Calculus II, Spring 2014

Quiz 3

April 17, 2014

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1. Give definition of lim_{(x,y)→(x_{0},y_{0})} f(x, y) = L. Let f(x, y) = (x^{3} − xy^{2})/(x^{2} + y^{2}) for (x, y) ≠ (0, 0). Does lim_{(x,y)→(0,0)} f(x, y) exist? Explain.

2. Give definition of continuity of g(x, y) at a point (x_{0}, y_{0}). Let g(x, y) = x^{2}y/(x^{4} + y^{2}) for (x, y) ≠ (0, 0) and g(0, 0) = 0. Is g(x, y) continuous at (0, 0)? Explain.

3. Give definition of differentiability of a function of two variables. Is g(x, y) differentiable at (0, 0)? Explain.

4. Define w(r, θ) = f(x, y) where x = r cos θ, y = r sin θ. Express w_{r}^{2} + w_{θ}^{2}/r^{2} in terms of partial derivatives of f.

5. Evaluate \frac{d}{dx} ∫_{x^{2}}^{1} \sqrt{t^{3} + x^{2}}dt

Calculus II, Spring 2014

Quiz 3

April 17, 2014

Show all details.

1. Give definition of lim_{(x,y)→(x_{0},y_{0})} f(x, y) = L. Let f(x, y) = (x^{3} − xy^{2})/(x^{2} + y^{2}) for (x, y) ≠ (0, 0). Does lim_{(x,y)→(0,0)} f(x, y) exist? Explain.

2. Give definition of continuity of g(x, y) at a point (x_{0}, y_{0}). Let g(x, y) = x^{2}y/(x^{4} + y^{2}) for (x, y) ≠ (0, 0) and g(0, 0) = 0. Is g(x, y) continuous at (0, 0)? Explain.

3. Give definition of differentiability of a function of two variables. Is g(x, y) differentiable at (0, 0)? Explain.

4. Define w(r, θ) = f(x, y) where x = r cos θ, y = r sin θ. Express w_{r}^{2} + w_{θ}^{2}/r^{2} in terms of partial derivatives of f.

5. Evaluate \frac{d}{dx} ∫_{x^{2}}^{1} \sqrt{t^{3} + x^{2}}dt