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diff --git a/Master/texmf-dist/doc/latex/probsoln/samples/easy2.tex b/Master/texmf-dist/doc/latex/probsoln/samples/easy2.tex deleted file mode 100644 index 9958efcb41c..00000000000 --- a/Master/texmf-dist/doc/latex/probsoln/samples/easy2.tex +++ /dev/null @@ -1,135 +0,0 @@ - % These are all easy differentiation problems - -\begin{defproblem}{diffeasy:gpowh}% -\begin{onlyproblem}% -\(f(x) = g(x)^{h(x)}.\)% -\end{onlyproblem}% -\begin{onlysolution}% -\begin{eqnarray*} -f(x) & = & e^{\ln g(x)^{h(x)}}\\ - & = & e^{h(x)\ln g(x)}\\ -f'(x) & = & e^{h(x)\ln g(x)}(h'(x)\ln g(x) + h(x)\frac{g'(x)}{g(x)})\\ - & = & g(x)^{h(x)}(h'(x)\ln g(x) + \frac{h(x)g'(x)}{g(x)}) -\end{eqnarray*}% -\end{onlysolution}% -\end{defproblem} - -\begin{defproblem}{diffeasy:arcsin}% -\begin{onlyproblem}% -\(y = \arcsin(x)\)% -\end{onlyproblem}% -\begin{onlysolution}% -\[\sin(y) = x\] -diff. w.r.t. $x$: -\begin{eqnarray*} -\cos y \frac{dy}{dx} & = & 1\\ -\frac{dy}{dx} & = & \frac{1}{\cos y}\\ - & = & \frac{1}{\sqrt{1 - \sin^2y}}\\ - & = & \frac{1}{\sqrt{1-x^2}}. -\end{eqnarray*} -\end{onlysolution}% -\end{defproblem} - -\begin{defproblem}{diffeasy:arccos}% -\begin{onlyproblem}% -$y = \arccos x$.% -\end{onlyproblem}% -\begin{onlysolution}% -\(\cos y = x\) -diff. w.r.t. $x$: -\begin{eqnarray*} --\sin y \frac{dy}{dx} & = & 1\\ -\frac{dy}{dx} & = & \frac{-1}{\sin y}\\ - & = & \frac{-1}{\sqrt{1-\cos^2y}}\\ - & = & \frac{-1}{\sqrt{1-x^2}} -\end{eqnarray*} -\end{onlysolution}% -\end{defproblem} - -\begin{defproblem}{diffeasy:tan}% -\begin{onlyproblem}% -\(y = \tan x\)% -\end{onlyproblem} -\begin{onlysolution}% -\begin{eqnarray*} -y & = & \tan x\\ - & = & \frac{\sin x}{\cos x}\\ -\frac{dy}{dx} & = & \frac{\cos x}{\cos x} + \sin x\times\frac{-1}{\cos^2x}\times -\sin x\\ - & = & 1 + \tan^2x\\ - & = & \sec^2x. -\end{eqnarray*} -\end{onlysolution}% -\end{defproblem} - -\begin{defproblem}{diffeasy:arctan}% -\begin{onlyproblem}% -\(y = \arctan x = \tan^{-1}x\)% -\end{onlyproblem}% -\begin{onlysolution}% -\[\tan y = x\] -diff w.r.t. $x$: -\begin{eqnarray*} -\sec^2y\frac{dy}{dx} & = & 1\\ -\frac{dy}{dx} & = & \frac{1}{\sec^2y}\\ - & = & \frac{1}{1+\tan^2y}\\ - & = & \frac{1}{1+x^2} -\end{eqnarray*} -\end{onlysolution}% -\end{defproblem} - -\begin{defproblem}{diffeasy:cot}% -\begin{onlyproblem}% -\(y = (\tan x)^{-1} = \cot x\)% -\end{onlyproblem} -\begin{onlysolution}% -\begin{eqnarray*} -\frac{dy}{dx} & = & -(\tan x)^{-2}\sec^2x\\ - & = & -\frac{\cos^2x}{\sin^2x}\cdot\frac{1}{\cos^2x}\\ - & = & \frac{-1}{\sin^2x}\\ - & = & -\csc^2x. -\end{eqnarray*} -\end{onlysolution}% -\end{defproblem} - -\begin{defproblem}{diffeasy:cosxsqsinx}% -\begin{onlyproblem}% -$y = \cos(x^2)\sin x$.% -\end{onlyproblem}% -\begin{onlysolution}% -\[\frac{dy}{dx} = -\sin(x^2)2x\sin x + \cos(x^2)\cos x\] -\end{onlysolution}% -\end{defproblem} - -\begin{defproblem}{diffeasy:xlnx}% -\begin{onlyproblem}% -$y = (x+1)\ln(x+1)$. -\end{onlyproblem}% -\begin{onlysolution}% -\begin{eqnarray*} -\frac{dy}{dx} & = & \ln(x+1) + \frac{x+1}{x+1}\\ - & = & 1 + \ln(x+1). -\end{eqnarray*} -\end{onlysolution} -\end{defproblem} - -\begin{defproblem}{diffeasy:glng}% -\begin{onlyproblem}% -$f(x) = g(x)\ln(g(x))$. -\end{onlyproblem} -\begin{onlysolution}% -\begin{eqnarray*} -f'(x) & = & g'(x)\ln(g(x)) + \frac{g(x)}{g(x)}g'(x)\\ - & = & g'(x)(1+\ln(g(x))). -\end{eqnarray*} -\end{onlysolution}% -\end{defproblem} - -\begin{defproblem}{diffeasy:sinx/x} -\begin{onlyproblem}% -$y = \frac{\sin x}{x}$. -\end{onlyproblem} -\begin{onlysolution}% -\[\frac{dy}{dx} = \frac{\cos x}{x} - \frac{\sin x}{x^2}\] -\end{onlysolution}% -\end{defproblem} - |