Graph the equation using a graphing utility.

$$x=y-3 y^3+y^5$$

Sketch the graph and check your work with a graphing utility.

$$y=\sin ^{-1} 2 x$$

Confirm your results in parts (a) and (b) using a graphing utility.

Graph the equation using a graphing utility.

$x=\cot y, 0<y<\pi$

A mixture of $1.00 \mathrm{~g} \mathrm{H}_2$ and $1.06 \mathrm{~g} \mathrm{H}_2 \mathrm{~S}$ in a $0.500 \mathrm{~L}$ flask comes to equilibrium at $1670 \mathrm{~K}: 2 \mathrm{H}_2(\mathrm{~g})+\mathrm{S}_2(\mathrm{~g}) \rightleftharpoons$ $2 \mathrm{H}_2 \mathrm{~S}(\mathrm{~g})$. The equilibrium amount of $\mathrm{S}_2(\mathrm{~g})$ found is $8.00 \times 10^{-6}$ mol. Determine the value of $K_p$ at $1670 \mathrm{~K}$ for pressures expressed in atmospheres.

Compute

$$s ^ { 2 } , \text { and } s$$

for each of the data sets listed. Where appropriate, specify the units in which your answer is expressed. a. 3, 1, 10, 10, 4 b. 8 feet, 10 feet, 32 feet, 5 feet c. −1, −4, −3, 1, −4, −4 d. 1/5 ounce, 1/5 ounce, 1/5 ounce, 2/5 ounce, 1/5 ounce, 4/5 ounce

If a thread is unwound from a fixed circle while being held taut (i.e., tangent to the circle), then the end of the thread traces a curve called an involute of a circle. Show that if the circle is centered at the origin, has radius $a$, and the end of the thread is initially at the point $(a, 0)$, then the involute can be expressed parametrically as

$$x=a(\cos \theta+\theta \sin \theta), \quad y=a(\sin \theta-\theta \cos \theta)$$

where $\theta$ is the angle shown in part (a)

The motion of a vibrating particle is defined by the position vector $\mathbf{r}=10\left(1-e^{-3 t}\right) \mathbf{i}+\left(4 e^{-2 t} \sin 15 t\right) \mathbf{j}$, where r and t are expressed in millimeters and seconds, respectively. Determine the velocity and acceleration when (a) t = 0, (b) t = 0.5 s.

The infinite continued fraction can be expressed as $n=2+\frac{3}{2+\frac{3}{2+\frac{3}{2+\cdots}}}$. The value of $n$ can be determined by observing that $n=2+\frac{3}{n}$.

a Solve for $\mathrm{n}$ and check your result. Why does only one answer work?

b Find the value of $n=1+\frac{6}{1+\frac{6}{1+\frac{6}{1+\frac{6}{1+\ldots}}}}$.

Convert the following numbers expressed in polar form into rectangular form: a. $50\angle45^{\circ}$. b. $100\angle60^{\circ}$. c. $250\angle-53.13^{\circ}$. d. $1000\angle-30^{\circ}$. e. $12\angle0^{\circ}$. f. $180\angle-78.5^{\circ}$. g. $5\angle36.87^{\circ}$. h. $25\angle15^{\circ}$. i. $45\angle100^{\circ}$. j. $60\angle-90^{\circ}$. k. $100\angle53.13^{\circ}$. l. $40\angle90^{\circ}$.

ABCD is a parallelogram. The coordinates of A, B, D are (4, 2, 3), (18, 4, 8) and (-1, 12, 13) respectively. The origin of coordinates is O. (a) Find the vectors $\overrightarrow{A B} \text { and } \overrightarrow{A D}$. Find the coordinates of C. (b) Show that $\overrightarrow{O A}$ can be expressed in the form $\lambda \overrightarrow{A B}+\mu \overrightarrow{A D}$, stating the values of $\lambda$ and $\mu$. What does this tell you about the plane ABCD?

The indicated function $y_1(x)$ is a solution of the given differential equation. Use formula

$$y _ { 2 } = y _ { 1 } ( x ) \int \frac { e ^ { - \int P ( x ) d x } } { y _ { 1 } ^ { 2 } ( x ) } d x$$

to find a second solution $y_2(x)$ of

$$y ^ { \prime \prime } + P ( x ) y ^ { \prime } + Q ( x ) y = 0$$

expressed in terms of an integral-defined function.

$$x ^ { 2 } y ^ { \prime \prime } + \left( x ^ { 2 } - x \right) y ^ { \prime } + ( 1 - x ) y = 0 ; \quad y _ { 1 } = x$$

Consider the eigenvalue problem $-\Delta v=\lambda v$ in the unit square $D=$ $\{0<x<1,0<y<1\}$ with the Dirichlet $\mathrm{BC} v=0$ on the bottom and both vertical sides, and the Robin BC $\partial v / \partial y=-v$ on the top $\{y=1\}$ (a) Show that all the eigenvalues are positive. (b) Find an equation for the eigenvalues $\lambda .$ Show that they can be expressed in terms of the roots of the equation $s+\tan s=0$ . (c) Find the solutions of the last equation graphically. Find an approximate formula for the $(m, n)$ th eigenvalue for large $(m, n) .$

Let $\overrightarrow{\mathbf{a}}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{b}}=4 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$. Let $\overrightarrow{\mathbf{c}}=$ $\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}$. (a) Find $\mathbf{c}$, expressed in unit vector notation. (b) Find the angle between $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$.