Evaluate the following limits. Use I Hópital's Rule when it is convenient and applicable. $\lim _{x \rightarrow \pi / 2^{-}} \frac{\tan x}{3 /(2 x-\pi)}$

Find a splitting field of $x^{4}-2$ (a) over $\mathbb{Q}$. (b) over $\mathbb{R}$.

Determine the location and value of the absolute extreme values of f on the given interval, if they exist. $f(x)=3 x^{5}-25 x^{3}+60 x \text { on }[-2,3]$

Find y' by using logarithmic differentiation. $y=\frac{\sqrt{1-x^{2}}}{1-2 x}$

If a = bc with $a \neq 0$ and b and c nonunits, show that a is not an associate of b.

Refer to the Early Childhood Education Journal (Mar. 2014) study of interactions in a children's museum. (Interactions by visitors to the museum included show-and-tell, learning, teaching, refocusing, participatory play, advocating, and disciplining interactions.) Recall that the researchers observed a sample of 170 meaningful interactions, of which 81 were led by children and 89 were led by adult caregivers. a. Give a point estimate of the true proportion of all meaningful interactions in a children's museum that are led by children. b. Form a 90% confidence interval around the proportion, part a. c. Interpret the interval, part b, in the words of the study. d. Suppose an educator claims that the true proportion of all meaningful interactions in a children's museum that are led by children is 35%. Make an inference about this claim.

Use the definition of the derivative to find each of the following. $\frac{d}{d x}(3-2 x)$

Prove that if r and s are relatively prime, then $\varphi(r s)=\varphi(r) \varphi(s).$

How many 5-card hands from a standard 52-card deck contain exactly 3 jacks and exactly 2 hearts?

Write the standard form of the equation of the ellipse centered at the origin. Vertices: (-8, 0), (8, 0) Co-vertices: (0, -3), (0, 3)

Let G be an abelian group, k a fixed positive integer, and

$$H=\{a \in G|| a | \text { divides } k\} .$$

Prove that H is a subgroup of G.

Why do we need the term "probability" when we explain the absence of free hydrogen in Earth's atmosphere?

Lichen has a high absorbance capacity for radiation fallout from nuclear accidents. Since lichen is a major food source for Alaskan caribou, and caribou are, in turn, a major food source for many Alaskan villagers, it is important to monitor the level of radioactivity in lichen. Researchers at the University of Alaska, Fairbanks, collected data on nine lichen specimens at various locations for this purpose. The amount of the radioactive element cesium-137 was measured (in microcuries per milliliter) for each specimen. The data values, converted to logarithms, are given in the following table (note that the closer the value is to zero, the greater is the amount of cesium in the specimen).

$$\begin{matrix} \text{Location} & \text{ } & \text{ } & \text{ }\\ \text{Bethel} & \text{-5.50} & \text{-5.00} & \text{ }\\ \text{Eagle Summit} & \text{-4.15} & \text{-4.85} & \text{ }\\ \text{Moose Pass} & \text{-6.05} & \text{ } & \text{ }\\ \text{Turnagain Pass} & \text{-5.00} & \text{ } & \text{ }\\ \text{Wickersham Dome} & \text{-4.10} & \text{-4.50} & \text{-4.60}\\ \end{matrix}$$

a. Construct a dot plot for the nine measurements. b. Construct a stem-and-leaf display for the nine measurements. c. Construct a histogram plot of the nine measurements. d. Which of the three graphs in parts a-c, respectively, is most informative? e. What proportion of the measurements has a radioactivity level of -5.00 or lower?

Let $S=\mathbb{N}$ and let $\rho$ be a binary relation on S defined by $x \rho y \leftrightarrow x^{2}-y^{2}$ is even. Show that $\rho$ is an equivalence relation on S and describe the resulting equivalence classes.