Differentiation Problems

Trivium

  1. Given a multivariable function \(f\colon \mathbf{R}^2 \to \mathbf{R}\) sketch the surface \(z = f(x,y)\) that it corresponds to, and sketch the contour plots of that surface.
  2. Given a multivariable function \(f\colon \mathbf{R}^2 \to \mathbf{R}\) such that \[ \lim\limits_{(x,y)\to(a,b)} f(x,y) \] doesn’t exist, exhibit two paths in \(\mathbf{R}^2\) terminating at \((a,b)\) along which the outputs of \(f(x,y)\) approach different values.
  3. Given a multivariable function \(f\colon \mathbf{R}^2 \to \mathbf{R},\) compute the first-order and second-order partial derivatives of \(f\) with respect to either input parameter.
  4. Given a multivariable function \(f\colon \mathbf{R}^2 \to \mathbf{R},\) its graph \(z = f(x,y),\) and a point on that graph, determine an equation of the plane tangent to the graph at that point.
  5. Given an equation involving the symbols \(x,\) \(y,\) \(z,\) etc, be able to compute the partial derivative of any of them with respect to any other, both in the case that those symbols represent variables and in the case that they represent functions of other variables (e.g. \(t\) or \(s\)). If possible, be able to solve this implicit partial derivative to express the partial derivative explicitly.
  6. Given a multivariable function \(f\colon \mathbf{R}^2 \to \mathbf{R}\) and a vector \(\bm{v},\) compute the gradient vector of \(f\) and the directional derivative of \(f\) in the direction of \(\bm{v}.\)
  7. Given a multivariable function \(f\colon \mathbf{R}^2 \to \mathbf{R}\) determine the local minimum and maximum values of that function on the interior of some open subset of its domain by analyzing its partial derivatives, gradient, and Hessian.
  8. Given a multivariable function \(f\colon \mathbf{R}^2 \to \mathbf{R}\) determine the local minimum and maximum values of that function on the boundary of some closed subset of its domain either by directly analyzing a parameterization of that boundary or by the method of Lagrange multipliers.

“Exercises” from Stewart that Look Like They Could Be the Basis for a Reasonable Exam Question

Problems & Challenges

  1. Given the function \(f\colon \mathbf{R}^2 \to \mathbf{R}\) defined by the formula \(f(x,y) = 1-x\mathrm{e}^{xy}\,,\) consider the graph \(z = f(x,y)\) and let \(p\) be the point \((1,1)\) in the domain of \(f.\)

    1. Calculate the gradient of \(f,\) and list all points such that the gradient is \(\bm{0}.\)
    2. Calculate the directional derivative of \(f\) at the point \(p\) in the direction \(\langle4,-3\rangle.\)
    3. Write down an equation for the plane tangent to the graph of \(f\) at \(p.\)
    4. Does there exist a direction \(\bm{u}\) in which the directional derivative of \(f\) at \(p\) in the direction \(\bm{u}\) is \(6\)?
    5. Explain why there must exist a direction \(\bm{u}\) in which the directional derivative of \(f\) at \(p\) in the direction \(\bm{u}\) is zero, and calculate a unit vector in that direction.

  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?