Points at which the graphs of r=f(

$$\theta$$

) and r=g(\theta) intersect must be determine carefully. Solving f(\theta)=g(\theta) identifies some-but perhaps not all-intersection points. The reason is that the curves may pass through the same point for different values of

$$\theta$$

. Use analytical methods and a graphing utility to find all the intersection points of the following curves. r=1-sin

$$\theta$$

and r=1+cos

$$\theta$$

Solve the following initial-value problem over the interval from t = 2 to 3:

$$\frac { d y } { d t } = - 0.4 y + e ^ { - 2 t }$$

Use the non-self-starting Heun method with a step size of 0.5 and initial conditions of y(1.5) = 5.800007 and y(2.0) = 4.762673. Iterate the corrector to

$$\varepsilon _ { s } = 0.1 \%.$$

Compute the true percent relative errors

$$\varepsilon_t$$

for your results based on the analytical solution.

There are four dimethylcyclopropane isomers, (a) Write three-dimensional formulas for these isomers. (b) Which of the isomers are chiral? (c) If a mixture consisting of 1 mol of each of these isomers were subjected to simple gas chromatography (an analytical method that can separate compounds according to boiling point), how many fractions would be obtained and which compounds would each fraction contain? (d) How many of these fractions would be optically active?

Mixing the chelating reagent $\mathrm{B}$ with $\mathrm{Ni}$ (II) forms the highly colored $\mathrm{NiB}_2{ }^{2+}$, whose solutions obey Beer's law at $395 \mathrm{~nm}$ over a wide range. Provided the analytical concentration of the chelating reagent exceeds that of Ni(II) by a factor of 5 (or more), the cation exists, within the limits of observation, entirely in the form of the complex. Use the accompanying data to evaluate the formation constant $K_{\mathrm{f}}$ for the process

$$\mathrm{Ni}^{2+}+2 \mathrm{~B} \rightleftharpoons \mathrm{NiB}_2{ }^{2+}$$

$$\begin{aligned} &\text { Analytical Concentration, M }\\ &\begin{array}{ccc} \hline \mathrm{Ni}^{2+} & \mathrm{B} & A_{395}(\mathbf{1 . 0 0}-\mathrm{cm} \text { cells }) \\ \hline 2.00 \times 10^{-4} & 2.20 \times 10^{-1} & 0.718 \\ 2.00 \times 10^{-4} & 1.50 \times 10^{-3} & 0.276 \end{array} \end{aligned}$$

Design a crank-rocker mechanism with a time ratio of $Q$, throw angle of $\left(\Delta \theta_4\right)_{\max }$, and time per cycle of $t$. Use either the graphical or analytical method. Specify the link lengths $L_1, L_2, L_3, L_4$, and the crank speed.

$$Q=1.15 ;\left(\Delta \theta_4\right)_{\max }=55^{\circ} ; t=0.45 \mathrm{~s}$$

The sum of all chemical reactions in a cell is referred to as?

Why is it important that glycolysis be tightly controlled by the cell?

The following table gives the top 12 countries in the world whose populations have the highest concentration of type $\mathrm{A}^{-}$blood.

Find the mean (average), median, and mode of these concentrations of type $\mathrm{A}^{-}$blood in these 12 countries.

Now i know that the cell theory describes ____

The stress-strain diagram for many metal alloys can be described analytically using the Ramberg-Osgood three parameter equation $\epsilon=\sigma / E+k \sigma^n$, where $E, k$, and $n$ are determined from measurements taken from the diagram. Using the stress-strain diagram shown in the figure, take $E=30\left(10^3\right)$ ksi and determine the other two parameters $k$ and $n$ and thereby obtain an analytical expression for the curve.

Express the complex number $\space -\sqrt{3}+i\space$, in the polar form.

Design a crank-shaper mechanism with a time ratio of $Q$, stroke of $\left|\Delta \mathbf{R}_{\mathbb{E}}\right|_{\max }$ and time per cycle of $t$. Use either the graphical or analytical method. Specify the link lengths $L_1$, $L_2, L_3, L_4$, and the crank speed.

$Q=1.80 ;\left|\Delta R_{\mathrm{E}}\right|_{\max }=1.2$ in.; $t=0.25 \mathrm{~s}$.

Assume that

$$\begin{aligned} &\frac{d x_{1}}{d t}=x_{1}\left(10-2 x_{1}-x_{2}\right)\\ &\frac{d x_{2}}{d t}=x_{2}\left(10-x_{1}-2 x_{2}\right) \end{aligned}$$

(a) Graph the zero isoclines. (b) Show that $\left(\frac{10}{3}, \frac{10}{3}\right)$ is an equilibrium, and use the analytical approach to determine its stability.

The following table gives the top ten countries in the world whose populations have the highest concentration of type $\mathrm{O}^{+}$blood:

Find the mean (average), median, and mode of these concentrations of type $\mathrm{O}^{+}$blood in these ten countries.