Determine the longest possible wavelength of a standing wave on a string that is 2 m long.

Determine whether the statement is true and give an explanation or counterexample. The sequence of partial sums associated with the series $\sum_{k=1}^{\infty} \frac{1}{k^2+1}$ converges.

Name the conditions required for two waves to interfere constructively.

Draw a standing wave, and label the nodes and anti nodes.

A chemist weighs samples obtained from a production run. The weights of the samples are $13 \mathrm{~g}, 14 \mathrm{~g}, 65 \mathrm{~g}, 11 \mathrm{~g}, 15 \mathrm{~g}, 14 \mathrm{~g}, 14 \mathrm{~g}, 12 \mathrm{~g}, 13 \mathrm{~g}, 15 \mathrm{~g}$, $14 \mathrm{~g}$, and $12 \mathrm{~g}$.

Describe how any outlier affects the mean and the standard deviation of this data set.

Draw a rectangle that is 3 cm long and 2 cm wide. Then answer a and b. a. What is the perimeter of the rectangle in millimeters? b. What is the area of the rectangle in square centimeters?

What is the area, to the nearest tenth, of a figure that is formed using a rectangle 10 feet long and 7 feet wide and a semicircle with a diameter of 7 feet?

Solve each equation for all values of

$$\theta \text { if } \theta$$

is measured in degrees.

$$\sin ^ { 2 } \theta - \sin \theta = 0$$

In the last investigation, you studied intersection, union, and set difference. Another useful set operation is set to complement. The complement of a set A is everything in the "universe" that is not in A. This is written A$^{\prime}$ and is read " A compliment" or "not A." In terms of set difference, you can think of A$^{\prime}$ as "the universe" -A. The "universe" is defined by the universal set, which is the set of all objects that are under consideration. That is, the universal set defines the overall context.

a. Consider the universal set U = {5, 10, 15, 20, 25, 30, 35}. Suppose A = {10, 20, 30} and B = {5, 10, 15, 20}.

i. Find A$^{\prime}$.

ii. Find (A $\cap$ B)$^{\prime}$.

b. Give a general description of A$^{\prime}$, for any set A, using set-builder notation.

c. Consider the universal set Z = {x | x is an integer }. Find the complement of the set E = {x | x is an even integer}.

d. You can illustrate the universal set in a Venn diagram by drawing a large rectangle around the circles that represent the sets on which you are operating. Draw Venn diagrams that illustrate the situations in Parts a and c.

One is equal to one joule per second._____________________________

Determine whether the equation : $\cos^{2}{x}(1+\cot^{2}{x})=\cot^{2}{x}$, is a conditional equation or an identity.

Find the second derivative of $y ={x^2}\sin 3x$

Below are two solution procedures for solving

$$\sin ^ { 2 } \theta - \frac { 1 } { 2 } \sin \theta = 0 \text { for } 0 ^ { \circ } \leq \theta < 360 ^ { \circ }.$$

Which is incorrect? Explain the error. A.

$$\begin{array} { l } { \sin ^ { 2 } \theta - \frac { 1 } { 2 } \sin \theta = 0 } \\ { \sin \theta \left( \sin \theta - \frac { 1 } { 2 } \right) = 0 } \\ { \sin \theta = 0 \text { or } \sin \theta = \frac { 1 } { 2 } } \\ { \theta = 0 ^ { \circ } \text { or } 180 ^ { \circ } \text { or } \theta = 30 ^ { \circ } \text { or } 150 ^ { \circ } } \end{array}$$

B.

$$\begin{aligned} \sin ^ { 2 } \theta - \frac { 1 } { 2 } \sin \theta & = 0 \\ \sin ^ { 2 } \theta & = \frac { 1 } { 2 } \sin \theta \\ \sin \theta & = \frac { 1 } { 2 } \\ \theta & = 30 ^ { \circ } \text { or } 150 ^ { \circ } \end{aligned}$$

An object with negligible air resistance is dropped from a plane. During the first second of fall, the object falls 16 feet; during the second second, it falls 48 feet; during the third second, it falls 80 feet; and during the fourth second, it falls 112 feet. Assume this pattern continues. How many feet will the object fall in 8 seconds?