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The problem set for test iii of the egr 265: math tools for engineering problem solving course. It includes six problems covering various topics such as partial derivatives, directional derivatives, tangent planes, normal lines, line integrals, and work done by force fields. Students are required to find solutions for given mathematical problems.
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EGR 265, Math Tools for Engineering Problem Solving November 20, 2008, 50 minutes
Name (Print last name first):.......................................... Student ID Number:........................... TEST III
Problem 1 (8+8 points) (a) Let f (x, y, z) = x^3 y^3 z^3. Find the third order partial derivative fxyz.
(b) Find the directional derivative ofdirection of the vector 3i − 4 j. g(x, y) = sin(xy^2 ) at the point (π, 1) in the
Problem 1 (10+5 points) (a) Find a unit vector in the direction of steepest descent of(1, 1). f (x, y) = (^) x (^21) +y at the point
(b) Also, find a unit vector parallel to the level curve ofpoint (1, 1). f (x, y) = (^) x (^21) +y through the
Problem 4 (15 points) Calculate the line integral ∫ C^ √1 + 4y ds, where C is the graph of y = x^2 , 0 ≤ x ≤ 2.
Problem 5 (15 points) Find the work done by the force field F (x, y) = ey^2 i + xyj along the curve parametrized by x = t^4 , y = t^2 , 0 ≤ t ≤ 1.
Problem 6 (16+6 points) (a) Is the force field F (x, y) = (1 + 2xy^2 )i + (3y^2 + 2yx^2 )j conservative? If yes, find a potential for F.
(b) For the force fieldparametrized by F (x, y) from part (a) find the work done by F along the curve x = 2 cos t, y = 2 sin t, 0 ≤ t ≤ π/ 2.
