangent plane to the graph of z = (^) xxy+y at the point (3, 6 , 2).
Problem 9 (8 points)
Find the moment of inertia about the y-axis of the lamina given by the region which is bounded by the x-axis and the parabola y = 4 โ x^2.
Problem 10 (8 points)
Find the double integral of the function f (x, y) = โ^1 x^2 +y^2 + over the washer-shaped
region in the xy-plane, centered at the origin, with inner radius 1 and outer radius 2.
Problem 11 (5 points Bonus)
There are two points (x 1 , y 1 ) and (x 2 , y 2 ) at which the function
f (x, y) = x^4 โ x^2 + y^2 โ 2 xy โ 4 x + 4y
takes its minimum value. Find these two points!
