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$$\begin{aligned} \text { Find the ratio of } x \text { to } y : \frac{4}{y}+\frac{3}{x} &=44 \\ & \frac{12}{y}-\frac{2}{x}=44 \end{aligned}$$

$\frac{x}{y}=\frac{5}{8}$

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Describe the end behavior of each polynomial. $$(a)y=x^{3}-8 x^{2}+2 x-15$$End behavior: $$y \rightarrow \frac{\text { as }}{\text { as }} x \rightarrow \infty$$$$y \rightarrow \frac{ }{\longrightarrow}$ as $x \rightarrow-\infty$$$$(b)y=-2 x^{4}+12 x+100$$$$\begin{aligned} \text { End behavior: } & y \rightarrow \\ & y \rightarrow \end{aligned}$$ $$as\quad x \rightarrow-\infty$$

Finding the Standard Equation of a a Hyperbola In Exercises $41-48,$ find the standard form of the equation of the hyperbola with the given characteristics.$\begin{aligned} \text { Vertices: }(0,2), &(6,2) \\ \text { Asymptotes: } y &=\frac{2}{3} x \\ y &=4-\frac{2}{3} x \end{aligned}$

A point object is 10 $\mathrm{cm}$ away from a plane mirror, and theeye of an observer (with pupil diameter 5.0 $\mathrm{mm}$ ) is 20 $\mathrm{cm}$ away.Assuming the eye and the object to be on the same line perpendicular to the mirror surface, find the area of the mirror used in observing the reflection of the point. (Hint: Adapt Fig. $34-4 . )$

In Exercises 47-56, (a) use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant over the intervals you identified in part (a).

$g(x) = x$

A granite block in the shape of a right rectangularprism has dimensions 30 centimeters by40 centimeters by 50 centimeters. The block has adensity of 2.8 grams per cubic centimeter. What isthe mass of the block, in grams? (Density is massper unit volume.)$$\begin{aligned} \text { A) } & 336 \\ \text { B) } & 3,360 \\ \text { C) } & 16,800 \\ \text { D) } & 168,000 \end{aligned}$$

Applying the First Derivative Test In Exercises $41-48$ , consider the function on the interval $(0,2 \pi) .(a)$ Find the open intervals on which the function is increasing or decreasing. (b) Apply the First Derivative Test to identify all relative extrema. (c) Use a graphing utility to confirm your results.$f(x)=\frac{\sin x}{1+\cos ^{2} x}$

In Exercises 53-60, (a) use a graphing utility to graph each side of the equation to determine whether the equation is an identity, (b) use the table feature of a graphing utility to determine whether the equation is an identity, and (c) confirm the results of parts (a) and (b) algebraically.

$ \dfrac{1 + \cos x}{\sin x} = \dfrac{\sin x}{1 - \cos x} $

In Exercises $33-36$ , use division to write the rational function in the form $q(x)+r(x) d(x),$ where the degree of $r(x)$ is less than the degree of $d(x) .$ Then find the partial fraction decomposition of $r(x) / d(x) .$ Compare the graphs of the rational function with the graphs of its terms in the partial fraction decomposition.$$\frac{x^{3}+2}{x^{2}-x}$$

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## Recommended Questions

Describe the end behavior of each polynomial.

$$(a)y=x^{3}-8 x^{2}+2 x-15$$

End behavior: $$y \rightarrow \frac{\text { as }}{\text { as }} x \rightarrow \infty$$

$$y \rightarrow \frac{ }{\longrightarrow}$ as $x \rightarrow-\infty$$

$$(b)y=-2 x^{4}+12 x+100$$

$$\begin{aligned} \text { End behavior: } & y \rightarrow \\ & y \rightarrow \end{aligned}$$ $$as\quad x \rightarrow-\infty$$

Finding the Standard Equation of a a Hyperbola In Exercises $41-48,$ find the standard form of the equation of the hyperbola with the given characteristics.

$\begin{aligned} \text { Vertices: }(0,2), &(6,2) \\ \text { Asymptotes: } y &=\frac{2}{3} x \\ y &=4-\frac{2}{3} x \end{aligned}$

A point object is 10 $\mathrm{cm}$ away from a plane mirror, and the

eye of an observer (with pupil diameter 5.0 $\mathrm{mm}$ ) is 20 $\mathrm{cm}$ away.

Assuming the eye and the object to be on the same line perpendicular to the mirror surface, find the area of the mirror used in observing the reflection of the point. (Hint: Adapt Fig. $34-4 . )$

In Exercises 47-56, (a) use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant over the intervals you identified in part (a).

$g(x) = x$

A granite block in the shape of a right rectangular

prism has dimensions 30 centimeters by

40 centimeters by 50 centimeters. The block has a

density of 2.8 grams per cubic centimeter. What is

the mass of the block, in grams? (Density is mass

per unit volume.)

$$

\begin{aligned} \text { A) } & 336 \\ \text { B) } & 3,360 \\ \text { C) } & 16,800 \\ \text { D) } & 168,000 \end{aligned}

$$

Applying the First Derivative Test In Exercises $41-48$ , consider the function on the interval $(0,2 \pi) .(a)$ Find the open intervals on which the function is increasing or decreasing. (b) Apply the First Derivative Test to identify all relative extrema. (c) Use a graphing utility to confirm your results.

$f(x)=\frac{\sin x}{1+\cos ^{2} x}$

In Exercises 53-60, (a) use a graphing utility to graph each side of the equation to determine whether the equation is an identity, (b) use the table feature of a graphing utility to determine whether the equation is an identity, and (c) confirm the results of parts (a) and (b) algebraically.

$ \dfrac{1 + \cos x}{\sin x} = \dfrac{\sin x}{1 - \cos x} $

In Exercises $33-36$ , use division to write the rational function in the form $q(x)+r(x) d(x),$ where the degree of $r(x)$ is less than the degree of $d(x) .$ Then find the partial fraction decomposition of $r(x) / d(x) .$ Compare the graphs of the rational function with the graphs of its terms in the partial fraction decomposition.

$$\frac{x^{3}+2}{x^{2}-x}$$