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The fall 2006 exam for math 2210-04, a calculus course. The exam includes 10 problems, each worth 10 points, covering topics such as finding slopes of tangents, sketching domains, calculating limits, and determining critical points. Students are required to justify all answers and may not use books, notes, or calculators.
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Exam # 2 Fall 2006 MATH 2210-
Instructor: Oana Veliche Time: 50 minutes
(1) Fill in your name and your student ID number. (2) There are 10 problems, each worth 10 points. (3) Justify all your answers. Correct answers with no justification will not be given any credit. (4) No books, notes or calculators may be used.
Problem 1 2 3 4 5 6 7 8 9 10 Total
Points
1
Problem 1. Find the slope of the tangent to the curve of intersection of the surface z = e−xy^2 and the plane y = 1 at the point (0, 1 , 1).
Problem 2. Consider the function of two variables: f (x, y) =
xy 4 x^2 − y^2
(a) Find and sketch the domain of the function z = f (x, y).
(b) For what points is the function f continuous?
Problem 4. Consider the function f (x, y) = 9x^2 + 4y^2. (a)Draw the level curve of that goes through the point (1, 2).
(b) In what direction is decreasing f most rapidly at (1, 2)?
Problem 5. If z = x^2 y + xy, x = st and y = s + t, find
∂z ∂t
s=1,t=− 1
Problem 6. Consider the surface given by: x^2 + 2y^2 + 3z^2 = 12. (a) Find the equation of the tangent plane to this surface at P (0, 0 , 2).
(b) Find a point on this surface where the tangent plane is perpendicular to the line with parametric equations x = 1 + 2t, y = 3 + 8t, z = 2 − 6 t.
Problem 9. Find the global minimum and global maximum values of
f (x, y) = 2x^2 − y^2 + 1 on S = {(x, y)|x^2 + y^2 ≤ 1 }. Hint: You may use that the only critical point interior to the set S is (0, 0).
Problem 10. Find the minimum of f (x, y, z) = x^2 + y^2 + z^2 subject to the constraint x + 3y − 2 z = 12.
