Course Notes for Calculus III

Section 12.5 Assignment 5

1. Prove that the following limits do not exist. (Assumed the functions are defined on maximum possible domains). (10)

   1. \begin{equation*} \lim_{(x,y) \rightarrow (0,0)} \frac{x^2 + xy - y^2}{|xy|} \end{equation*}

   2. \begin{equation*} \lim_{(x,y) \rightarrow (3,3)} \frac{y-3}{x-3} \end{equation*}

2. Consider this function. (6)

   \begin{equation*} f(x,y) = \frac{x^3 + y^3}{x+y} \end{equation*}

   This function is not defined on the line \(x = -y\text{.}\) What values would you define on that line to make the function continuous?

3. Let \(a\) and \(b\) be real numbers. Consider this function. (4)

   \begin{equation*} f(x,y) = \frac{x-a}{y-b} \end{equation*}

   For what values of \(a\) and \(b\) does this limit exist?

4. Calculate all the partial derivatives of the following functions. (12)

   1. \(\displaystyle f(x,y,z) = z \sin (x + y) + x \cos (y + z)\)

   2. \(\displaystyle f(x,y,z) = x^2yz + y^z + z^2 (x^2 - y^2)\)

   3. \(\displaystyle f(x,y,z) = \frac{e^{xy} - e^{yz}}{y^2}\)

5. Draw a contour graph for the function \(f(x,y) = \frac{1}{(x^2 + 1)(y^2 + 1)}\text{.}\) Identify some points where both of the partial derivatives vanish. Describe the level curves close to these points. (8)

6. Draw a contour graph for the function \(f(x,y) = \frac{\sin x}{y^2 + 1}\text{.}\) Identify some points where both of the partial derivatives vanish. Describe the level curves close to these points. (8)

7. Let \(f(x,y)\) be a differentiable function. Prove that if \(\frac{\del f}{\del x}\) and \(\frac{\del f}{\del y}\) are constant, then the graph of \(f\) must be a plane in \(\RR^3\text{.}\) (4)

8. Let \(f(x,y)\) be a differentiable function where \(\frac{\del f}{\del x} = g(y)\) and \(\frac{\del f}{\del y} = h(x)\) for some single variable functions \(g\) and \(h\text{.}\) What does this say about the function \(f\text{?}\) (4)