Symmetry: For $y^{\prime}=f(t,y)$ do the following: $(a)$ Sketch the direction fields and identify visual symmetries. $(b)$Conjecture how these graphical symmetries relate to algebraic symmetries in $f(t,y)$.$y^{\prime}=\frac{y^{2}}{t}$

Elemental Li and Na are prepared by electrolysis of a molten salt, whereas K, Rb, and Cs are prepared by chemical reduction.

In general terms, explain why the alkali metals cannot be prepared by electrolysis of their aqueous salt solutions,

Miley and Damian determined formulas they could use to compute the first $n-$terms of the series Show that both representations for $S_{n}$ are equivalent.

Find the exact value of each quadrantal angle. If any are not defined, write undefined. $\sin \left( - 1530 ^ { \circ } \right)$

Find the period of the function and use the language of transformations to describe how the graph of the function is related to the graph of y= cos x. y = cos x/5

Besides anxiety, how might you realize that you are suffering from a somatoform or dissociative disorder?

Solve each equation and check your solution. $\frac{6}{5} x=-1$

Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients.

Consider the function $f(x)=\frac{3 x-2}{x+3}$. Find $f^{\prime}(x)$ and draw its sign diagram.

Solve the following cryptarithms. Assume that no leading digit is a 0.

$$AT \\ EAST \\ \frac{+WEST}{SOUTH}$$

Describe each of the events below as impossible, unlikely, as likely as not, likely, or certain a. Joshua rolls an odd number on a standard number cube. b. Maria's birthday is September 31st. c. The basketball team has won 11 of their last 12 games. The team will win the next game.

Guess the requested limit.

$$\lim _{n \rightarrow \infty} \frac{n+1}{2 n}$$

The table shows the results of a customer satisfaction survey of 100 randomly selected shoppers at a mall. They were asked if they would shop at an earlier time if the mall opened earlier.

$$\begin{matrix} & \mathrm{Ages\ 10-20} & \mathrm{Ages\ 21-45} & \mathrm{Ages\ 46-65} & \mathrm{65\ and\ Older} & \mathrm{Total}\\ \mathrm{Yes} & \mathrm{0.13} & \mathrm{0.02} & \mathrm{0.08} & \mathrm{0.24} & \mathrm{0.47}\\ \mathrm{No} & \mathrm{0.25} & \mathrm{0.1} & \mathrm{0.15} & \mathrm{0.03} & \mathrm{0.53}\\ \mathrm{Total} & \mathrm{0.38} & \mathrm{0.12} & \mathrm{0.23} & \mathrm{0.27} & \mathrm{1}\\ \end{matrix}$$

What is the probability that a person aged 46-65 answered yes?

Differentiate the given function. $y=\frac{x^{5}-4 x^{2}}{x^{3}}$ [Hint: Divide first.]