Calculus of Multivariable Functions: Surfaces

Exercises

  1. Sketch the level curves of each of the following functions \(f\colon \mathbf{R}^2 \to \mathbf{R}.\)

    \(\displaystyle f(x,y) = xy\)
    \(\displaystyle f(x,y) = \frac{1}{xy}\)
    \(\displaystyle f(x,y) = \frac{1}{x-y}\)
    \(\displaystyle f(x,y) = 6-3x-2y\)
    \(\displaystyle f(x,y) = y\mathrm{e}^x\)
    \(\displaystyle f(x,y) = \sqrt{9-x^2-y^2}\)
    \(\displaystyle f(x,y) = y^2-x^2\)
    \(\displaystyle f(x,y) = 9-x^2-y^2\)
    \(\displaystyle f(x,y) = 4x^2+y^2+1\)
    \(\displaystyle f(x,y) = 3x^2+y^2\)
    \(\displaystyle f(x,y) = x\ln(y)\)
    \(\displaystyle f(x,y) = y\mathrm{e}^x\)
    \(\displaystyle f(x,y) = \cos(x)\sin(y)\)
    \(\displaystyle f(x,y) = \arctan(y/x)\)

  2. Sketch the graphs of the following functions \(f\colon \mathbf{R}^2 \to \mathbf{R}.\) It may help to consider the level curves first, or it may help to remember that the graph is defined as \(z = f(x,y),\) replace the \(“f(x,y)”\) with a \(“z”\) and see if the resulting equation looks familiar.

    \(\displaystyle f(x,y) = x\)
    \(\displaystyle f(x,y) = x^2\)
    \(\displaystyle f(x,y) = \cos(x)\)
    \(\displaystyle f(x,y) = 1/y\)
    \(\displaystyle f(x,y) = 6-3x-2y\)
    \(\displaystyle f(x,y) = y^2-x^2\)
    \(\displaystyle f(x,y) = 9-x^2-y^2\)
    \(\displaystyle f(x,y) = \sqrt{9-(x-1)^2-(y-2)^2}\)
    \(\displaystyle f(x,y) = 3x^2+y^2\)
    \(\displaystyle f(x,y) = x\ln(y)\)
    \(\displaystyle f(x,y) = y\mathrm{e}^x\)
    \(\displaystyle f(x,y) = \cos(x)\sin(y)\)
    \(\displaystyle f(x,y) = \cos(x)+\sin(y)\)
    \(\displaystyle f(x,y) = \arctan(y/x)\)
    \(\displaystyle f(x,y) = x^2-3xy+y^2 \)
    \(\displaystyle f(x,y) = xy^2-x^3\)
    \(\displaystyle f(x,y) = xy^3-x^3y \)

  3. Describe the level surfaces for each of the following functions \(f\colon \mathbf{R}^3 \to \mathbf{R}.\)

    \(\displaystyle f(x,y,z) = x^2+y^2+z^2\)
    \(\displaystyle f(x,y,z) = x^2-y-z^2\)
    \(\displaystyle f(x,y,z) = x^2+y^2-z^2\)
    \(\displaystyle f(x,y,z) = 2x-3y+4z\)
    \(\displaystyle f(x,y,z) = xyz\)
  4. Show that \(\lim\limits_{(x,y) \to (0,0)}\frac{x^2-y^2}{x^2+y^2}\) doesn’t exist.

  5. For each of the following limits, either prove the limit doesn’t exist, or convince yourself the limit does exist and determine its value.

    \(\displaystyle \lim\limits_{(x,y) \to (0,0)}\frac{xy}{x^2+y^2}\)
    \(\displaystyle \lim\limits_{(x,y) \to (0,0)}\frac{xy^2}{x^2+y^4}\)
    \(\displaystyle \lim\limits_{(x,y) \to (1,2)} x^2y^3-x^3y^2+3x+2y\)
    \(\displaystyle \lim\limits_{(x,y) \to (-2,3)}\frac{x^2y+1}{x^3y^2-2x}\)
    \(\displaystyle \lim\limits_{(x,y) \to (0,0)}\frac{3x^2y}{x^2+y^2}\)
    \(\displaystyle \lim\limits_{(x,y) \to (\pi,\pi/2)} y\sin(x-y)\)
    \(\displaystyle \lim\limits_{(x,y) \to (1,2)} \frac{2x-y}{4x^2-y^2}\)
    \(\displaystyle \lim\limits_{(x,y) \to (0,0)} \frac{5xy}{x^2+3y^2}\)
    \(\displaystyle \lim\limits_{(x,y) \to (0,0)} \frac{(x+7y)^2}{x^2+49y^2} \)

  6. For each of the following limits, prove that the limit exists and determine its value. Hint: describe the domain in polar coordinates.

    \(\displaystyle \lim\limits_{(x,y) \to (0,0)} \frac{x^3+y^3}{x^2+y^2}\)
    \(\displaystyle \lim\limits_{(x,y) \to (0,0)} \bigl(x^2+y^2\bigr)\ln\bigl(x^2+y^2\bigr)\)
  7. For \(f(x,y) = x^3+x^2y^3-2y^2,\) calculate \(f_x(2,1)\) and \(f_y(2,1).\)
  8. For \(f(x,y) = \sin\bigl(\frac{x}{1+y}\bigr)\) calculate \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}.\)
  9. For \(f(x,y) = 4-x^2+-2y^2\) calculate \(f_x(1,1)\) and \(f_y(1,1)\) and interpret these numbers as slopes.
  10. Suppose \(z\) is defined implicitly as a function of \(x\) and \(y\) by the equation \(x^3+y^3+z^3+6xyz+4=0.\) Calculate \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) and evaluate these partial derivatives at \(\bigl(-1,1,2\bigr).\)
  11. For \(f(x,y,z) = \mathrm{e}^{xy}\ln(z),\) determine formulas for \(f_x\) and \(f_y\) and \(f_z.\)

  12. Differentiable Functions For each of the following functions \(f\colon \mathbf{R}^2 \to \mathbf{R},\) determine formulas for the partial derivatives \(f_x(x,y)\) and \(f_y(x,y).\)

    \(\displaystyle f(x,y) = 3x^2+y^2\)
    \(\displaystyle f(x,y) = x^7y^2-2\sqrt{x}y^2+7\)
    \(\displaystyle f(x,y) = x^2-3xy+y^2 \)
    \(\displaystyle f(x,y) = xy^2-x^3\)
    \(\displaystyle f(x,y) = xy^3-x^3y \)
    \(\displaystyle f(x,y) = x\ln(y)\)
    \(\displaystyle f(x,y) = (x+y)\ln(xy)\)
    \(\displaystyle f(x,y) = y\mathrm{e}^x\)
    \(\displaystyle f(x,y) = \cos(x)\sin(y)\)
    \(\displaystyle f(x,y) = \cos^2(xy)\)
    \(\displaystyle f(x,y) = \arctan(y/x)\)
    \(\displaystyle f(x,y) = \frac{1}{x^2-y^2} \)
    \(\displaystyle f(x,y) = \frac{\ln(xy)}{x+y} \)
    \(\displaystyle f(x,y) = \frac{\sin(xy)}{\cos(x+y)} \)
  13. Determine formulas for all the second-order partial derivatives of the function \(f(x,y) = x^3+x^2y^3-2y^2.\)
  14. For \(f(x,y,z) = \sin(3x+yz),\) determine a formula for \(f_{xxyz}(x,y,z).\)
  15. For each of the functions \(f\) defined in the problem labelled “Differentiable Functions”, compute formulas for the second-order partial derivatives \(f_{xx}(x,y)\) and \(f_{yy}(x,y)\) and \(f_{xy}(x,y)\) and \(f_{yx}(x,y).\)
  16. Determine a formula for the plane tangent to the elliptic paraboloid \(z = 2x^2+y^2\) at the point \((1,1,3).\)
  17. Determine equations for (1) the tangent plane and (2) the normal line to the ellipsoid \(\frac{1}{4}x^2 + y^2 + \frac{1}{9}z^2 = 3\) at the point \((-2,1,-3).\)
  18. What is a formula for the plane tangent to the surface defined parametrically as \(\bm{r}(s,t) = \bigl\langle s^2, t^2, s+2t \bigr\rangle \) at the point \(\bigl(1,1,3\bigr)?\)

Problems & Challenges

  1. James Stewart For what values of the number \(r\) is the following function continuous on \(\mathbf{R}^3?\)

    \(\displaystyle f(x,y,z) = \begin{cases} \frac{(x+y+z)^r}{x^2+y^2+z^2} \quad&\text{ if } (x,y,z) \neq 0 \\ 0 \quad&\text{ if } (x,y,z) = 0 \end{cases} \)

  2. James Stewart Consider Laplace’s equation

    \(\displaystyle \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0\,. \)
    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 \[ \frac{\partial^2 u}{\partial \rho^2} + \frac{2}{\rho} \frac{\partial u}{\partial \rho} + \frac{\cot(\varphi)}{\rho^2} \frac{\partial u}{\partial \varphi} + \frac{1}{\rho^2} \frac{\partial^2 u}{\partial \varphi^2} + \frac{1}{\rho^2\sin^2(\varphi)} \frac{\partial^2 u}{\partial \theta^2} = 0 \,. \]
  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 Among all planes that are tangent to the surface \(xy^2z^2 = 1,\) find the ones that are farthest from the origin.