Genes produce molecules called mRNA that then produce proteins. High levels of protein can inhibit the production of mRNA, resulting in a feedback that regulates gene expression. Using m and p to denote the amounts of mRNA and protein in a cell $\left(\times 10^{2} \text { copies } / \mathrm{cell}\right),$ a simple model of gene regulation is

$$\begin{array}{l} {d m / d t=1-p-m} \\ {d p / d t=m-p} \end{array}$$

Construct the phase plane, including the nullclines.

Find the domain of the function. $f(x)=\frac{1}{3+e^{x}}$

True or False. $\frac{d^{2}}{d x^{2}} \sin x=-\sin x$

Indicate whether the statement is true or false. If the marginal cost function of a product is $C^{\prime}(q)=0.03 q+0.1$ dollars, and the fixed costs are $1500, then the total cost to produce 1000 items is$15,100.

A physics student uses a converging lens with a focal length of 14 cm to read a small measurement scale. (a) Maximum magnification is achieved if the image is at (1) the near point, (2) infinity, (3) the far point. Explain. (b) What are the magnifications when the image is at the near point and infinity, respectively?

Show that $A=P D P^{-1},$ where P is a matrix whose columns are the eigenvectors of A, and D is a diagonal matrix with the corresponding eigenvalues.

$$A=\left[\begin{array}{lll} {1} & {0} & {1} \\ {2} & {1} & {0} \\ {3} & {0} & {1} \end{array}\right]$$

Find the maximum and minimum values of $f(x, y)=x-x^{2}+y^{2}$ on the rectangle $0 \leq x \leq 2$, $0 \leq y \leq 1$

Let $f(x)=x^{2}+2$. Find all points on the graph of f for which the tangent line passes through the origin.

True/False. $\mathbf{F}=\left(e^{x} \cos y+3\right) \mathbf{i}-e^{x} \sin y \mathbf{j}$ is a conservative vector field on $R^2.$

Consider the line integral $\int_{C} z^{2} d x+2 y d y+x z d z$. (a) Evaluate this integral, where C is the line segment from (0, 0, 0) to (1, 1, 1). (b) Evaluate this integral, where C is the path from (0, 0, 0) to (1, 1, 1) parametrized by $\mathbf{x}(t)=\left(t, t^{2}, t^{3}\right), 0 \leq t \leq 1$. (c) Is the vector field $\mathbf{F}=z^{2} \mathbf{i}+2 y \mathbf{j}+x z \mathbf{k}$ conservative? Why or why not?

Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point. $y^{2}=\frac{x^{2}-49}{x^{2}+49}$, (7, 0)

Locate and classify all local extreme values of the given function. Determine whether any of these extreme values are absolute. Sketch the graph of the function. $f(x)=\left(x^{2}-4\right)^{2}$

Differentiate the given function and simplify the answer whenever possible. $y=\sin ^{-1} \frac{a}{x}$

Find the maximum rate of change of $f$ at the given point and the direction in which it occurs. $f(p, q)=q e^{-p}+p e^{-q}, \quad(0,0)$