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

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?

n. Asexual reproduction What are the four criteria for life?

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.

This exercise concerns a 5-card hand from a standard 52-card deck. A standard deck has 13 cards from each of 4 suits (clubs, diamonds, hearts, spades). The 13 cards have face value 2 through10, jack, queen, king, or ace. Each face value is a “kind” of card. The jack, queen, and king are “face cards.” How many hands contain 4 queens?

Prove that any amount of postage greater than or equal to 2 cents can be built using only 2-cent and 3-cent stamps.

Use Karnaugh maps to find all possible minimum sum-of-products expressions for each function. (a) $F(a, b, c)=\Pi M(3,4)$ (b) $g(d, e, f)=\Sigma m(1,4,6)+\Sigma d(0,2,7)$ (c) $F(p, q, r)=\left(p+q^{\prime}+r\right)\left(p^{\prime}+q+r^{\prime}\right)$ (d) $F(s, t, u)=\Sigma m(1,2,3)+\Sigma d(0,5,7)$ (e) $f(a, b, c)=\Pi M(2,3,4)$ (f) $G(D, E, F)=\Sigma m(1,6)+\Sigma d(0,3,5)$

Use a 4-to-1 multiplexer and a minimum number of external gates to realize the function $F(w, x, y, z)=\Sigma m(3,4,5,7,10,14)+\Sigma d(1,6,15)$. The inputs are only available uncomplemented.

Round to the nearest thousandth. 3.4953