Differentiation Problems

  1. TK

  2. James Stewart

    For what values of the number \(r\) is the following function continuous on \(\mathbb{R}^3?\) \[ f(x,y,z) = \begin{cases} \frac{(x+y+z)^r}{x^2+y^2+z^2} \quad&\text{ if } (x,y,z) \neq 0 \\ 0 \quad&\text{ if } (x,y,z) = 0 \end{cases} \]

  3. James Stewart

    Suppose \(f\) is a differentiable function of one variable. Show that all tangent planes to the surface \(z = xf(y/x)\) intersect in a common point.

  4. James Stewart

    Consider Laplace’s equation \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0\,. \]

    1. Show that, when written in cylindrical coordinates, Laplace’s equation becomes \[ \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} + \frac{\partial^2 u}{\partial z^2} = 0 \,. \]
    2. Show that, when written in spherical coordinates, Laplace’s equation becomes \[ \frac{\partial^2 u}{\partial \rho^2} + \frac{2}{\rho} \frac{\partial u}{\partial \rho} + \frac{\cot(\varphi)}{\rho^2} \frac{\partial u}{\partial \varphi} + \frac{1}{\rho^2} \frac{\partial^2 u}{\partial \varphi^2} + \frac{1}{\rho^2\sin^2(\varphi)} \frac{\partial^2 u}{\partial \theta^2} = 0 \,. \]

  5. James Stewart

    Among all planes that are tangent to the surface \(xy^2z^2 = 1,\) find the ones that are farthest from the origin.

  6. James Stewart

    If the ellipse \(x^2/a^2 + y^2/b^2 = 1\) is to enclose the circle \(x^2 + y^2 = 2y,\) what values of \(a\) and \(b\) minimize the area of the ellipse?