Modern Computing for Mathematicians (Math 587)
Written Assignment No. 4

1.  Directions

Use GELLMU to write solutions for the following Calculus III exercises.

Re-state each exercise before presenting its solution. Write each solution carefully as if it were to be included as an example in a calculus textbook.

Submit in writing:

• A verbatim listing of your generalized LaTeX source.

• A printout of the PDF output.

• A verbatim listing of the <math> element in the XHTML + MathML output file that has the greatest length as a string.

• The URL in your website at www.albany.edu where your XHTML + MathML output may be found.

Note that this assignment sheet originated with the GELLMU source amcm090407.glm.

2.  Exercises

1. Find the equation of the plane in ${\mathbf{R}}^3$ passing through the point $\left(1,-2,2\right)$ that is parallel to the plane given by the equation $$2x-3y+z=0\text{.}$$

2. When $$g\left(x,y,z\right)=xz+y{e}^x$$ find the second order partial derivatives: $$\text{(a)}\frac{{\partial }^{2}g}{\partial x^2}\text{(b)}\frac{{\partial }^{2}g}{\partial z\partial x}\text{(c)}\frac{{\partial }^{2}g}{\partial y^2}$$

3. Find the equation of the plane in ${\mathbf{R}}^3$ that is tangent at the point $\left(6,-3,1\right)$ to the ellipsoid given by the equation $$\frac{x^2}{4}+\frac{y^2}{9}+z^2=11\text{.}$$