Find the points on the surface

$$x y + y z + z x - x - z ^ { 2 } = 0$$

where the tangent plane is parallel to the xy-plane.

Determine whether the following statements are true and give an explanation or counterexample.

The planes tangent to the cylinder $x ^ { 2 } + z ^ { 2 } = 1$ in $\mathbb { R } ^ { 3 }$ all have the form $ax+bz+c=0$.

Define the term melting point.

Use Lagrange multipliers to find three positive numbers whose sum is $100$ and whose product is a maximum.

Find three positive numbers whose sum is 100 and whose product is a maximum.

Find two integers that make each equation true.

$$a^2=9$$

A gas turbine combustion chamber may be approximated as a long tube of 0.4-m diameter. The combustion gas is at a pressure and temperature of 1 atm and $1000^{\circ} \mathrm{C}$, respectively, while the chamber surface temperature is $500^{\circ} \mathrm{C}$. If the combustion gas contains $\mathrm{CO}_{2}$ and water vapor, each with a mole fraction of 0.15, what is the net radiative heat flux between the gas and the chamber surface, which may be approximated as a blackbody?

Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n=0, 1, and 2.

f(x)=$\sqrt [ 3 ] { x }$, a=8

Find each limit algebraically. $\lim _{x \rightarrow 2} \frac{x^{3}-8}{x^{2}-4}$

Find the sum. −3 + 9

Given the following ANOVA output, answer the questions that follow.

$$\begin{matrix} \text{Analysis of Variance for Response}\\ \text{Source} & \text{df} & \text{SS} & \text{MS} & \text{F} & \text{P}\\ \text{Factor A} & \text{2} & \text{1209.7} & \text{604.8} & \text{8.32} & \text{0.002}\\ \text{Factor B} & \text{2} & \text{577.6} & \text{288.8} & \text{3.97} & \text{0.031}\\ \text{Interaction} & \text{4} & \text{1474.9} & \text{368.7} & \text{5.07} & \text{0.004}\\ \text{Error} & \text{27} & \text{1962.2} & \text{72.7}\\ \text{Total} & \text{35} & \text{5224.3}\\ \end{matrix}$$

Is there evidence of an interaction effect? Why or why not?

Find a counterexample for the conjecture. Three coplanar lines always make a triangle.

Find an equation of the tangent plane to the given surface at the specified point. $z=x e^{x y}, \quad(2,0,2)$

Write the repeating decimal as a fraction in simplest form.

$$0.888 \dots$$