exercise:De51de9d34: Difference between revisions
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(Created page with "<div class="d-none"><math> \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow}...") |
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\newcommand{\conj}[1]{\overline{#1}} | \newcommand{\conj}[1]{\overline{#1}} | ||
\newcommand{\mathds}{\mathbb} | \newcommand{\mathds}{\mathbb} | ||
\newcommand{\dilemma}[2] { | |||
\left\{ | |||
\begin{array}{ll} | |||
#1 \\ | |||
#2 | |||
\end{array} | |||
\right. | |||
} | |||
\newcommand{\trilemma}[3] { | |||
\left\{ | |||
\begin{array}{ll} | |||
#1 \\ | |||
#2 \\ | |||
#3 | |||
\end{array} | |||
\right. | |||
} | |||
</math></div> | </math></div> | ||
Draw the graph of <math>f</math>, and evaluate | Draw the graph of <math>f</math>, and evaluate | ||
<math>\int_a^b f(x) \; dx</math> in each of the | <math>\int_a^b f(x) \; dx</math> in each of the | ||
following examples. | following examples. | ||
<ul style{{=}}"list-style-type:lower-alpha"><li><math>\trilemma{1 & \mbox{if | <ul style{{=}}"list-style-type:lower-alpha"><li><math>\trilemma{1 & \mbox{if $-\infty < x \leq 0 $},} | ||
{5 & \mbox{if | {5 & \mbox{if $0 < x < 2$},} | ||
{3 & \mbox{if | {3 & \mbox{if $2 \leq x < \infty $},}</math> | ||
and <math>[a,b] = [-3,3]</math>.</li> | and <math>[a,b] = [-3,3]</math>.</li> | ||
<li><math>f(x) = \dilemma{x^2&\mbox{if | <li><math>f(x) = \dilemma{x^2&\mbox{if $-\infty < x < 0 $},} | ||
{2-x^2&\mbox{if | {2-x^2&\mbox{if $0\leq x < \infty$},}</math> | ||
and <math>[a,b] = [-2,2]</math>.</li> | and <math>[a,b] = [-2,2]</math>.</li> | ||
<li><math>f(x) = n</math> | <li><math>f(x) = n \quad</math> if <math>n \leq x < n+1</math> | ||
where <math>n</math> is any integer, | where <math>n</math> is any integer, | ||
and <math>[a,b] = [0,5]</math>.</li> | and <math>[a,b] = [0,5]</math>.</li> | ||
</ul> | </ul> | ||
Latest revision as of 02:45, 25 November 2024
[math]
\newcommand{\ex}[1]{\item }
\newcommand{\sx}{\item}
\newcommand{\x}{\sx}
\newcommand{\sxlab}[1]{}
\newcommand{\xlab}{\sxlab}
\newcommand{\prov}[1] {\quad #1}
\newcommand{\provx}[1] {\quad \mbox{#1}}
\newcommand{\intext}[1]{\quad \mbox{#1} \quad}
\newcommand{\R}{\mathrm{\bf R}}
\newcommand{\Q}{\mathrm{\bf Q}}
\newcommand{\Z}{\mathrm{\bf Z}}
\newcommand{\C}{\mathrm{\bf C}}
\newcommand{\dt}{\textbf}
\newcommand{\goesto}{\rightarrow}
\newcommand{\ddxof}[1]{\frac{d #1}{d x}}
\newcommand{\ddx}{\frac{d}{dx}}
\newcommand{\ddt}{\frac{d}{dt}}
\newcommand{\dydx}{\ddxof y}
\newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}}
\newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}}
\newcommand{\dist}{\mathrm{distance}}
\newcommand{\arccot}{\mathrm{arccot\:}}
\newcommand{\arccsc}{\mathrm{arccsc\:}}
\newcommand{\arcsec}{\mathrm{arcsec\:}}
\newcommand{\arctanh}{\mathrm{arctanh\:}}
\newcommand{\arcsinh}{\mathrm{arcsinh\:}}
\newcommand{\arccosh}{\mathrm{arccosh\:}}
\newcommand{\sech}{\mathrm{sech\:}}
\newcommand{\csch}{\mathrm{csch\:}}
\newcommand{\conj}[1]{\overline{#1}}
\newcommand{\mathds}{\mathbb}
\newcommand{\dilemma}[2] {
\left\{
\begin{array}{ll}
#1 \\
#2
\end{array}
\right.
}
\newcommand{\trilemma}[3] {
\left\{
\begin{array}{ll}
#1 \\
#2 \\
#3
\end{array}
\right.
}
[/math]
Draw the graph of [math]f[/math], and evaluate [math]\int_a^b f(x) \; dx[/math] in each of the following examples.
- [math]\trilemma{1 & \mbox{if $-\infty \lt x \leq 0 $},} {5 & \mbox{if $0 \lt x \lt 2$},} {3 & \mbox{if $2 \leq x \lt \infty $},}[/math] and [math][a,b] = [-3,3][/math].
- [math]f(x) = \dilemma{x^2&\mbox{if $-\infty \lt x \lt 0 $},} {2-x^2&\mbox{if $0\leq x \lt \infty$},}[/math] and [math][a,b] = [-2,2][/math].
- [math]f(x) = n \quad[/math] if [math]n \leq x \lt n+1[/math] where [math]n[/math] is any integer, and [math][a,b] = [0,5][/math].