Use the formula

$$\sin h+\sin 2 h+\sin 3 h+\cdots+\sin m h =\frac{\cos (h / 2)-\cos ((m+(1 / 2)) h)}{2 \sin (h / 2)}$$

to find the area under the curve y = sin x from x = 0 to $x=\pi / 2$ in two steps: Partition the interval $[0, \pi / 2]$ into n subintervals of equal length and calculate the corresponding upper sum U

Complete the following table to compare polynomial approximations of values of cosine x to the actual values of cosine x.

$$\begin{matrix} \text{x} & \ {1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}} & \ {\cos x}\\ \text{0.1} & \text{ } & \text{ }\\ \text{0.2} & \text{ } & \text{ }\\ \text{0.3} & \text{ } & \text{ }\\ \end{matrix}$$

Nettles, King, and Tanaka are partners sharing income 3:2:1. After the firm’s loss from liquidation is distributed, the capital account balances were: Nettles, $15,000 Dr.; King,$46,000 Cr.; and Tanaka, $71,000 Cr. If Nettles is personally bankrupt and unable to pay any of the$15,000, what will be the amount of cash received by King and Tanaka upon liquidation?

A train's horn produces a $148-\mathrm{Hz}$ sound while moving at $32.0 \mathrm{~m} / \mathrm{s}$. What frequencies are heard by a person standing still beside the tracks as the train approaches and after it passes?

According to the Old Farmer's Almanac, in Detroit, Michigan, the number of hours of sunlight on the summer solstice is $13.65$ and the number of hours of sunlight on the winter solstice is 9.067. Look up the number of hours of sunlight for April 1 in the Old Farmer's Almanac and compare the actual hours of daylight to the results found in the previous part.

The diameter of Mars is about 6800 km. Approximate the surface area of Mars using the formula for the surface area of a sphere, $S=4 \pi r^{2}$. Write your answer in scientific notation.

If you want to factor $a x^2+b x+c$ by grouping, you look for factors of $a c$ whose sum is $b$. In the following exercise, information is given about the signs of $b$ and $c$ in the trinomial $a x^2+b x+c, a>0$. In each case, state whether the factors of $a c$ should be two positive numbers, two negative numbers, or one positive and one negative number.

$b>0$ and $c>0$

As a biologist, you wish to notice if there is a relationship between the heights of tall trees and their diameters. You see the following data for the diameter (in inches) of the tree at $4.5$ feet from the ground and the corresponding heights (in feet).

2. What type of nonparametric analysis could be used to answer the question?

$$\begin{array} { l } { \text { Determine the symmetry group of the tessellation of the plane ex- } } \\ { \text { emplified by the brickwork shown. } } \end{array}$$

Both "The Story of an Hour" and "The Yellow Wallpaper" were written in the 1890s. How would you compare the themes of the two stories? Which story do you prefer, and why?

$\mathrm{B}_2 \mathrm{Br}_4$ has the following staggered structure: Show that $\mathrm{B}_2 \mathrm{Br}_4$ has one less plane of symmetry than $\mathrm{B}_2 \mathrm{~F}_4$ which is planar.

Hilliard, Downey, and Petrov are partners sharing income $3$:$2$:$1$. After the firm’s loss from liquidation is distributed, the capital account balances were: Hilliard, $\$24,000$ Dr.; Downey, $\$90,000$ Cr.; and Petrov, $\$64,000$ Cr. If Hilliard is personally bankrupt and unable to pay any of the $\$24,000$, what will be the amount of cash received by Downey and Petrov upon liquidation?

Answer each question about the surface area S on a surface given by a positive function z = f(x, y) over a region R in the xy-plane. Explain each answer. (a) Is it possible for S to equal the area of R?

The following data relate to labor cost for production of $5,500$ cellular telephones:

$$\begin{array}{lll} \text { Actual: } & 3,650\ \mathrm{hrs} \text {. at } \$ 15.20 & \$ 55,480 \\ \text { Standard: } & 3,710\ \mathrm{hrs} \text {. at } \$ 15.00 & \$ 55,650 \end{array}$$

Discuss what might have caused these variances.