Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. $\lim _{x \rightarrow 0} \frac{x}{e^{x}}=1$

Solve the system of first-order linear differential equations. $y_{1}^{\prime}=2 y_{1} \\ y_{2}^{\prime}= y_{2}$

A person running a fever has a body temperature of $40^{\circ} \mathrm{C} .$ What is this temperature on the Fahrenheit scale?

The Michaelis-Menten equation models the rate v of an enzymatic reaction as a function of the concentration [S] of a substrate S. In the case of the enzyme chymotrypsin the equation is $v=\frac{0.14[\mathrm{S}]}{0.015+[\mathrm{S}]}$ Use a graphing calculator or computer to graph v as a function of [S].

Solve the system of linear equations.

$$\begin{align*} 0.05x_1 - 0.03x_2 &= 0.21 \\ 0.07x_1+0.02x_2 &= 0.17 \\ \end{align*}$$

Sketch the image of the unit square [a square with vertices at (0, 0), (1, 0), (1, 1), and (0, 1)] under the specified transformation. T is the shear represented by T(x, y) = (x + 2y, y).

Use the bond dissociation enthalpy data in the table to calculate $\Delta H^{\circ}$ for (a) the monofluorination and (b) the monoiodination of methane. What is the main reason the values are so different?

A company has sales (measured in millions) of $\$50$, $\$60$, and $\$75$ during three consecutive years. Find the quadratic function that fits the data, and use it to predict the sales during the fourth year.

Perform the operations when

$A=$

$$\begin{bmatrix} 1 & 2 \\ 0 & -1 \\ \end{bmatrix}$$

.

Evaluate

$$A^2$$

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. $f(x, y)=x y+\frac{1}{x}+\frac{1}{y}$

Use a software program or a graphing utility to solve the system of linear equations.

$$\begin{align*} x+2y+z+3w &= 0\\ x-y+w &= 0 \\ 5y-z+2w &=0 \end{align*}$$

Suppose that the population dynamics of a species obeys a modified version of the logistic differential equation having the following form: $\frac{d N}{d t}=r\left(1-\frac{N}{K}\right)^{2} N$ where $r \neq 0 \text { and } K>0$ Apply the local stability criterion to the equilibrium $\hat{N}=K.$ What do you think your answer means about the stability of this equilibrium?

Differentiate the function. $f(t)=\frac{1}{2} t^{6}-3 t^{4}+t$

Obtain the sinusoids corresponding to each of the following phasors:

$$\begin{array}{l}{\text { (a) } \mathbf{V}_{1}=60 \angle 15^{\circ} \mathrm{V}, \omega=1} \\ {\text { (b) } \mathrm{V}_{2}=6+j 8 \mathrm{V}, \omega=40} \\ {\text { (c) } \mathrm{I}_{1}=2.8 e^{-j \pi / 3} \mathrm{A}, \omega=377} \\ {\text { (d) } \mathrm{I}_{2}=-0.5-j 1.2 \mathrm{A}, \omega=10^{3}}\end{array}$$