Find an equation for the plane containing the line in the $x y$-plane where $y=1$, and the line in the $x z$-plane where $z=2$.

Sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced. f(x, y)=3x+3y

Find an equation for the contour of $f(x, y)=2x^2y+6x+25$ that goes through the point $(4, −2)$.

Find the equation of the surface.

A cylinder of radius $\sqrt{7}$ with its axis along the $y$-axis.

Find the equation of the vertical plane perpendicular to the $y$-axis and through the point $(7,8,9)$.

Find a point on both the planes (that is, on their line of intersection) 2 x+6 y-3 z=10 and 5 x+2 y-z=12. Find a vector parallel to the line of intersection. Write the equations of the line of intersection of the planes. Find the distance from the origin to the line.

The line of intersection of the planes x + 2y + 3z = 1 and x - y + z = 1

In the following exercise, which of (I)-(IV) lie on the graph of the equation?

I. $(2,2,4)$

II. $(-1,1,0)$

III. $(-3,-2,-1)$

IV. $(-2,-2,4)$

$$z=4$$

Calculate the mole fraction of each compound in a mixture containing $15.0$ $\mathrm{g}$ of cyclohexane and $5.0$ $\mathrm{g}$ of toluene.

What is kurtosis?

Compute each of these double sums:

a) $\sum_{i=1}^2\sum_{j+1}^3(i+j)$

b) $\sum_{i=0}^2\sum_{j=0}^3(2i+3j)$

c) $\sum_{i=1}^3\sum_{j=0}^2 i$

d) $\sum_{i=0}^2\sum_{j=1}^3 if$

Find the distance between the skew lines with parametric equations $x = 1 + t, y = 1 + 6t, z = 2t,$ and $x = 1 + 2s, y = 5 + 15s, z = -2 + 6s$.

Find the points of intersection of the line given with the coordinate planes.

$$\frac{x-4}{4}=\frac{y+3}{3}=\frac{z+2}{1}$$

Show that in a simple graph with at least two vertices there must be two vertices that have the same degree.