Find an equation of the tangent line to the given curve $2\left(x^2+y^2\right)^2=25\left(x^2-y^2\right)$ at the point $(3,1)$.

How long (in seconds) will it take you to swim 100 meters if you swim at $1.25 \mathrm{~m} / \mathrm{sec}$ ?

For this exercise, you will draw a circle and locate the point $\left(\cos 220^{\circ}, \sin 220^{\circ}\right)$ on it.

Suppose $f$ is a real function on $(-x, \infty) .$ Call $x$ a fixed point of $f$ if $f(x)=x$ . (a) If $f$ is differentiable and $f^{\prime}(t) \neq 1$ for every real $t,$ prove that $f$ hast one fixed point. (b) Show that the function $f$ defined by

$$f(t)=t+\left(1+e^{t}\right)^{-1}$$

has no fixed point, although $0<f^{\prime}(t)<1$ for all real $t$ (c) However, if there is a constant $A<1$ such that $\left|f^{\prime}(t)\right| \leq A$ for all real $t$ , prove that a fixed point $x$ of $f$ exists, and that $x=\lim x_{n},$ where $x_{1}$ is an arbitrary real number and $x_{n+1}=f\left(x_{n}\right)$ for $n=1,2,3, \ldots$ (d) Show that the process described in (c) can be visualized by the zig-zag path

$$\left(x_{1}, x_{2}\right) \rightarrow\left(x_{2}, x_{2}\right) \rightarrow\left(x_{2}, x_{3}\right) \rightarrow\left(x_{3}, x_{3}\right) \rightarrow\left(x_{3}, x_{4}\right) \rightarrow \cdots$$

Complete each sentence by writing the form of the verb indicated in parentheses.

The dance troupe ________ the crowd with its performance. (present tense of stun)

Complete each sentence by writing the form of the verb indicated in parentheses.

The lost hikers ________ to the cleared path. (past tense of retreat)

In this given problem, plot the point with the given polar coordinates.

$$\left(-\frac{1}{2}, \pi / 2\right)$$

In this exercise, find the first through the fourth derivatives. Be sure to simplify each derivative before proceeding.

$$f(x)=\frac{x+3}{x-2}$$

Complete of the following mini-dialogues by writing in the question that was asked.

(a qui / tu) - $\rule{5cm}{1pt}$

• J'ai téléphoné a ma cousine.

The smell of a burning match is due to the formation of sulfur oxides. We can represent the relevant reaction as $\mathrm{S}+\mathrm{O}_2 \rightarrow \mathrm{SO}_2$. If 4.8 g of sulfur is burned in this manner, what mass of sulfur dioxide is produced?

A human being can swim at a maximum rate of 7.4 feet per second. The function y = 7.4x describes how many feet y a person can swim in x seconds. Graph the function. Use the graph to estimate the maximum number of feet a person can swim in 4.5 seconds.

In given exercise, the matrix product AX performs the translation of the point (x, y) to the point (x+h, y+k), when

$$A=\left[\begin{array}{lll}1 & 0 & h \\0 & 1 & k \\0 & 0 & 1\end{array}\right] \text { and } X=\left[\begin{array}{l}x \\y \\1\end{array}\right] \text {. }$$

Predict the coordinates of the translated point.

$$A=\left[\begin{array}{rrr}1 & 0 & -3 \\ 0 & 1 & 4 \\ 0 & 0 & 1\end{array}\right], \quad X=\left[\begin{array}{r}2 \\ -4 \\ 1\end{array}\right]$$

If a compressive force of $3.3 \times 10^4 \mathrm{~N}$ is exerted on the end of a $22 \mathrm{~cm}$ long bone of cross-sectional area $3.6 \mathrm{~cm}^2$, $(a)$ will the bone break, and $(b)$ if not, by how much does it shorten?

Factor. Assume that variables used as exponents represent positive integers.

$$2 x^{2 n}+11 x^n+5$$