A musical cent is a unit in a logarithmic scale of relative pitch or intervals. One octave is equal to 1200 cents. The formula

$$n = 1200 \left( \log _ { 2 } \frac { a } { b } \right)$$

can be used to determine the difference in cents between two notes with frequencies a and b. Find the interval in cents when the frequency changes from 443 Hertz (Hz) to 415 Hz.

The series converge by the alternating series test. Use Theorem 9.9 to find how many terms give a partial sum, $S_n,$ within 0.01 of the sum, S, of the series. $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{(2 n) !}$

For each item below, identify the letter of the word or phrase whose meaning is closest to that of the first word. benevolently: (a) wealthily, (b) attractively, (c) in a kind way

Write the equation of the line parallel to the given line and passing through the given point, which is the y-intercept. Check your answer by graphing the lines. $y = - \frac { 1 } { 2 } x _ { ; } ( 0 , - 3 )$

Find each indicated sum.

$$\sum _ { i = 1 } ^ { 5 } \frac { ( i + 2 ) ! } { i ! }$$

Evaluate the integral: $\int_{1}^{e} x^{(\ln 2)-1} d x$

$$\begin{array}{l} \text{ Find the derivative of the function. } \end{array}$$

$$g ( t ) = \frac { t ^ { 2 } } { 2 t ^ { 2 } + 1 }$$

For languages $A, B \subseteq \Sigma^{\star},$ does $A^{*} \subseteq B^{*} \Rightarrow A \subseteq B ?$

$$\begin{array}{c}{\text { A coin is tossed twice, the probability of heads on the first toss being } p_{1} \text { and that on the second }} \\ {\text { toss } p_{2} . \text { Consider this a compound experiment determined by two stochastically independent }} \\ {\text { experiments, and let the sample space be }} \\ {S=\{(H, H),(H, T),(T, H),(T, T)\}}\end{array}$$

(a) Compute the probability of each element of $S$

$$\begin{array}{c}{\text { (b) Can } p_{1} \text { and } p_{2} \text { be assigned so that }} \\ {P(H, H)=\frac{1}{9}, \quad P(H, T)=P(T, H)=\frac{2}{\vartheta}, \quad P(T, T)=\frac{4}{9} ?}\end{array}$$

$$\begin{array}{c}{\text { (c) Can } p_{1} \text { and } p_{2} \text { be assigned so that }} \\ {P(H, H)=P(T, T)=\frac{1}{3}, \quad P(H, T)=P(T, H)=\frac{1}{6} ?}\end{array}$$

$$\begin{array}{c}{\text { (d) Consider the following four events (subsets of S): }} \\ {H_{1} : \text { heads on the first toss, }} \\ {H_{2} : \text { heads on the second toss, }} \\ {T_{1} : \text { tails on the first toss, }} \\ {T_{2} : \text { tails on the second toss. }}\end{array}$$

Determine which pairs of these four events are independent. In each of Exercises 2 through 12 describe your sample space, your assignment of probabilities, and the event whose probability you are computing.

Simplify each expression. -3-7(r+0.9)

Combustion reactions involve reacting a substance with oxygen. When compounds containing carbon and hydrogen are combusted, carbon dioxide and water are the products. Using the enthalpies of combustion for $\mathrm { C } _ { 4 } \mathrm { H } _ { 4 } ( - 2341\ \mathrm { kJ }/ \mathrm { mol } ) , \mathrm { C } _ { 4 } \mathrm { H } _ { 8 } ( - 2755\ \mathrm { kJ } / \mathrm { mol } ) ,$ and $\mathrm { H } _ { 2 }$ $( - 286\ \mathrm { kJ } / \mathrm { mol } ) ,$ calculate $\Delta H$ for the reaction $\mathrm { C } _ { 4 } \mathrm { H } _ { 4 } ( g ) + 2 \mathrm { H } _ { 2 } ( g ) \longrightarrow \mathrm { C } _ { 4 } \mathrm { H } _ { 8 } ( g )$

The chemical formula of glucose is C6H12O6. What are the names of the elements in glucose, and how many atoms of each are present in a glucose molecule ?

use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is in the correct mode.) csc(- 15π / 14)

In a natural decay series, how many $\alpha$ and $\beta -$ emissions per atom of uranium-235 result in an atom of lead-207?