calc -3

Tennessee Technological University Fall 2014 Department of Mathematics

FINAL EXAM: Math 2110-001, Calculus III

Dr. R. Le Borne

Wednesday, December 10, 2014, 8:00 - 10:00 a.m.

Please fill in your first and last name:

Exercise 1 2 3 4 ∑

Potential Points 10 10 10 10 40 Knowledge (Points)

Performance (Points)

Actual Points

Solve the following: Of course, you must show ALL your work to receive full credit (answers are worth 1pt).

Exercise 1 (10 points)

a) (5 pts) Determine whether or not the vector field F(x, y, z) = i + sin(z)j + y cos(z)k is conservative. If it is conservative, find a function f such that F = ∇f. b) (5 pts) Find equations of (i) the tangent plane and (ii) the normal line to the surface xyz2 = 6 at the point (3, 2, 1).

Exercise 2 (10 points) Consider the following minimization problem: Find the dimensions of the rectangular box with largest volume if the surface area is given as 64cm2. Solve using EITHER the method of Min/Max. or Lagrange Multipliers.

Exercise 3 (10 points)

Find the surface area of the part of the surface z = xy that lies within the cylinder x2 + y2 = 1.

Exercise 4 (10 points)

Use Green’s Theorem to evaluate the line integral ∮ C cos(y)dx + x2 sin(y)dy, along the

positively oriented curve C that is the rectangle with vertices (0, 0), (5, 0), (5, 2), and(0, 2).

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