MATH 105 101
Assignment 2
Due date: October 2, 2014
MATH 105 101 Assignment 2
All work must be shown for full marks.
1. Find both
first-order partial derivatives of the function g(x, y) = y sin (x
√
point ( 2π , 49). Simplify your answers. (4 marks)
2− π )
12
at the
2. Let f (x, y) be a two-variable function defined on R2 such that all first-, second- and
third- order partial derivatives are continuous on R2 . Suppose that at the point (3, 2),
the third order partial derivatives of f satisfy the equation:
2(fxyx (3, 2))2 − 5fyxx (3, 2) = 7
Find all possible values for fxxy (3, 2). Clearly state any theorem that is used.(4 marks)
3. Find and classify all critical points of the function f (x, y) = x3 − 12x + y 3 + 3y 2 − 9y.
(6 marks)
1 3
2
4. Find the absolute maximum and minimum values of the function f (x, y) = e− 3 x +x−y
in the square S = {(x, y) ∈ R2 : −2 ≤ x ≤ 2, −2 ≤ y ≤ 2}. Indicate those points at
which f attains those values. (6 marks)
5. Find the largest area of a rectangle that can be inscribed in the ellipse x2 + 2y 2 = 1,
and that its sides are parallel to the axes. (5 marks)
Total: 25 marks.
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