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Exam questions covering topics such as directional derivatives, level sets, partial differential equations, and critical points of real-valued functions. Students are expected to demonstrate their understanding of these concepts through problem-solving. The questions include drawing vectors and regions in the coordinate plane, calculating partial derivatives, and finding tangent lines.
Typology: Exams
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Exam 2- In-Class Portion
Show all your work to receive full credit for a problem.
(a) (3 pts) At the point (1/2,1/2), draw the vector v = 12 i โ 12 j. Is the directional derivative in the direction of v going to be positive, negative, or zero? Explain your reasoning.
(b) (3 pts) At the same point, draw the approximate direction of the gradient, and briefly explain your reasoning.
(a) (3 pts) Draw S in the coordinate axes below.
(b) (3 pts) Is S open, closed or neither?
(c) (3 pts) Is S bounded?
(d) (3 pts) Let f (x, y) = x^4 y^5 sin x cos y. Does f attain a maximum value on S? Does f attain a minimum value on S? For both questions, explain why or why not.
Jf (0, 0) =
Explain how you would use these values to find an approximation for f (x, y) where (x, y) is very close to the origin.
(b) (6 pts) Now suppose g : R^3 โ R^2 is differentiable at (1, 0 , โ1), satisfies g(1, 0 , โ1) = (0, 0) and has Jacobian matrix
Jg(1, 0 , โ1) =
Explain how you can use these values and the values in part (a) to find an ap- proximation for (f โฆ g)(x, y, z) where (x, y, z) is very close to (1, 0 , โ1).
and at these points, the Hessians are
Hf (1, 0) =
Hf (โ 1 , 0) =
Hf (1, 4) =
Hf (โ 1 , 4) =
(a) (6 pts) Fill in the following equalities, and briefly explain your reasoning. โ^2 f โx^2
โ^2 f โyโx
โf (โ 1 , 4) =
(b) (6 pts) Classify the critical points as local maxima, local minima, saddle points, or none of these.
