Find the orthogonal trajectories of each given family of curves. In each case sketch several members of the family and several of the orthogonal trajectories on the same set of axes.

$$y^{2}=c x$$

Consider the tangent line linearization $L(x)$ to the graph of a function $f(x)$ of one variable, and discuss its relative predictive value for the behavior of $f$ in the cases $L^{\prime}\left(x_{0}\right)>0, L^{\prime}\left(x_{0}\right)=0,$ and $L^{\prime}\left(x_{0}\right)<0 .$ Can you draw an analogy to the linearization of an autonomous system of DEs?

Find all real solutions of the equation.

$$x ^ { 5 } = 5 x ^ { 3 }$$

If $B_{n}$ represents the bacteria count after n quarter-hours, then there are several different formulas that can be used to model this situation. (a.) Write a formula using the words NOW and NEXT that shows how the bacteria count increases as time passes. (b.) Write a recursive formula beginning "$B_{n}=\ldots$" for the sequence of bacteria counts. Use the recursive formula to predict the bacteria count after 18 quarter-hours. (c.) Write a function formula beginning "$B_{n}=\ldots$" for the sequence of bacteria counts. Use the function formula to predict the bacteria count after 18 quarter-hours. (d.) Describe the difference between the processes of using the recursive formula and using the function formula to compute the bacteria count after 18 quarter-hours.

Solve the initial-value problem. In the case assume $x>0$

$$x^2y''-5xy'+8y=2x^3, y(2)=0, y'(2)=-8$$

What if the outcome of the French and Indian War had been different and France had won? How might this have affected the 13 colonies?

How would you say 'calendar' in Spanish?

State the period of: $y=\sin \left(\frac{x}{3}\right)$

Use a fourth-degree Taylor approximation for $e^h,$ for h near 0, to evaluate the following limits. Would your answer be different if you used a Taylor polynomial of higher degree? (a) $\lim _{h \rightarrow 0} \frac{e^{h}-1-h}{h^{2}}$ (b) $\lim _{h \rightarrow 0} \frac{e^{h}-1-h-\frac{h^{2}}{2}}{h^{3}}$

$\sin \theta$ and $\cos \theta$ are given. Find the exact value of each of the four remaining trigonometric functions. $\sin \theta=\frac{1}{2},\ \cos \theta=\frac{\sqrt{3}}{2}$

Factor each polynomial by removing any common monomial factor. $9 x ^ { 3 } - 18 x ^ { 4 }$

Suppose that the displacement of an object is related to time according to the expression $x = Bt ^2$. What are the dimensions of B?

If we are given an acute angle and a side in a right triangle, what unknown part of the triangle requires the least work to find?

About how many microorganisms are found in the human intestinal tract? (A typical bacterial length scale is one micron = $10^{-6}$ m. Estimate the intestinal volume and assume bacteria occupy one hundredth of it.)