Calculus I, Spring 2022

Brief solutions to Quiz 4

Apr 12, 2022.

1. (40 pts) Give formal definition of lim_{(x,y)→(x_{0},y_{0})}f(x, y) = L. Does f(x, y) = \frac{x^{2}y}{x^{4} + y^{2}} have a limit at (0, 0)? Explain. Ans: Definition: See page 816 of the textbook. No. See Section 14.2, Example 6 on page 820 of the textbook.

2. (40 pts)

(a) Give formal definition of ‘f(x, y) is differentiable at (x_{0}, y_{0})’.

(b) True or False? Explain: If f(x, y) is differentiable at (x_{0}, y_{0}), then it is continuous at (x_{0}, y_{0}).

Ans: (a): See page 832 of the textbook, or page 2 of Lecture 13 note. Either one will do. (b): True. See page 832 of the textbook, or page 9 of Lecture 13 note.

3. (20 pts) Find \frac{∂z}{∂x} and \frac{∂z}{∂y} at (x, y, z) = (1, 2, 3) if z(x, y) is implicitly defined by

x^{2} + \frac{y^{2}}{4} + \frac{z^{2}}{9} = 3.

Ans:

\frac{∂}{∂x} ⟹ 2x + \frac{2z}{9} z_{x} = 0 ⟹ z_{x} = \frac{−9x}{z} = −3

\frac{∂}{∂y} ⟹ \frac{y}{2} + \frac{2z}{9} z_{y} = 0 ⟹ z_{y} = \frac{−9y}{4z} = −\frac{3}{2}