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The test questions for egr 265, math tools for engineering problem solving, held on april 18, 2011. The test covers various topics in multivariable calculus, including finding partial derivatives, directional derivatives, tangent planes, normal lines, and evaluating integrals and work. Students are required to solve problems using the given functions and curves.
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EGR 265, Math Tools for Engineering Problem Solving April 18, 2011, 50 minutes
Name (Print last name first):..........................................
Problem 1 (9+9 points) (a) Let f (x, y) = 2x^2 y^2 + 3y^3. Find fxx + fyy.
(b) For the function g(x, y) = xexy^ find gx, gy and gyx.
Problem 2 (9+9 points) (a) For the functionat the point P (1, 1). h(x, y) = ln(x^2 + y^2 ) find its direction and rate of steepest descent
(b) Find the directional derivative of v = − 4 i + 3j. h(x, y) at P (1, 1) in the direction of the vector
Problem 4 (12 points) Evaluate 0 ≤ t ≤ π/ ∫2. C xy^4 ds, where the curve C is the quarter circle x = 2 cos t, y = 2 sin t,
Problem 5 (12 points) Find the work done by the force field F (x, y) = yex^3 i + 4xyj along the curve C given by the graph of y = x^2 , 0 ≤ x ≤ 1.
Problem 6 (5+5 points) Determine for each of the following force fields if it is conservative. (a) F (x, y) = yexi + xeyj
(b) F (x, y) = 2xyi + (x^2 + y)j
Problem 7 (12 points) For the conservative force fieldand calculate the work done by the force field along the curve traced by the vector F (x, y) from Problem 6 find a potential function φ(x, y) function r(t) = t^2 sin(t)i + t^2 cos(t)j, 0 ≤ t ≤ π.
