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The solutions to exam ii of the multivariable calculus course taught by prof. P. Wong, held on november 6, 2007. The exam covers topics such as continuity, vector calculus, directional derivatives, critical points, and surface integrals.
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EXAM II - NOVEMBER 6, 2007
Instruction: Read each question carefully. Explain ALL your work and give reasons to support your answers. Advice: DON’T spend too much time on a single problem.
Problems Maximum Score Your Score
1
2 EXAM II - NOVEMBER 6, 2007
1.(7 pts) (i) Consider the function
f (x, y) =
3 xy x^4 +y^2 ,^ if (x, y)^6 = (0,^ 0); 0 , otherwise.
Determine whether f (x, y) is a continuous at (0, 0). Justify your answer. [Hint: try approaching (0, 0) from different directions]
(8 pts.)(ii) Consider the vector field F (x, y, z) = (2xz, −xy, −z). Find div F and curl F.
4 EXAM II - NOVEMBER 6, 2007
(8 pts) (ii) Find an equation for the line tangent to the level curve f (x, y) = 1 at the point (1, 0).
MATH206A MULTIVARIABLE CALCULUS - PROF. P. WONG 5
(8 pts) (ii) For each of the critical point(s) a found in part (i), find the corresponding Hessian matrix Hf (a).
(8 pts) (iii) Use the second derivative test to classify each of the critical point(s) in part (i), i.e., determine whether the critical point is a local max, local min, or saddle point.
MATH206A MULTIVARIABLE CALCULUS - PROF. P. WONG 7
(9 pts.) Find the work done by the force F over the curve C. That is, find
C F^ (x)^ ·^ dx.
