<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\usepackage{epsfig}
\begin{document}


{\bf Question}

Sketch each of the following regions and define them by
inequalities of the form

$$f_1(x) \leq y \leq f_2(x)$$

$$a_1 \leq x \leq a_2$$

where $f_1, f_2$ are functions of $x$ and $a_1, a_2$ are real
constants, and also of the form

$$g_1(y) \leq x \leq g_2(y)$$

$$b_1 \leq y \leq b_2$$

where $g_1, g_2$ are functions of $y$ and $b_1, b_2$ are real
constants:
\begin{description}
\item[(i)]
the circle with centre $(1, 2)$ and radius 3;
\item[(ii)]
the triangle with vertices $(1,1)$, $(4,1)$ and $(4,7)$;
\item[(iii)]
the region defined by the inequalities $y \leq 4-x^2$ and $y \geq
(2-x)^2$.
\end{description}



{\bf Answer}

\begin{description}
\item{(i)}

\begin{center}
$\begin{array}{c}
\epsfig{file=158-9-1.eps, width=55mm}
\end{array}
\begin{array}{c}
\rm{The\ circle\ has\ equation}\\
(x-1)^2+(y-2)^2=3^2=9
\end{array} $
\end{center}

\end{description}

\begin{center}
$\begin{array}{c}
\epsfig{file=158-9-2.eps, width=40mm}
\end{array}
\ 
\begin{array}{c}
\textrm{Think of vertical lines (x-fixed)}\\
\textrm{and let y vary from bottom to top:}\\
2- \sqrt{9 -(x-1)^2} \le y \le 2 + \sqrt{9- (x-1)^2}\\
\textrm{Then let the line move from left to right:}\\
-2 \le x \le 4 
\end{array}$
\end{center}

\begin{center}
$ \begin{array}{c}
\epsfig{file=158-9-3.eps, width=40mm}\\
\end{array}
\ 
\begin{array}{c}
\textrm{Think of horizontal lines (y-fixed)}\\
\textrm{and let x vary from left to right:}\\
1- \sqrt{9 -(y-2)^2} \le x \le 1 + \sqrt{9 -(y-2)^2}\\
\textrm{Then let the line move from bottom to top:} \\
-1 \le y \le 5
\end{array}$
\end{center}

\begin{description}
\item{(ii)}
$$\\$$
\begin{center}
\epsfig{file=158-9-4.eps, width=60mm}
\end{center}
\end{description}

\begin{center}
$ \begin{array}{c}
\epsfig{file=158-9-5.eps, width=40mm}
\end{array}
\
\begin{array}{c}
\textrm{Think of vertical lines} \\
\textrm{and let y vary from bottom to top:} \\
1 \le y \le 2x-1 \\
\textrm{Then let the line move from left to right:} \\
1 \le x \le 4 
\end{array} $
\end{center}
 
\begin{center}
$ \begin{array}{c}
\epsfig{file=158-9-6.eps, width=40mm}
\end{array}
\begin{array}{c}
\textrm{Think of horizontal lines}\\
\textrm{and let x vary from left to right:} \\
\frac{y+1}{2} \le x \le 4 \\
\textrm{Then let the line move from bottom to top:} \\
1 \le y \le 7
\end{array} $
\end{center}

\begin{description}
\item{(iii)}
\end{description}
$$\\$$
\begin{center}
$\begin{array}{cc}
\epsfig{file=158-9-7.eps, width=50mm} \ &amp;
\ \epsfig{file=158-9-8.eps, width=50mm}\\
y \le 4-x^2 &amp;
y \ge (2-x)^2
\end{array} $
\end{center}

\begin{center}
\begin{tabular}{c}
The two inequalities above\\
define the region $\longrightarrow$
\end{tabular}
\begin{tabular}{c}
\epsfig{file=158-9-9.eps, width=50mm}
\end{tabular}
\end{center}


\begin{center}
$ \begin{array}{c|c}
\textrm{Vertical lines} &amp;
\textrm{Horizontal lines}\\
\epsfig{file=158-9-10.eps, width=50mm} \ \ \ &amp;
\ \ \ \epsfig{file=158-9-11.eps, width=50mm}\\
(2-x)^2 \le y \le 4-x^2 &amp;
2- \sqrt{y} \le x \le \sqrt{4-y}\\
0 \le x \le 2 &amp;
0 \le y \le 4
\end{array} $
\end{center}


\end{document}
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