What number do you associate with the following? Une octave _________

How many moles of hydrogen gas are required for complete hydrogenation of 1 mol of linolenic acid, whose structure is shown below? Write a balanced equation for the hydrogenation reaction. $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}=\mathrm{CHCH}_{2} \mathrm{CH}=\mathrm{CHCH}_{2} \mathrm{CH}=\mathrm{CH}$ $\left(\mathrm{CH}_{2}\right)_{7} \mathrm{COOH}$

Suppose you set up a proportion so that each ratio compares parts of one triangle, as shown below.

$$\begin{aligned} & \text { height of } \triangle A B C \longrightarrow \frac{B C}{A B}=\frac{D E}{A D} \rightleftarrows \text { height of } \triangle A D E \\ & \text { base of } \triangle A B C \longrightarrow \text { base of } \triangle A D E \end{aligned}$$

Show that this proportion leads to the same value for h as in Example 2.

Indicate the best answer by writing the appropriate letter.

Two equilateral triangles have perimeters 6 and $9 \sqrt{3}$. The ratio of their areas is: (A) $2: 3 \sqrt{3}$ (B) $2 \sqrt{3}: 9$ (C) $4: 27$ (D) $4: 9$ (E) $8: 81 \sqrt{3}$

The 50-kg crate of previous problem is now projected down an incline as shown with an initial speed of 7 m/s. Investigate the time t required for the crate to come to rest and the corresponding distance x traveled if (a) $\theta$ = 150 and (b) $\theta$ = 300.

(a) Prove that for all x>0,

$$\frac{1}{x \sqrt{2 \pi}}\left(1-\frac{1}{x^2}\right) e^{-x^2 / 2}<1-\Phi(x)<\frac{1}{x \sqrt{2 \pi}} e^{-x^2 / 2}$$

Hint: Integrate the following inequalities:

$$\left(1-3 y^{-4}\right) e^{-y^2 / 2}<e^{-y^2 / 2}<\left(1+y^{-2}\right) e^{-y^2 / 2} .$$

(b) Use part (a) to prove that $1-\Phi(x) \sim \frac{1}{x \sqrt{2 \pi}} e^{-x^2 / 2}$. That is, as $x \rightarrow \infty$, the ratio of the two sides approaches 1 .

The ratio of the measures of the angles in a triangle is $4: 5: 6$. What is the measure of the smallest angle of the triangle? A. 72 B. 60 C. 48 D. 24 E. 12

The area of a right triangle is 12 square inches. The ratio of the lengths of its legs is 2:3. Find the length of the hypotenuse. A $\sqrt{13}$ in. B $26$ in. C $2 \sqrt{13}$ in. D $52$ in. E $4\sqrt{13}$ in.

How were the Minoans and Mycenaeans similar?

Identify Minoans, Mycenaeans, Homer.

The formula for calculating solutions of quadratic equations is a complex of directions. You can begin to make sense of the formula, by connecting $\mathrm{ith}_b$ patterns in the graphs of quadratic functions. Consider the related quadratic functions:

$$\begin{aligned} & y=2 x^2-12 x \\ & y=2 x^2-12 x+10 \\ & y=2 x^2-12 x-14 \\ & y=2 x^2-12 x+24 \end{aligned}$$

a. Match each function with its graph.

b. What are the $x$-coordinates of the minimum points on each graph? How (if at all) are those $x$-coordinates related to the $x$-intercepts of the graphs?

c. What are the coordinates of the $x$-intercepts on the graph of a quadratic function with rule $y=a x^2+b x$ ? What is the $x$-coordinate of the minimum or maximum point for such a graph?

d. How will the $x$-coordinate of the maximum or minimum point of $y=a x^2+b x+c$ be related to that of $y=a x^2+b x ?$

Evaluate the determinant of each matrix. Describe any patterns. a.

$$\left[\begin{array}{lll}{1} & {2} & {3} \\ {1} & {2} & {3} \\ {1} & {2} & {3}\end{array}\right]$$

b.

$$\left[\begin{array}{rrr}{-1} & {-2} & {-3} \\ {-3} & {-2} & {-1} \\ {-1} & {-2} & {-3}\end{array}\right]$$

c.

$$\left[\begin{array}{lll}{1} & {2} & {3} \\ {2} & {3} & {1} \\ {1} & {2} & {3}\end{array}\right]$$

d.

$$\left[\begin{array}{rrr}{-1} & {2} & {-3} \\ {2} & {-3} & {-1} \\ {-1} & {2} & {-3}\end{array}\right]$$

Consider the following array:

$$\begin{array}{rrrrrrrrrr|} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ \hline 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 \\ \hline 31 & 32 & 33 & 34 & 35 & 36 & 37 & 38 & 39 & 40 \\ 41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 & 49 & 50 \\ 51 & 52 & 53 & 54 & 55 & 56 & 57 & 58 & 59 & 60 \end{array}$$

Compare the pair $26,37$ with the pair $36,27$ .

d. What similar patterns can you find?

In parts (a) through (d) below, look for patterns as you calculate the sum and the product for each pair of complex numbers. Use what you find to answer parts (e) through (g). $a. 2 + i , 2 – i$ $b. 3 - 5 i , 3 + 5 i$ $c. - 4 + i , - 4 – i$ $d. 1 + i \sqrt { 3 } , 1 - i \sqrt { 3 }$ e. What complex number can you multiply 3 + 2i by to get a real number? f. What happens when you multiply (-4 + 5i)(-4 + 3i)? g. What complex number can you multiply a + bi by to get a real number?