exercise:D7bea31ae5: 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> | ||
The same curve can be defined by | The same curve can be defined by more than one parametrization: | ||
more than one parametrization: | |||
<ul style{{=}}"list-style-type:lower-alpha" | <ul style{{=}}"list-style-type:lower-alpha"> | ||
<li> | <li> | ||
Draw the curve defined parametrically by | Draw the curve defined parametrically by | ||
| Line 41: | Line 59: | ||
\dilemma{x(t) = t,}{y(t) = t, & 0 \leq t \leq 1.} | \dilemma{x(t) = t,}{y(t) = t, & 0 \leq t \leq 1.} | ||
</math></li> | </math></li> | ||
<li> | <li> | ||
Draw the curve defined parametrically by | Draw the curve defined parametrically by | ||
| Line 48: | Line 65: | ||
\dilemma{x(t) = \sin\pi t,}{y(t) = \sin\pi t, & 0 \leq t \leq 1.} | \dilemma{x(t) = \sin\pi t,}{y(t) = \sin\pi t, & 0 \leq t \leq 1.} | ||
</math></li> | </math></li> | ||
<li>Compute the arc lengths from <math>t=0</math> to <math>t=1</math> | <li>Compute the arc lengths from <math>t=0</math> to <math>t=1</math> for the parametrizations in (a) and (b).</li> | ||
for the parametrizations in | <li>Give a geometric interpretation which explains the difference between the arc lengths obtained for the two parametrizations.</li> | ||
and | |||
<li>Give a geometric interpretation which explains | |||
the difference between the arc lengths | |||
obtained for the two parametrizations.</li> | |||
</ul> | </ul> | ||
Latest revision as of 22:28, 25 November 2024
[math]
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\newcommand{\conj}[1]{\overline{#1}}
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\newcommand{\dilemma}[2] {
\left\{
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#1 \\
#2
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}
\newcommand{\trilemma}[3] {
\left\{
\begin{array}{ll}
#1 \\
#2 \\
#3
\end{array}
\right.
}
[/math]
The same curve can be defined by more than one parametrization:
-
Draw the curve defined parametrically by
[[math]] \dilemma{x(t) = t,}{y(t) = t, & 0 \leq t \leq 1.} [[/math]]
-
Draw the curve defined parametrically by
[[math]] \dilemma{x(t) = \sin\pi t,}{y(t) = \sin\pi t, & 0 \leq t \leq 1.} [[/math]]
- Compute the arc lengths from [math]t=0[/math] to [math]t=1[/math] for the parametrizations in (a) and (b).
- Give a geometric interpretation which explains the difference between the arc lengths obtained for the two parametrizations.