Find the first seven terms of the Maclaurin expansion of $f(z)=\sin ^{2}(z)$ in four ways, as follows. (a) First, compute the Taylor coefficients at 0. (b) Find the first seven terms of the product of the Maclaurin series for sin(z) with itself. (c) Write $\sin ^{2}(z)$ in terms of the exponential function and use the Maclaurin expansion of this function. (d) Write $\sin ^{2}(z)=(1-\cos (2 z)) / 2$ and use the Maclaurin expansion of cos(z).

Approximate the value of the given expression using the indicated number of terms of a Maclaurin series. $\sin \left(\frac{1+i}{10}\right)$, two terms

The coefficient of $x^6$ in the Maclaurin series expansion of $f(x)=\sin x^2$ is -?-.

Complete each statement by writing the correct word or words. Rules of sequence and meaning are required for the form of communication called ____.

find the general solution of the given system of equations. x'='114−2x+'e−2t−2e

In his book Outliers, author Malcolm Gladwell argues that more American-born baseba]] players have birth dates in the months immediately following July 31 because that was the age cutoff date for nonschool baseball leagues. The table below lists month of births for a sample of American-born basebal l players and foreign-born baseball players. Using a 0.05 significance level, is there sufficient evidence to warrant rejection of the claim that months of births of baseball players are independent of whether they are born in America? Do the data appear to support Gladwell 's claim?

$$\begin{matrix} \text{ } & \text{Jan. } & \text{Feb.} & \text{March} & \text{April} & \text{May} & \text{June} & \text{July} & \text{Aug.} & \text{Sep.} & \text{Oct.} & \text{Nov.} & \text{Dec.}\\ \text{Born in America} & \text{387} & \text{329} & \text{366} & \text{344} & \text{336} & \text{313} & \text{313} & \text{503} & \text{421} & \text{434} & \text{398} & \text{371}\\ \text{Foreign Born} & \text{101} & \text{82} & \text{85} & \text{82} & \text{94} & \text{83} & \text{59} & \text{91} & \text{70} & \text{100} & \text{103} & \text{82}\\ \end{matrix}$$

Use Bairstow’s method to determine the roots of

$$(a). f(x) = -2 + 6.2x - 4x^2 + 0.7x^3$$

$$(b). f(x) = 9.34 - 21.97x + 16.3x^2 - 3.704x^3$$

$$(c). f(x) = x^4 - 2x^3 + 6x^2 - 2x + 5.$$

Members of a certain club are required to register for one of three games: badminton, table tennis or darts. The number of club members of each gender choosing each game in a particular year is shown in the table.

$$\begin{matrix} \text{ } & \text{Badminton} & \text{Table tennis} & \text{Darts}\\ \text{Male} & \text{37} & \text{16} & \text{28}\\ \text{Female} & \text{32} & \text{10} & \text{19}\\ \end{matrix}$$

Use a chi-squared test, at the 5% significance level, to test whether choice of game is independent of gender. State clearly the null and alternative hypotheses, the expected values and the number of degrees of freedom.

Jose started at his home, H(0, 0) and walked to a park located at P(4, 2). He then went to Jacintha's house located at F(8, 0). Later, he walked straight home. If each unit on the coordinate plane represents 50 feet, what is the distance that Jose walked, to the nearest foot?

The damped spring-mass oscillator is initially stationary with the spring unstretched. At $t=p$, a constant force $F(t)=6 \mathrm{~N}$ is applied to the mass. (a) What is the steady-state (particular) solution? (b) Determine the position of the mass as a function of time.

a. Write a brief definition for each of the following terms: county and sheriff. b. Describe the organization of county government c. What are some of the duties of county government?

In a carbonated beverage, the main ingredients are water and carbon dioxide. Carbon dioxide gas is released when the bottle is opened. Why is water a liquid but carbon dioxide a gas at room temperature?

Identify the carcinogens found in tobacco. a. benzene b. formaldehyde c. vinyl chloride d. all of the above

Why would a large number of offspring in a cross be more likely to match Punnett-square ratios than a small number would?