Describe two main differences between a water molecule (H2O) and a methane molecule (CH4).

find approximate values of the solution of the given initial value problem at t=0.1, 0.2, 0.3, and 0.4. Compare the results with those obtained by using other methods and with the exact solution (if available).(a) Use the Runge-Kutta method with h=0.1.(b) Use the Runge-Kutta method with h=0.05. y'=5t−3√y,y(0)=2

Prove the following. a) $\lim _ { z \rightarrow 0 } \frac { \sin z } { z } = 1$ b) $\lim _ { z \rightarrow 0 } \frac { \cos z - 1 } { z } = 0$ [HINT: Use the fact that $f ^ { \prime } ( 0 ) = \lim _ { z \rightarrow 0 } [ f ( z ) - f ( 0 ) ] / z$.]

$$\text { Let } S \text { be a nonempty set with an asso- }\left. \begin{array} { l } { \text { ciative operation that is left and right cancellative } ( x y = x z \text { implies } } \\ { y = z , \text { and } y x = z x \text { implies } y = z ) . \text { Assume that for every } a \text { in } S \text { the } } \\ { \text { set } \left\{ a ^ { n } | n = 1,2,3 , \ldots \right\} \text { is finite. Must } S \text { be a group? } } \end{array} \right.$$

The resolving power of a microscope is proportional to the wavelength used. A resolution of $1.0 \times 10 ^ { - 11 }$ m (0.010 nm) would be required in order to “see” an atom. If photons were used, what minimum photon energy would be needed to obtain $1.0 \times 10 ^ { - 11 }$ m resolution?

Solve each equation in the complex number system. $x ^ { 3 } + x ^ { 2 } - 8 x - 6 = 0$

The percentage of alcohol in a person's bloodstream $t$ after drinking 8 fluid oz of whiskey is given by

$$A(t)=0.23e^{-0.4t},\,\,\, (0\leq t\leq 12)$$

a. What is the percentage of alcohol in a person's bloodstream after $\frac{1}{2}$ hr? After 8 hr? b. How fast is the percentage of alcohol in a person's bloodstream changing after $\frac{1}{2}$ hr? After 8 hr?

sketch the graph of the function. (Include two full periods.) g(t) = 5/2 sin (t - π)

A particle moves along a curved path

$$y(x) = (10 m)\{1 + cos[(0.1 m^{-1})x]\},$$

from x = 0 to x = 10π m, subject to a tangential force of variable magnitude

$$F(x) = (10 N)sin[(0.1 m^{-1})x].$$

How much work does the force do? (Hint: Consult a table of integrals or use a numerical integration program.)

p = S(q) is the price (dollars per unit) at which q units of a particular commodity will be supplied to the market by producers, and $q_0$ is a specified level of production. Find the price $p_0 = S(q_0)$ at which $q_0$ units will be supplied and compute the corresponding producers' surplus PS. Sketch the supply curve y = S(q) and shade the region whose area represents the producers' surplus. $S(q)=10+15 e^{0.03 q} ; q_{0}=3$ units

Use a calculator to find a decimal approximation for each value. Give as many digits as your calculator displays. Simplify the expression before using the calculator. $\frac{\cos 77^{\circ}}{\sin 77^{\circ}}$

Although DNA transposons are abundant in the genomes of multicellular eukaryotes, class 1 elements usually make up the largest fraction of very large genomes such as those from humans (~2500 Mb), maize (~2500 Mb), and barley (~5000 Mb). Given what you know about class 1 and class 2 elements, what is it about their distinct mechanisms of transposition that would account for this consistent difference in abundance?

Suppose $\mathbf{x}$ and $\mathbf{y}$ are vectors in $\mathbb{R}^{n}$, the angle between $\mathbf{x}$ and $\mathbf{y}$ is acute, and $\left\|\mathbf{p r o j}_{\mathbf{x}} \mathbf{y}\right\| \neq \mathbf{x} \cdot \mathbf{y}$. Prove by contradiction that x is not a unit vector.

what is the length of the side of the only equilateral triangle whose perimeter has the same numerical value in units as its area in square units?. express your answer as a decimal to the nearest tenth