A vocabulary word appears in italics in the sentence or short passage below. Think about how the word is used in the passage. Then write a definition for the vocabulary word.

The principal formed a $council$ of students and teachers to make suggestions for improvements at the school.

The scatterplot shows the batting average (BA) during April and the batting average during May-September for 66 Major League Baseball players who played regularly throughout the 2010 season. Also included on the scatterplot are the line $y=x$ and a vertical line showing the average batting average in April 2010.

a) Of the 36 hitters who were better than average in April, what percentage of these players had worse PERFORMANCES in May-September?

b) Of the 30 hitters who were worse than average in April, what percentage of these players had better PERFORMANCES in May-September?

c) Explain how your answers to parts (a) and (b) illustrate regression to the mean.

Find the unit tangent and principal unit normal vectors at the given points. $\mathbf{r}(t)=\langle\cos 2 t, t, \sin 2 t\rangle$ at t=0, $t=\frac{\pi}{2}$

Let $X=\operatorname{Spec} R$ and let $X_{f}$ be the principal open set corresponding to f $\in$ R. Prove that $X_{f} \cap X_{g}=X_{f g}$. Prove that $X_{f}=X$ if and only if f is a unit in R, and that $X_{f}=\emptyset$ if and only if f is nilpotent.

Find the exact interest, to the nearest cent. $(P=$ principal, $R=$ rate, $T=$ time $)$

$$\begin{array}{lllllllll} \boldsymbol{P} & \boldsymbol{R} & \boldsymbol{T} & \text { Interest }\\ \$ 1,000 & 12 \% & 60\ \text{days} \end{array}$$

Use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. $x^{2}+4 x y+y^{2}-9=0$

Find the exact interest, to the nearest cent. $(P=$ principal, $R=$ rate, $T=$ time $)$

$$\begin{array}{lllllllll} \boldsymbol{P} & \boldsymbol{R} & \boldsymbol{T} & \text { Interest }\\ \$ 2,800 & 14 \% & 180\ \text{days} \end{array}$$

Imagine that you are cruising down $I-75$ headed to spring break in Florida. a. Suppose that you cruise at $70 \mathrm{mph}$ for $3$ hours and then at $60\mathrm{ mph}$ for $1$ hour. What is your average speed for those $4$ hours? When is the average speed over two time intervals actually the average of the two speeds? Explain your answer. b. Suppose that you cruise at $60\mathrm{ mph}$ for $180$ miles and then cruise at $40\mathrm{ mph}$ for $160$ miles. What is your average speed? When is the average speed over two distance intervals actually the average of the two speeds? Explain your answer.

A hydrogen atom is in its fifth excited state, with principal quantum number 6. The atom emits a photon with a wavelength of 1 090 nm. Determine the maximum possible magnitude of the orbital angular momentum of the atom after emission.

Vanillin (4-hydroxy-3-methoxybenzaldehyde), the principal component of vanilla, occurs in vanilla beans and other natural sources as a $\beta-D-$glucopyranoside. Draw a structural formula for this glycoside, showing the d-glucose unit as a chair conformation.

Evaluate the Cauchy principal value of the given improper integral. $\int_{-\infty}^{\infty} \frac{\sin x}{x+i} d x$ [Hint: First substitute $\sin x=\left(e^{i x}-e^{-i x}\right) / 2 i, x$ real.]

$$\begin{array} { l } { \text { Let } D \text { be a principal ideal domain and let } p \in D . \text { Prove that } \langle p \rangle \text { is a } } \\ { \text { maximal ideal in } D \text { if and only if } p \text { is irreducible. } } \end{array}$$

Vasectomy is a procedure for birth control in which the principal ducts carrying sperm out of the testes are tied off. Basing your answer on the cellular structure of the human testis, explain why a man’s testosterone level is not affected by vasectomy.

If a machine part is made of tool L2 steel and a critical point in the material is subjected to in-plane principal stresses σ₁ and σ₂ = -0.5σ₁, determine the magnitude of σ₁ in ksi that will cause yielding according to the maximum shear stress theory.