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[Stack-cvs] stack-1-0/lang/en/doc author_commonsyntax.php,1.4,1.5 author_factsformula.php,1.4,1.5 author_gettingstarted.php,1.6,1.7 author_potresp.php,1.2,1.3 student_factsformula.php,1.4,1.5
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From: Chris S. <san...@us...> - 2005-10-07 06:26:46
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Update of /cvsroot/stack/stack-1-0/lang/en/doc In directory sc8-pr-cvs1.sourceforge.net:/tmp/cvs-serv27355/lang/en/doc Modified Files: author_commonsyntax.php author_factsformula.php author_gettingstarted.php author_potresp.php student_factsformula.php Log Message: Index: author_factsformula.php =================================================================== RCS file: /cvsroot/stack/stack-1-0/lang/en/doc/author_factsformula.php,v retrieving revision 1.4 retrieving revision 1.5 diff -C2 -d -r1.4 -r1.5 *** author_factsformula.php 14 Jul 2005 18:05:04 -0000 1.4 --- author_factsformula.php 7 Oct 2005 06:26:38 -0000 1.5 *************** *** 21,33 **** again, STACK had provided a mechanism to link to them. To use any of the facts and formulae, type ! <pre> ! <hint>key</hint> ! </pre> in any of the CAS-text fields, eg a worked solution.</p> ! <p>For example, the CAS-text</p> <p><tt>The chain rule <hint>calc_chain_rule</hint> can be used ....</tt></p> <?php --- 21,35 ---- again, STACK had provided a mechanism to link to them. To use any of the facts and formulae, type ! ! <p><hint>key</hint></p> ! in any of the CAS-text fields, eg a worked solution.</p> ! <p>For example, if the the CAS-text</p> <p><tt>The chain rule <hint>calc_chain_rule</hint> can be used ....</tt></p> + is used, then the result displayed to the student will be as follows. Try clicking on the question mark. + <?php *************** *** 53,57 **** $errors = ''; ! echo ' <h3>'.$fact['name'].' <tt>('.$key.')</tt></h3>'; echo ' <p> '.stack_castext_to_display($fact['fact'],$locals,$options['Display'],$errors).'</p>'; } --- 55,59 ---- $errors = ''; ! echo ' <table width=100%><tr bgcolor=\'#DDDDDD\'><td>'.$fact['name'].' <tt>('.$key.')</tt></td></tr></table>'; echo ' <p> '.stack_castext_to_display($fact['fact'],$locals,$options['Display'],$errors).'</p>'; } Index: author_commonsyntax.php =================================================================== RCS file: /cvsroot/stack/stack-1-0/lang/en/doc/author_commonsyntax.php,v retrieving revision 1.4 retrieving revision 1.5 diff -C2 -d -r1.4 -r1.5 *** author_commonsyntax.php 12 Sep 2005 08:07:47 -0000 1.4 --- author_commonsyntax.php 7 Oct 2005 06:26:38 -0000 1.5 *************** *** 13,17 **** ?> ! <h1>Common syntax, and STACK commands</h1> <p>Every CAS system is unique, and they all have different a syntax for doing --- 13,17 ---- ?> ! <h1>STACK commands</h1> <p>Every CAS system is unique, and they all have different a syntax for doing *************** *** 20,29 **** details of which are given in the invaluable <a href="doc/dev/Rosetta.pdf">Rosetta.pdf</a>. ! This makes moving from one to the other a difficult task! ! Since many CAS systems allow a synonym (or alias) to be set ! up, adapting the function name is not an overwhelming task. ! Furthermore, the vast majority of tasks we need in single variable calculus and ! algebra CAA can be reduced to these tasks. ! So by adopting this common syntax more than expected can be achieved. </p> --- 20,28 ---- details of which are given in the invaluable <a href="doc/dev/Rosetta.pdf">Rosetta.pdf</a>. ! This makes moving from one to the other a difficult task. ! </p> ! <p>This document details functions defined in Maxima unique to STACK, and ! synonyms (or alias) which have been set ! up. </p> *************** *** 54,58 **** if ($type == $vals['type']) { echo "<tr><td nowrap='nowrap' align='top'><b><a NAME=\"STACK_command_$fun\">$fun</a></b></td> ! <td nowrap='nowrap' align='top'>{$vals['syntax']}</tt></td> <td>{$vals['doc']}</td></tr>\n"; } --- 53,57 ---- if ($type == $vals['type']) { echo "<tr><td nowrap='nowrap' align='top'><b><a NAME=\"STACK_command_$fun\">$fun</a></b></td> ! <td nowrap='nowrap' align='top'><tt>{$vals['syntax']}</tt></td> <td>{$vals['doc']}</td></tr>\n"; } *************** *** 70,74 **** <h2>Sets, lists and matrices</h2> ! <p> Lists are entered using brackets, eg <tt>[1,2,3]</tt>. These are native to Maxima.</p> <p> Sets are implemented using the Maxima package <tt>nset</tt> which is automatically loaded by STACK. See the <a href="maximadocs/maxdoc/nset.html">specific documentation</a>. --- 69,78 ---- <h2>Sets, lists and matrices</h2> ! <p> Lists are entered using brackets, eg <tt>[1,2,3]</tt>. These are native to Maxima. ! ! <?php disp_cas('list') ?> ! ! </p> ! <p> Sets are implemented using the Maxima package <tt>nset</tt> which is automatically loaded by STACK. See the <a href="maximadocs/maxdoc/nset.html">specific documentation</a>. *************** *** 92,100 **** CAS, but probably not best suited for more elementary school assessment. Similarly <tt>log</tt> is a natural logarithm. </p> - <table> ! <tr> <td> <b>Trig </br> </td> <td><tt>sin(x), cos(x), tan(x)</tt></td> <td> The angular measure is always assumed to be <em>radians</em>.</td> </tr> ! <tr> <td> <b>Hyperbolic </br> </td> <td><tt>sinh(x), cosh(x), tanh(x)</tt></td></tr> ! <tr> <td> <b>Logarithms </br> </td> <td><tt>log(x), exp(x)</tt></td> <td> Natural logarithms, and exponential function.</td></tr> </table> --- 96,106 ---- CAS, but probably not best suited for more elementary school assessment. Similarly <tt>log</tt> is a natural logarithm. </p> <table> ! <tr> <td> <b>Trig </b> </td> <td><tt>sin(x), cos(x), tan(x)</tt></td> <td> The angular measure is always assumed to be <em>radians</em>.</td> </tr> ! <tr> <td> <b>Inverse trig </b> </td> <td><tt>asin(x)</tt> etc</td> <td> The angular measure is always assumed to be <em>radians</em>. These functions are displayed as sin<sup>-1</sup>(x) etc.</td> </tr> ! <tr> <td> <b>Hyperbolic </b> </td> <td><tt>sinh(x), cosh(x), tanh(x)</tt></td></tr> ! <tr> <td> <b>Logarithms </b> </td> <td><tt>log(x), exp(x)</tt> </td> <td> Natural logarithms, and exponential function.</td></tr> ! <tr> <td> <b> </b> </td> <td><tt>ln(x)</tt> </td> <td> Natural logarithm.</td></tr> ! <tr> <td> <b> </b> </td> <td><tt>sqrt(x)</tt> </td> <td> Square root.</td></tr> </table> *************** *** 103,110 **** <?php disp_cas('algebra') ?> - <h2>Calculus</h2> - - <?php disp_cas('calculus') ?> - <h2>Random numbers</h2> --- 109,112 ---- *************** *** 146,147 **** --- 148,153 ---- <a href='maximadocs/maxdoc/maxima_38.html#IDX948'>MAKELIST</a> command <pre>@(p(k):=x^k,pl:makelist(p(k),k,1,5),plot(pl,[x,-1,1]))@</pre> + + <h2>Calculus and Analysis</h2> + + <?php disp_cas('calculus') ?> Index: author_potresp.php =================================================================== RCS file: /cvsroot/stack/stack-1-0/lang/en/doc/author_potresp.php,v retrieving revision 1.2 retrieving revision 1.3 diff -C2 -d -r1.2 -r1.3 *** author_potresp.php 14 Jul 2005 18:05:05 -0000 1.2 --- author_potresp.php 7 Oct 2005 06:26:38 -0000 1.3 *************** *** 33,39 **** The student's answer is assigned to variable named as Student's answer key. ! This variable name can appear in the list of Feedback variables, together with ! the question variables. Hence, quite sophisticated ! processing is possible before any of the answer tests are applied in the potential response tree itself.</p> --- 33,39 ---- The student's answer is assigned to variable named as Student's answer key. ! This variable can be used in the list of Feedback variables and using it together with ! the question variables allows quite sophisticated ! processing before any of the answer tests are applied in the potential response tree itself.</p> Index: author_gettingstarted.php =================================================================== RCS file: /cvsroot/stack/stack-1-0/lang/en/doc/author_gettingstarted.php,v retrieving revision 1.6 retrieving revision 1.7 diff -C2 -d -r1.6 -r1.7 *** author_gettingstarted.php 12 Sep 2005 08:07:47 -0000 1.6 --- author_gettingstarted.php 7 Oct 2005 06:26:38 -0000 1.7 *************** *** 15,20 **** include_once("{$stack_root}/scripts/maxima/maximafun.php"); - - ?> --- 15,18 ---- *************** *** 93,96 **** --- 91,99 ---- <p>The following things to remember about <tt>castext</tt>: <ul> + <li>Anything enclosed beteeen $ symbols is treated as an <em>inline equation</em>, as is the + case with normal LaTeX.</li> + <li>Anything enclosed beteeen $$ symbols, or matching \[ and \] is treated as a + <em>displayed equation</em>, in the centre of a new line. Again, this is the + case with LaTeX.</li> <li>Anything enclosed between @ symbols is evaluated by the CAS, and displayed using the display option as an inline equation. This is analogous to using LaTeX's $'s. *************** *** 101,104 **** --- 104,109 ---- <li> Anything between <tt><html> stuff </html></tt> structures is kept as HTML, and not interpreted as LaTeX. This enables URL's to be included in the text, as well as external images etc. </li> + <li> A specific key between <tt><hint> key </hint></tt> will allow access to the facts + and formula sheet. See <a href="documentation.php?action=author_factsformula&expand=20">here</a> for more details. </li> </ul></p> *************** *** 395,400 **** ?> - - <p>Simple assignments are of the form <tt>key = value</tt>, for example, --- 400,403 ---- *************** *** 406,409 **** --- 409,428 ---- <pre>p = expand( (x-3)*(x-4) )</pre></p> + <p>Another common task is that of <em>substitution</em>. This can be performed with + Maxima's <?php + $url = $maxima_cmd['SUBST']['urls'][0]; + echo " <a href='maximadocs/maxdoc/$url'>SUBST</a>"; + ?> + command. This is quite useful, for example if we define <tt>p</tt> as follows, + in the <?php stack_doc_dispstackqfield('questionAnsVarsRaw'); ?> + then we can use this in reqponse processing to determine if the student's answer is odd. + <pre>p = ans1 + subst(-x,x,ans1)</pre></p> + All sorts of properties can be checked for in this way. For example, interpolates. Another + example is a stationary point of <tt>f(x)</tt> at <tt>x=a</tt>, which can be checked for using + <pre>p = subst(a,x,diff(ans1,x))</pre></p> + Here we have assumed <tt>a</tt> is some point given to the student, <tt>ans1</tt> is + the <?php stack_doc_dispstackqfield('questionAnsKey'); ?> and that <tt>p</tt> + will be used in the response processing tree. + <p>You can use Maxima's looping structures within Question variables, although the syntax requires this to be of the form *************** *** 450,452 **** --- 469,472 ---- $url = $stack_web_url."maximadocs/maxdoc/maxima_39.html#SEC120"; echo " <a href='$url'>functions</a>.</p> "; + ?> Index: student_factsformula.php =================================================================== RCS file: /cvsroot/stack/stack-1-0/lang/en/doc/student_factsformula.php,v retrieving revision 1.4 retrieving revision 1.5 diff -C2 -d -r1.4 -r1.5 *** student_factsformula.php 29 Jul 2005 15:59:31 -0000 1.4 --- student_factsformula.php 7 Oct 2005 06:26:38 -0000 1.5 *************** *** 13,50 **** */ ! $stackFact['calc_product_rule']['name'] = 'The Product Rule'; ! $stackFact['calc_product_rule']['fact'] = 'The following rule allows one to differentiate functions built up ! by compositions. Let us assume that we have some function which we ! can choose to write as a composition $g(f(x))$. Let $u=f(x)$, then ! $$ ! \frac{\mathrm{d}}{\mathrm{d}{x}} (g(f(x)) = \left(\frac{\mathrm{d}}{\mathrm{d}{x}} f(x)\right)\left(\frac{\mathrm{d}}{\mathrm{d}{u}} g(u)\right), ! $$ or, using alternative notation, $$ (g(f(x)))\' = f\'(x)g\'(f(x)). $$'; ! ! $stackFact['calc_quotient_rule']['name'] = 'The quotient rule'; ! $stackFact['calc_quotient_rule']['fact'] = 'The quotient rule for differentiation states that for any two differentiable functions $f(x)$ and $g(x)$, ! \[\frac{d}{dx}\left(\frac{f}{g}\right)=\frac{\frac{df}{dx}\cdot g - f\cdot \frac{dg}{dx}}{g^2}. \]'; ! ! // . concatinates (joins) strings together. ! //$stackFact['calc_rules']['name'] = 'Rules of Differentiation'; ! //$stackFact['calc_rules']['fact'] = $stackFact['calc_product_rule']['fact'].$stackFact['calc_quotient_rule']['fact'] ! //.$stackFact['calc_chain_rule']['fact'].$stackFact[calc_diff_linearity_rule']['fact'].$stackFact['calc_diff_standard_derivatives']['fact']; ! ! $stackFact['calc_rules']['name'] = 'Calculus rules'; ! $stackFact['calc_rules']['fact'] = $stackFact['calc_product_rule']['fact'].'Another rule'.$stackFact['calc_quotient_rule']['fact']; ! ! ! //show_array($stackFact); ! $stackFact['calc_chain_rule']['name'] = 'The Chain Rule'; ! $stackFact['calc_chain_rule']['fact'] = 'The following rule allows one to find the derivative of a composition of two functions. ! Assume we have a function $f(x) = g(h(x))$, then the derivative of $f(x)$, $f\'(x)$, is given by: ! $$f\'(x) = \frac{df(x)}{dx} = f\'(g(x))\cdot g\'(x).$$ ! Alternatively, we can write: ! $$\frac{df(g(x))}{dx} = \frac{df(g(x))}{dg}\cdot\frac{dg}{dx}.$$'; $stackFact['alg_quadratic_formula']['name'] = 'The Quadratic Formula'; $stackFact['alg_quadratic_formula']['fact'] = 'If we have a quadratic equation of the form: $$ax^2 + bx + c = 0,$$ ! then the solution to that equation given by the quadratic formula is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$'; --- 13,65 ---- */ ! $stackFact['greek_alphabet']['name'] = 'The Greek Alphabet'; ! $stackFact['greek_alphabet']['fact'] = ' ! \begin{tabular}{ccl} ! Upper case, $\quad$& lower case, $\quad$& name \\\\ ! $A$ & $\alpha$ & alpha \\\\ ! $B$ & $\beta$ & beta \\\\ ! $\Gamma$ & $\gamma$ & gamma \\\\ ! $\Delta$ & $\delta$ & delta \\\\ ! $E$ & $\epsilon$ & epsilon \\\\ ! $Z$ & $\zeta$ & zeta \\\\ ! $H$ & $\eta$ & eta \\\\ ! $\Theta$ & $\theta$ & theta \\\\ ! $K$ & $\kappa$ & kappa \\\\ ! $M$ & $\mu$ & mu \\\\ ! $N$ & $\nu$ & nu \\\\ ! $\Xi$ & $\xi$ & xi \\\\ ! $O$ & $o$ & omicron \\\\ ! $\Pi$ & $\pi$ & pi \\\\ ! $I$ & $\iota$ & iota \\\\ ! $P$ & $\rho$& rho \\\\ ! $\Sigma$ & $\sigma$ & sigma \\\\ ! $\Lambda$ & $\lambda$ & lambda \\\\ ! $T$ & $\tau$ & tau \\\\ ! $\Upsilon$ & $\upsilon$ & upsilon \\\\ ! $\Phi$ & $\phi$ & phi \\\\ ! $X$ & $\chi$ & chi \\\\ ! $\Psi$ & $\psi$ &psi \\\\ ! $\Omega$ & $\omega$ & omega ! \end{tabular}'; ! $stackFact['alg_inequalities']['name'] = 'Inequalities'; ! $stackFact['alg_inequalities']['fact'] = '$$ ! a>b \hbox{ means } a \hbox{ is greater than } b ! $$ ! \vskip-0.2in ! $$ ! a<b \hbox{ means } a \hbox{ is less than } b ! $$ ! $$ ! a\geq b \hbox{ means } a \hbox{ is greater than or equal to } b ! $$ ! $$ ! a\leq b \hbox{ means } a \hbox{ is less than or equal to } b ! $$'; $stackFact['alg_quadratic_formula']['name'] = 'The Quadratic Formula'; $stackFact['alg_quadratic_formula']['fact'] = 'If we have a quadratic equation of the form: $$ax^2 + bx + c = 0,$$ ! then the solution(s) to that equation given by the quadratic formula are: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$'; *************** *** 57,259 **** $$a^{-m} = \frac{1}{a^m}$$ $$a^{\frac{1}{n}} = \sqrt[n]{a}$$ ! $$a^{\frac{m}{n}} = (\sqrt[n]{a})^m$$'; $stackFact['alg_logarithms']['name'] = 'The Laws of Logarithms'; $stackFact['alg_logarithms']['fact'] = 'For any positive base $b$ (with $b \neq 1$): ! $$\log_bA = c \mbox{ means } A = b^c$$ ! $$\log_bA + \log_bB = \log_bAB$$ ! $$\log_bA - \log_bB = \log_b\frac{A}{B}$$ ! $$n\log_bA = \log_bA^n$$ ! $$\log_b1 = 0$$ ! $$\log_bb = 1$$ The formula for a change of base is: ! $$\log_ax = \frac{\log_bx}{\log_ba}$$ Logarithms to base $e$, denoted $\log_e$ or alternatively $\ln$ are called natural logarithms. The letter $e$ represents the exponential constant which is approximately 2.718.'; - $stackFact['alg_inequalities']['name'] = 'Inequalities'; - $stackFact['alg_inequalities']['fact'] = '$$ - a>b \hbox{ means } a \hbox{ is greater than } b - $$ - \vskip-0.2in - $$ - a<b \hbox{ means } a \hbox{ is less than } b - $$ - $$ - a\geq b \hbox{ means } a \hbox{ is greater than or equal to } b - $$ - $$ - a\leq b \hbox{ means } a \hbox{ is less than or equal to } b - $$'; $stackFact['alg_partial_fractions']['name'] = 'Partial Fractions'; ! $stackFact['alg_partial_fractions']['fact'] = '\noindent For {\sl proper fractions} ${P(x)\over Q(x)}$ ! where $P$ and $Q$ are polynomials with the degree of $P$ less than the degree of $Q$: \smallbreak \noindent a {\sl linear factor} $ax+b$ in the denominator ! produces a partial fraction of the form ${A\over ax+b}$ \smallbreak \noindent {\sl repeated linear factors} $(ax+b)^2$ in the denominator ! produce partial fractions of the form ${A\over ax+b}+{B\over (ax+b)^2}$ \smallbreak \noindent a {\sl quadratic factor} $ax^2+bx+c$ in the denominator produces a partial fraction of ! the form ${Ax+B\over ax^2+bx+c}$ \smallbreak \noindent{\sl Improper fractions} require an additional term which is a polynomial of degree $n-d$ where $n$ is ! the degree of the numerator and $d$ is the degree of ! the denominator.'; ! $stackFact['hyp_functions']['name'] = 'Hyperbolic Functions'; ! $stackFact['hyp_functions']['fact'] = 'Hyperbolic functions have similar properties to trigonometric functions but can be represented in exponential form as follows:$$ ! \cosh x={{\rm e}^x+{\rm e}^{-x}\over 2}, ! \qquad \sinh x={{\rm e}^x-{\rm e}^{-x}\over 2} ! $$ ! $$ ! \tanh x= {\sinh x\over \cosh x}={{\rm e}^x-{\rm e}^{-x} ! \over {\rm e}^x+{\rm e}^{-x}} $$ $$ ! {\rm sech\,}x={1\over \cosh x}={2\over {\rm e}^x+{\rm e}^{-x}} $$ $$ ! {\rm cosech\,} x= {1\over \sinh x}={2\over {\rm e}^x-{\rm e}^{-x}} $$ $$ ! {\rm coth\,}x ={\cosh x\over \sinh x}= ! {1\over {\rm tanh\,} x} ={{\rm e}^x+{\rm e}^{-x} ! \over {\rm e}^x-{\rm e}^{-x}} $$'; ! $stackFact['hyp_identities']['name'] = 'Hyperbolic Identities'; ! $stackFact['hyp_identities']['fact'] = 'The similarity between the way hyperbolic and trigonometric functions behave is apparent when observing some basic hyperbolic identities: ! \noindent ${\rm e}^x=\cosh x+\sinh x, \quad {\rm e}^{-x}=\cosh x-\sinh x$ ! \noindent $\cosh^2x -\sinh^2x =1$ ! \noindent $1-{\rm tanh}^2x={\rm sech}^2 x$ ! \noindent ${\rm coth}^2 x-1={\rm cosech}^2x$ - \noindent $\sinh(x\pm y)=\sinh x\cosh y\pm \cosh x\sinh y$ ! \noindent $\cosh(x\pm y)=\cosh x\cosh y\pm \sinh x\sinh y$ ! \noindent $\sinh 2x=2\sinh x\cosh x$ - \noindent $\cosh 2x=\cosh^2x+\sinh^2x$ - \noindent $\cosh^2x={\cosh 2x+1\over 2}$ - \vskip0.07in - \noindent $\sinh^2x={\cosh 2x-1\over 2}$'; ! $stackFact['hyp_inverse_functions']['name'] = 'Inverse Hyperbolic Functions'; ! $stackFact['hyp_inverse_functions']['fact'] = '\noindent $\cosh^{-1}x=\ln(x+\sqrt{x^2-1})$\quad for $x\geq 1$ ! \noindent $\sinh^{-1}x=\ln(x+\sqrt{x^2+1})$ ! \noindent ${\rm tanh}^{-1}x = {1\over 2}\ln\left({1+x\over ! 1-x}\right)$ \quad for $-1<x<1$'; ! $stackFact['greek_alphabet']['name'] = 'The Greek Alphabet'; ! $stackFact['greek_alphabet']['fact'] = '\noindent\begin{tabular}{cclcclccl} \hline ! &&&&&&&&\\[-8pt] ! $A$ & $\alpha$& alpha & $I$ & $\iota$& iota & $P$ & $\rho$& rho \\ ! $B$ &$\beta$ & beta & $K$ & $\kappa$ & kappa & $\Sigma$ & $\sigma$ & sigma \\ ! $\Gamma$ & $\gamma$ & gamma & $\Lambda$ & $\lambda$ & lambda & $T$ ! & ! $\tau$ & tau \\ ! $\Delta$ & $\delta$ & delta & $M$ & $\mu$ & mu & $\Upsilon$ & ! $\upsilon$ ! & upsilon \\ ! $E$ & $\epsilon$ & epsilon & $N$ & $\nu$ & nu & $\Phi$ & $\phi$ & ! phi \\ ! $Z$ & $\zeta$ & zeta & $\Xi$ & $\xi$ & xi & $X$ & $\chi$ & chi \\ ! $H$ & $\eta$ & eta & $O$ & $o$ & omicron & $\Psi$ & $\psi$ ! &psi \\ ! $\Theta$ & $\theta$ & theta & $\Pi$ & $\pi$ & pi & $\Omega$ & $\omega$ & omega\\ ! \hline ! \end{tabular}'; ! $stackFact['calc_diff_standard_derivatives']['name'] = 'Standard Derivatives'; ! $stackFact['calc_diff_standard_derivatives']['fact'] = 'The following table displays the derivatives of some standard functions. It is useful to learn these standard derivatives as they are used frequently in calculus. ! $y=f(x)$ $${{\rm d}y\over {\rm d}x}=f\'(x)$$\\ ! $k$, constant 0 \\ ! $x^n$, any constant $n$ $$nx^{n-1}$$\\ ! $e^x$ $e^x$\\ ! $\ln x=\log_{\rm e}x$ $$\frac{1}{x}$$\\ ! $\sin x$ $\cos x$\\ ! $\cos x$ $-\sin x$\\ ! $\tan x = {\sin x\over \cos x}$ $\sec^2 x$\\ ! cosec$\,x={1\over \sin x}$ $-$cosec$\, x\,$cot $x$\\ ! $\sec x={1\over \cos x}$ $\sec x\tan x$\\ ! cot$\, x={\cos x\over \sin x}$ $-$cosec$^2x$\\ ! $\sin^{-1}x$ ${1\over \sqrt{1-x^2}}$\\ ! $\cos^{-1}x$ ${-1\over \sqrt{1-x^2}}$\\ ! $\tan^{-1}x$ ${1\over 1+x^2}$\\ ! $\cosh x$ $\sinh x$\\ ! $\sinh x$ $\cosh x$\\ ! $\tanh x$ sech$^2x$\\ ! sech$\,x$ $-$sech$\,x\tanh x$\\ ! cosech$\,x$ $-$cosech\,$x$\,coth\,$x$\\ ! coth$\,x$ $-$cosech$^2x$\\ ! $\cosh^{-1}x$ ${1\over \sqrt{x^2-1}}$\\ ! $\sinh^{-1}x$ ${1\over \sqrt{x^2+1}}$\\ ! $\tanh^{-1}x$ ${1\over 1-x^2}$\\ ! \hline '; ! $stackFact['calc_diff_linearity_rule']['name'] = 'The Linearity Rule for Differentiation'; ! $stackFact['calc_diff_linearity_rule']['fact'] = '$$ ! {{\rm d}\,\over {\rm d}x}(au+bv)=a{{\rm d}u\over {\rm d}x} ! +b{{\rm d}v\over {\rm d}x}\quad a,b {\rm\ constant} ! $$'; $stackFact['calc_int_standard_integrals']['name'] = 'Standard Integrals'; ! $stackFact['calc_int_standard_integrals']['fact'] = '\begin{table}[h] ! \begin{tabular}{lll}\hline ! &\\[-8pt] ! $f(x)$ & $\int f(x)\,{\rm d}x=F(x)+c$&\\[4pt] ! \hline ! $k$, constant &$kx +c$&\\ ! $x^n$, ($n\ne -1)$ & ${x^{n+1}\over n+1}+c$& ! \\ ! $x^{-1}={\frac{1}{x}}$ & $\left\{\matrix{\ln x+c & x>0\cr ! \ln(-x)+c & x<0\cr}\right.$&\\ ! e$^x$& e$^x+c$&\\ ! $\cos x$ & $\sin x+c$ &\\ ! $\sin x$ & $-\cos x+c$ &\\ ! $\tan x$ & $\ln(\sec x)+c$ & $ ! -{\pi\over 2}<x<{\pi\over 2}$\\ ! $\sec x$ & $\ln (\sec x+\tan x)+c$ & $ ! -{\pi\over 2}<x<{\pi\over 2}$\\ ! cosec$\, x$ & $\ln ($cosec$\,x-$cot$\,x)+c$ & ! \,\,\,\, $0<x<\pi$\\ ! cot$\,x$ & $\ln(\sin x)+c$ & \,\,\,\, $0<x<\pi$\\ ! $\cosh x$ & $\sinh x+c$& ! \\ ! $\sinh x$ & $\cosh x + c$ &\\ ! $\tanh x$ & $\ln \cosh x+c$&\\ ! coth$\,x$ & $\ln \sinh x+c $& $x>0$\\ ! ${1\over x^2+a^2}$ & ${1\over a}\tan^{-1}{x\over a}+c$&$a>0$\\[2pt] ${1\over x^2-a^2}$ & ${1\over 2a}\ln{x-a\over x+a}+c$ ! & $|x|>a>0$\\[2pt] ! ${1\over a^2-x^2}$ & ${1\over 2a}\ln{a+x\over a-x}+c$ & $|x|<a$\\[3pt] ! ${1\over \sqrt{x^2+a^2}}$ & $\sinh^{-1}{x\over a} + c$ &$a>0$ \\ ! ${1\over \sqrt{x^2-a^2}}$ & $\cosh^{-1}{x\over a} + c$ & $x\geq a>0$ \\ ! ${1\over \sqrt{x^2+k}}$ & $\ln (x+\sqrt{x^2+k})+c$&\\ ! ${1\over \sqrt{a^2-x^2}}$ & $\sin^{-1}{x\over a}+c$& $-a\leq ! x\leq a$ \\[4pt] ! \hline ! &\\[-8pt] ! $f(ax+b)$ & ${1\over a}F(ax+b) +c$ & $a\ne 0$ \\[4pt] ! e.g. $\cos(2x-3)$ & ${1\over 2}\sin(2x-3)+c$ & \\[4pt] ! \hline ! ! \end{tabular} ! \end{table}'; $stackFact['calc_int_linearity_rule']['name'] = 'The Linearity Rule for Integration'; --- 72,273 ---- $$a^{-m} = \frac{1}{a^m}$$ $$a^{\frac{1}{n}} = \sqrt[n]{a}$$ ! $$a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$$'; $stackFact['alg_logarithms']['name'] = 'The Laws of Logarithms'; $stackFact['alg_logarithms']['fact'] = 'For any positive base $b$ (with $b \neq 1$): ! $$\log_b(a) = c \mbox{, means } a = b^c$$ ! $$\log_b(a) + \log_b(b) = \log_b(ab)$$ ! $$\log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right)$$ ! $$n\log_b(a) = \log_b\left(a^n\right)$$ ! $$\log_b(1) = 0$$ ! $$\log_b(b) = 1$$ The formula for a change of base is: ! $$\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$$ Logarithms to base $e$, denoted $\log_e$ or alternatively $\ln$ are called natural logarithms. The letter $e$ represents the exponential constant which is approximately 2.718.'; $stackFact['alg_partial_fractions']['name'] = 'Partial Fractions'; ! $stackFact['alg_partial_fractions']['fact'] = '\noindent {\sl Proper fractions} occur with $${\frac{P(x)}{Q(x)}}$$ ! when $P$ and $Q$ are polynomials with the degree of $P$ less than the degree of $Q$. This this case, we proceed ! as follows: write $Q(x)$ in factored form, \smallbreak \noindent a {\sl linear factor} $ax+b$ in the denominator ! produces a partial fraction of the form $${A\over ax+b}$$ \smallbreak \noindent {\sl repeated linear factors} $(ax+b)^2$ in the denominator ! produce partial fractions of the form $${A\over ax+b}+{B\over (ax+b)^2}$$ \smallbreak \noindent a {\sl quadratic factor} $ax^2+bx+c$ in the denominator produces a partial fraction of ! the form $${Ax+B\over ax^2+bx+c}$$ \smallbreak \noindent{\sl Improper fractions} require an additional term which is a polynomial of degree $n-d$ where $n$ is ! the degree of the numerator (ie $P(x)$) and $d$ is the degree of ! the denominator (ie $Q(x)$).'; ! $stackFact['trig_degrees_radians']['name'] = 'Degrees and Radians'; ! $stackFact['trig_degrees_radians']['fact'] = '$$ ! 360^\circ= 2\pi \hbox{ radians},\quad ! 1^\circ={2\pi\over 360}={\pi\over 180}\hbox{ radians} $$ $$ ! 1 \hbox{ radian} = {180\over \pi} \hbox{ degrees} ! \approx 57.3^\circ ! $$'; ! ! $stackFact['trig_standard_values']['name'] = 'Standard Trigonometric Values'; ! $stackFact['trig_standard_values']['fact'] = ' ! $$\sin(45^\circ)={1\over \sqrt{2}}, \qquad \cos(45^\circ) = {1\over \sqrt{2}},\qquad ! \tan( 45^\circ)=1 $$ $$ ! \sin (30^\circ)={1\over 2}, \qquad \cos (30^\circ)={\sqrt{3}\over 2},\qquad ! \tan (30^\circ)={1\over \sqrt{3}} $$ $$ ! \sin (60^\circ)={\sqrt{3}\over 2}, \qquad \cos (60^\circ)={1\over 2},\qquad ! \tan (60^\circ)={ \sqrt{3}} $$'; ! $stackFact['trig_standard_identities']['name'] = 'Standard Trigonometric Identities'; ! $stackFact['trig_standard_identities']['fact'] = '$$\sin(a\pm b)\ = \ \sin(a)\cos(b)\ \pm\ \cos(a)\sin(b)$$ ! $$\cos(a\ \pm\ b)\ = \ \cos(a)\cos(b)\ \mp \\sin(a)\sin(b)$$ ! $$\tan (a\ \pm\ b)\ = \ {\tan (a)\ \pm\ \tan (b)\over1\ \mp\ \tan (a)\tan (b)}$$ ! $$ 2\sin(a)\cos(b)\ = \ \sin(a+b)\ +\ \sin(a-b)$$ ! $$ 2\cos(a)\cos(b)\ = \ \cos(a-b)\ +\ \cos(a+b)$$ ! $$ 2\sin(a)\sin(b) \ = \ \cos(a-b)\ -\ \cos(a+b)$$ ! $$ \sin^2(a)+\cos^2(a)\ = \ 1$$ ! $$ 1+{\rm cot}^2(a)\ = \ {\rm cosec}^2(a),\quad \tan^2(a) +1 \ = \ \sec^2(a)$$ ! $$ \cos(2a)\ = \ \cos^2(a)-\sin^2(a)\ = \ 2\cos^2(a)-1\ = \ 1-2\sin^2(a)$$ ! $$ \sin(2a)\ = \ 2\sin(a)\cos(a)$$ ! $$ \sin^2(a) \ = \ {1-\cos (2a)\over 2}, \qquad \cos^2(a)\ = \ {1+\cos(2a)\over 2}$$'; ! $stackFact['hyp_functions']['name'] = 'Hyperbolic Functions'; ! $stackFact['hyp_functions']['fact'] = 'Hyperbolic functions have similar properties to trigonometric functions but can be represented in exponential form as follows: ! $$ \cosh(x) = \frac{e^x+e^{-x}}{2}, \qquad \sinh(x)=\frac{e^x-e^{-x}}{2} $$ ! $$ \tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{{e^x-e^{-x}}}{e^x+e^{-x}} $$ ! $$ {\rm sech}(x) ={1\over \cosh(x)}={2\over {\rm e}^x+{\rm e}^{-x}}, \qquad {\rm cosech}(x)= {1\over \sinh(x)}={2\over {\rm e}^x-{\rm e}^{-x}} $$ ! $$ {\rm coth}(x) ={\cosh(x)\over \sinh(x)} = {1\over {\rm tanh}(x)} ={{\rm e}^x+{\rm e}^{-x}\over {\rm e}^x-{\rm e}^{-x}}$$'; ! $stackFact['hyp_identities']['name'] = 'Hyperbolic Identities'; ! $stackFact['hyp_identities']['fact'] = 'The similarity between the way hyperbolic and trigonometric functions behave is apparent when observing some basic hyperbolic identities: ! $${\rm e}^x=\cosh(x)+\sinh(x), \quad {\rm e}^{-x}=\cosh(x)-\sinh(x)$$ ! $$\cosh^2(x) -\sinh^2(x) = 1$$ ! $$1-{\rm tanh}^2(x)={\rm sech}^2(x)$$ ! $${\rm coth}^2(x)-1={\rm cosech}^2(x)$$ ! $$\sinh(x\pm y)=\sinh(x)\ \cosh(y)\ \pm\ \cosh(x)\ \sinh(y)$$ ! $$\cosh(x\pm y)=\cosh(x)\ \cosh(y)\ \pm\ \sinh(x)\ \sinh(y)$$ ! $$\sinh(2x)=2\,\sinh(x)\cosh(x)$$ ! $$\cosh(2x)=\cosh^2(x)+\sinh^2(x)$$ ! $$\cosh^2(x)={\cosh(2x)+1\over 2}$$ ! $$\sinh^2(x)={\cosh(2x)-1\over 2}$$'; ! $stackFact['hyp_inverse_functions']['name'] = 'Inverse Hyperbolic Functions'; ! $stackFact['hyp_inverse_functions']['fact'] = '$$\cosh^{-1}(x)=\ln\left(x+\sqrt{x^2-1}\right) \quad \mbox{ for } x\geq 1$$ ! $$\sinh^{-1}(x)=\ln\left(x+\sqrt{x^2+1}\right)$$ ! $${\rm tanh}^{-1}(x) = {1\over 2}\ln\left({1+x\over 1-x}\right) \quad \mbox{ for } -1<x<1$$'; ! $stackFact['calc_diff_standard_derivatives']['name'] = 'Standard Derivatives'; ! $stackFact['calc_diff_standard_derivatives']['fact'] = 'The following table displays the derivatives of some standard functions. It is useful to learn these standard derivatives as they are used frequently in calculus. ! \begin{tabular}{ll} ! {\large\bf $f(x)$} & {\large\bf $f\'(x)$}\\\\ ! $k$, constant & $0$ \\\\ ! $x^n$, any constant $n$ & $nx^{n-1}$\\\\ ! $e^x$ & $e^x$\\\\ ! $\ln(x)=\log_{\rm e}(x)$ & $\frac{1}{x}$ \\\\ ! $\sin(x)$ & $\cos(x)$ \\\\ ! $\cos(x)$ & $-\sin(x)$ \\\\ ! $\tan(x) = \frac{\sin(x)}{\cos(x)}$ & $\sec^2(x)$ \\\\ ! $cosec(x)=\frac{1}{\sin(x)}$ & $-cosec(x)\cot(x)$ \\\\ ! $\sec(x)=\frac{1}{\cos(x)}$ & $\sec(x)\tan(x)$ \\\\ ! $\cot(x)=\frac{\cos(x)}{\sin(x)}$ & $-cosec^2(x)$ \\\\ ! $\cosh(x)$ & $\sinh(x)$ \\\\ ! $\sinh(x)$ & $\cosh(x)$ \\\\ ! $\tanh(x)$ & $sech^2(x)$ \\\\ ! $sech(x)$ & $-sech(x)\tanh(x)$ \\\\ ! $cosech(x)$ & $-cosech(x)\coth(x)$ \\\\ ! $coth(x)$ & $-cosech^2(x)$ \\\\ ! \end{tabular} ! $$ \frac{d}{dx}\left(\sin^{-1}(x)\right) = \frac{1}{\sqrt{1-x^2}}$$ ! $$ \frac{d}{dx}\left(\cos^{-1}(x)\right) = \frac{-1}{\sqrt{1-x^2}}$$ ! $$ \frac{d}{dx}\left(\tan^{-1}(x)\right) = \frac{1}{1+x^2}$$ ! $$ \frac{d}{dx}\left(\cosh^{-1}(x)\right) = \frac{1}{\sqrt{x^2-1}}$$ ! $$ \frac{d}{dx}\left(\sinh^{-1}(x)\right) = \frac{1}{\sqrt{x^2+1}}$$ ! $$ \frac{d}{dx}\left(\tanh^{-1}(x)\right) = \frac{1}{1-x^2}$$ ! '; ! $stackFact['calc_diff_linearity_rule']['name'] = 'The Linearity Rule for Differentiation'; ! $stackFact['calc_diff_linearity_rule']['fact'] = '$$ ! {{\rm d}\,\over {\rm d}x}\big(af(x)+bg(x)\big)=a{{\rm d}f(x)\over {\rm d}x} ! +b{{\rm d}g(x)\over {\rm d}x}\quad a,b {\rm\ constant} ! $$'; ! $stackFact['calc_product_rule']['name'] = 'The Product Rule'; ! $stackFact['calc_product_rule']['fact'] = 'The following rule allows one to differentiate functions which are ! multiplied together. Assume that we wish to differentiate $f(x)g(x)$ with respect to $x$. ! $$ \frac{\mathrm{d}}{\mathrm{d}{x}} \big(f(x)g(x)\big) = f(x) \cdot \frac{\mathrm{d} g(x)}{\mathrm{d}{x}} + g(x)\cdot \frac{\mathrm{d} f(x)}{\mathrm{d}{x}},$$ or, using alternative notation, $$ (f(x)g(x))\' = f\'(x)g(x)+f(x)g\'(x). $$'; ! $stackFact['calc_quotient_rule']['name'] = 'The Quotient Rule'; ! $stackFact['calc_quotient_rule']['fact'] = 'The quotient rule for differentiation states that for any two differentiable functions $f(x)$ and $g(x)$, ! \[\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{g(x)\cdot\frac{df(x)}{dx}\ \ - \ \ f(x)\cdot \frac{dg(x)}{dx}}{g(x)^2}. \]'; ! //show_array($stackFact); ! $stackFact['calc_chain_rule']['name'] = 'The Chain Rule'; ! $stackFact['calc_chain_rule']['fact'] = 'The following rule allows one to find the derivative of a composition of two functions. ! Assume we have a function $f(g(x))$, then defining $u=g(x)$, the derivative with respect to $x$ is given by: ! $$\frac{df(g(x))}{dx} = \frac{dg(x)}{dx}\cdot\frac{df(u)}{du}.$$ ! Alternatively, we can write: ! $$\frac{df(x)}{dx} = f\'(g(x))\cdot g\'(x).$$ '; ! $stackFact['calc_rules']['name'] = 'Calculus rules'; ! $stackFact['calc_rules']['fact'] = '{\bf '.$stackFact['calc_product_rule']['name'].'}\par '.$stackFact['calc_product_rule']['fact']; ! $stackFact['calc_rules']['fact'] .= '{\bf '.$stackFact['calc_quotient_rule']['name'].'}\par '.$stackFact['calc_quotient_rule']['fact']; ! $stackFact['calc_rules']['fact'] .= '{\bf '.$stackFact['calc_chain_rule']['name'].'}\par '.$stackFact['calc_chain_rule']['fact']; $stackFact['calc_int_standard_integrals']['name'] = 'Standard Integrals'; ! $stackFact['calc_int_standard_integrals']['fact'] = ' ! ! $$\int k\ dx = kx +c, \mbox{ where k is constant.}$$ ! $$\int x^n\ dx = \frac{x^{n+1}}{n+1}+c, \quad (n\ne -1)$$ ! $$\int x^{-1}\ dx = \int {\frac{1}{x}}\ dx = \left\{\matrix{\ln(x)+c & x>0\cr ! \ln(-x)+c & x<0\cr}\right.$$ ! ! \begin{tabular}{lll} ! {\large\bf $f(x)$} & {\large\bf $\int f(x)\ dx$}\\\\ ! e$^x$& e$^x+c$&\\\\ ! $\cos(x)$ & $\sin(x)+c$ &\\\\ ! $\sin(x)$ & $-\cos(x)+c$ &\\\\ ! $\tan(x)$ & $\ln(\sec(x))+c$ & $ ! -{\pi\over 2}<x<{\pi\over 2}$\\\\ ! $\sec x$ & $\ln (\sec(x)+\tan(x))+c$ & $ ! -{\pi\over 2}<x<{\pi\over 2}$\\\\ ! cosec$\, x$ & $\ln ($cosec$(x)-\cot(x))+c$ & ! \,\,\,\, $0<x<\pi$\\\\ ! cot$\,x$ & $\ln(\sin(x))+c$ & \,\,\,\, $0<x<\pi$\\\\ ! $\cosh(x)$ & $\sinh(x)+c$& ! \\\\ ! $\sinh(x)$ & $\cosh(x) + c$ &\\\\ ! $\tanh(x)$ & $\ln(\cosh(x))+c$&\\\\ ! coth$(x)$ & $\ln(\sinh(x))+c $& $x>0$\\\\ ! ${1\over x^2+a^2}$ & ${1\over a}\tan^{-1}{x\over a}+c$&$a>0$\\\\[2pt] ${1\over x^2-a^2}$ & ${1\over 2a}\ln{x-a\over x+a}+c$ ! & $|x|>a>0$\\\\[2pt] ! ${1\over a^2-x^2}$ & ${1\over 2a}\ln{a+x\over a-x}+c$ & $|x|<a$\\\\[3pt] ! ${1\over \sqrt{x^2+a^2}}$ & $\sinh^{-1}\left(\frac{x}{a}\right) + c$ &$a>0$ \\\\ ! ${1\over \sqrt{x^2-a^2}}$ & $\cosh^{-1}\left(\frac{x}{a}\right) + c$ & $x\geq a>0$ \\\\ ! ${1\over \sqrt{x^2+k}}$ & $\ln (x+\sqrt{x^2+k})+c$&\\\\ ! ${1\over \sqrt{a^2-x^2}}$ & $\sin^{-1}\left(\frac{x}{a}\right)+c$& $-a\leq ! x\leq a$ \\\\ ! \end{tabular}'; $stackFact['calc_int_linearity_rule']['name'] = 'The Linearity Rule for Integration'; *************** *** 278,355 **** d}x}\left\{\int g(x){\rm d}x\right\}{\rm d}x $$'; - $stackFact['trig_degrees_radians']['name'] = 'Degrees and Radians'; - $stackFact['trig_degrees_radians']['fact'] = '$$ - 360^\circ= 2\pi \hbox{ radians},\quad - 1^\circ={2\pi\over 360}={\pi\over 180}\hbox{ radians} - $$ - $$ - 1 \hbox{ radian} = {180\over \pi} \hbox{ degrees} - \approx 57.3^\circ - $$'; - - $stackFact['trig_standard_values']['name'] = 'Standard Trigonometric Values'; - $stackFact['trig_standard_values']['fact'] = '$$ - \sin 45^\circ={1\over \sqrt{2}}, \qquad \cos 45^\circ = {1\over \sqrt{2}},\qquad - \tan 45^\circ=1 - $$ - $$ - \sin 30^\circ={1\over 2}, \qquad \cos 30^\circ={\sqrt{3}\over 2},\qquad - \tan 30^\circ={1\over \sqrt{3}} - $$ - $$ - \sin 60^\circ={\sqrt{3}\over 2}, \qquad \cos 60^\circ={1\over 2},\qquad - \tan 60^\circ={ \sqrt{3}} - $$'; - - $stackFact['trig_standard_identities']['name'] = 'Standard Trigonometric Identities'; - $stackFact['trig_standard_identities']['fact'] = '\noindent $\sin(A\pm B)=\sin A\cos B\pm \cos A\sin B$ - \smallbreak - \noindent $\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B - $ - \smallbreak - \noindent $\tan (A\pm B)={\tan A\pm \tan B\over - 1\mp \tan A\tan B} - $ - \vskip0.05in - \noindent $2\sin A\cos B=\sin(A+B)+\sin(A-B)$ - \smallbreak - \noindent $2\cos A\cos B=\cos(A-B)+\cos(A+B)$ - \smallbreak - \noindent $2\sin A\sin B =\cos(A-B)-\cos(A+B)$ - \smallbreak - \noindent $\sin^2 A+\cos^2 A=1$ - \smallbreak - \noindent $1+{\rm cot}^2 A={\rm cosec}^2 A,\quad - \tan^2 A +1 = \sec^2A$ - \smallbreak - \noindent $\cos 2A=\cos^2A-\sin^2A=2\cos^2A-1=1-2\sin^2A$ - \smallbreak - \noindent $\sin 2A=2\sin A\cos A$ - \smallbreak - \noindent $\sin^2A ={1-\cos 2A\over 2}, \qquad - \cos^2A={1+\cos 2A\over 2}$'; - - $stackFact['part_frac']['name'] = 'Partial Fraction Form'; - $stackFact['part_frac']['fact'] = '\noindent - Partial Fraction Form is the result of reversing the process of adding fractions. - It involves splitting a complicated rational expression into the sum of two or - more simpler ones.\smallbreak \noindent - Example: Find the partial form of - $$\frac{5x + 14}{(x+1)(x+4)}$$ - $$\frac{5x + 14}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x+4}$$ - $$5X+14=A(x+4) + B(x+1)$$ - Let $$x=-4$$ - Then $$5(-4) + 14 = 0 + B(-3)$$ - $$-6 = -3B$$ - $$B=2$$ - Let $$x =-1$$ - Then $$5(-1) + 14 = A(3) + 0$$ - $$9 = 3A$$ - $$A=3$$ - Hence the partial fraction form of - $$\frac{5x + 14}{(x+1)(x+4)}$$ is - $$\frac{3}{x+1} + \frac{2}{x+4}$$.'; - - ?> --- 292,295 ---- |
