Suppose that $z$ is a linear function of $x$ and $y$ with slope $2$ in the $x$-direction and slope $3$ in the $y$-direction.

(a) A change of $0.5$ in $x$ and $-0.2$ in $y$ produces what change in $z?$

Find the equation of the linear function $z=c+m x+n y$ whose graph contains the points $(0,0,0),(0,2,-1)$, and $(-3,0,-4).$

Sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced. $f(x, y)=y-x^2$

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

While doing a pedigree analysis of a royal family from a Pacific Island nation, you notice two parents with the same genetic disease who have four children. All four of the girls have the disease, but when they marry unaffected individuals, none of the 16 grandchildren has the disease. If this disease is controlled by a single gene, then the most likely explanation would be that the disease is

a. a spontaneous mutation

b. recessive and is only seen in homozygous recessive individuals such as the affected girls

c. dominant and is only seen in homozygous dominant individuals

d. dominant and is seen in heterozygous or homozygous dominant Individuals

Which of the following bases are strong enough to deprotonate

$$\mathrm { CH } _ { 3 } \mathrm { CH } _ { 2 } \mathrm { CH } _ { 2 } \mathrm { C } \equiv \mathrm { CH } \left( \mathrm { p } K _ { \mathrm { a } } = 25 \right),$$

so that equilibrium favors the products:

$$\text { (a) } \mathrm { H } _ { 2 } \mathrm { O } ; ( \mathrm { b } ) \mathrm { NaOH };( \mathrm { c } ) \mathrm { NaNH } _ { 2 } ; ( \mathrm { d } ) \mathrm { NH } _ { 3 }; \text { (e) } \mathrm { NaH } ; \text { (f) } \mathrm { CH } _ { 3 } \mathrm { Li } \text { ? }$$

The figure shows a contour map of a hill with two paths, A and B.

On which path, A or B, will you have to climb more steeply?

Sketch a graph of the surface and briefly describe it in words.

$$x^2+z^2=4$$

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 $(7,8,9)$.

A manufacturer sells two goods, one at a price of $\$ 3000$ a unit and the other at a price of $\$ 12,000$ a unit. A quantity $q_1$ of the first good and $q_2$ of the second good are sold at a total cost of $\$ 4000$ to the manufacturer.

(b) Sketch curves of constant profit in the $q_1 q_2$-plane for $\pi=10,000, \pi=20,000$, and $\pi=30,000$ and the break-even curve $\pi=0$.

Match the function with the shapes of their level curves (I)-(IV). Sketch each contour diagram.

(a) $f(x, y)=x^2$

$\mathrm{I.}$ Lines $\quad\mathrm{II.}$ Parabolas

$\mathrm{III.}$ Hyperbolas $\quad\mathrm{IV.}$ Ellipses

A manufacturer sells two goods, one at a price of $\$ 3000$ a unit and the other at a price of $\$ 12,000$ a unit. A quantity $q_1$ of the first good and $q_2$ of the second good are sold at a total cost of $\$ 4000$ to the manufacturer.

(a) Express the manufacturer's profit, $\pi$, as a function of $q_1$ and $q_2$.

Match the function with the shapes of their level curves (I)-(IV). Sketch each contour diagram.

(b) $f(x, y)=x^2+2 y^2$

$\mathrm{I.}$ Lines $\quad\mathrm{II.}$ Parabolas

$\mathrm{III.}$ Hyperbolas $\quad\mathrm{IV.}$ Ellipses