Calculus II, Spring 2016

Quiz 3

Apr 14, 2016

1. Give formal definitions of (1): lim_{(x,y)→(x_{0},y_{0})} f(x, y) = L, and (2): The function z = f(x, y) is differentiable at (x_{0}, y_{0}).

2. Does lim_{(x,y)→(0,0)} \frac{x^{3} − xy^{2}}{x^{2} + y^{2}} exist? Explain.

3. Let f(x, y) = \frac{xy^{2}}{x^{2} + y^{4}} for (x, y) ≠ (0, 0) and f(0, 0) = 0. Do f_{x}(0, 0) and f_{y}(0, 0) exist? Explain. Is f continuous at (0, 0)? Explain.

4. Let F (x) = ∫_{0}^{x} cos(x^{2} − t^{2})dt. Evaluate F′(0). Give details.

5. Let f(x, y) = x^{2} − xy + 2y^{2}. Find the direction u (a unit vector) for which the directional derivative (\frac{df}{ds})_{u,(1,1)} (that is, D_{u}f(1, 1)), is largest.

Calculus II, Spring 2016

Quiz 3

Apr 14, 2016

1. Give formal definitions of (1): lim_{(x,y)→(x_{0},y_{0})} f(x, y) = L, and (2): The function z = f(x, y) is differentiable at (x_{0}, y_{0}).

2. Does lim_{(x,y)→(0,0)} \frac{x^{3} − xy^{2}}{x^{2} + y^{2}} exist? Explain.

3. Let f(x, y) = \frac{xy^{2}}{x^{2} + y^{4}} for (x, y) ≠ (0, 0) and f(0, 0) = 0. Do f_{x}(0, 0) and f_{y}(0, 0) exist? Explain. Is f continuous at (0, 0)? Explain.

4. Let F (x) = ∫_{0}^{x} cos(x^{2} − t^{2})dt. Evaluate F′(0). Give details.

5. Let f(x, y) = x^{2} − xy + 2y^{2}. Find the direction u (a unit vector) for which the directional derivative (\frac{df}{ds})_{u,(1,1)} (that is, D_{u}f(1, 1)), is largest.