Prove each identity. State any restrictions on the variables. a)

$$\frac { \sin x } { \tan x } = \cos x$$

b)

$$\frac { \tan \theta } { \cos \theta } = \frac { \sin \theta } { 1 - \sin ^ { 2 } \theta }$$

c)

$$\frac { 1 } { \cos \alpha } + \tan \alpha = \frac { 1 + \sin \alpha } { \cos \alpha }$$

d)

$$1 - \cos ^ { 2 } \theta = \sin \theta \cos \theta \tan \theta$$

Suppose that an observational study showed that students who got at least 7 hours of sleep performed better on exams than students who got less than 7 hours of sleep. Which of the following are possible confounding variables, and which are not? Explain why in each case.

Number of courses the student took that term.

Simplify the expression. Write answers using positive exponents.

$$\left( \frac { y ^ { 8 } } { z ^ { 6 } } \right) ^ { - 2 }$$

Find the product. (-7/12)(-4/21)

Suppose that you wanted to know whether the mean arm span is different for U.S. Census at School seventh-grade girls and U.S. Census at School seventh-grade boys. A random sample of 10 students from the population of U.S. Census at School seventh-grade girls was selected using the random sampler on the Census at School website. The arm spans (in cm) for these 10 students were as follows: 171 152 176 147 167 165 152 161 152 147 A random sample of 10 students from the population of U.S. Census at School seventh-grade boys was also selected. The arm spans for these 10 students were as follows: 176 168 176 155 154 160 170 172 179 190. Do the dot plots suggest that there is about the same amount of variability in the arm span data for U.S. seventh-grade girls and the arm span data for U.S. seventh-grade boys?

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. For any matrices A and B of the same dimensions, rank(A + B) = rank(A) + rank(B).

Solve.

$$x ^ { 4 } + x ^ { 2 } - 12 = 0$$

Complete e each statement by writing the correct word or words. The amount that freezing point is depressed or boiling point is elevated is proportional to the ______ of the solution.

Explain the importance of each term, person, or place: faction; Marquis de Lafayette; Olympe de Gouges; Marie Antoinette; émigré; sans-culotte; republic; Jacobins.

When a pilot takes a tight turn at high speed in a modern fighter airplane, the blood pressure at the brain level decreases, blood no longer perfuses the brain, and the blood in the brain drains. If the heart maintains the (hydrostatic) gauge pressure in the aorta at 120 torr (or mm Hg) when the pilot undergoes a horizontal centripetal acceleration of 4.5g, what is the blood pressure (in torr) at the brain, 30 cm radially inward from the heart? The perfusion in the brain is small enough that the vision switches to black and white and narrows to "tunnel vision" and the pilot can undergo g-LOC ("g-induced loss of consciousness"). Blood density is

$$1.06 \times 10 ^ { 3 } \mathrm { kg } / \mathrm { m } ^ { 3 }$$

Evaluate the expression.

$$3 - \left[ 3 ^ { 3 } + ( 3 - 1 ) ^ { 3 } \right]$$

What year came one decade after 1802?

Convert the given number to base ten.

$$735_{e i g h t}$$

sketch the graph of y = x^n and each transformation. y = x^6 , f(x) = - 1/8 x^6