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author | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
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committer | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
commit | e0c6872cf40896c7be36b11dcc744620f10adf1d (patch) | |
tree | 60335e10d2f4354b0674ec22d7b53f0f8abee672 /macros/latex/contrib/probsoln/samples/prob-easy.tex |
Initial commit
Diffstat (limited to 'macros/latex/contrib/probsoln/samples/prob-easy.tex')
-rw-r--r-- | macros/latex/contrib/probsoln/samples/prob-easy.tex | 114 |
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diff --git a/macros/latex/contrib/probsoln/samples/prob-easy.tex b/macros/latex/contrib/probsoln/samples/prob-easy.tex new file mode 100644 index 0000000000..0527e3e464 --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/prob-easy.tex @@ -0,0 +1,114 @@ + % This file is public domain + % + % These are all easy differentiation problems + +\newproblem{diffeasy:gpowh}{% +\(f(x) = g(x)^{h(x)}.\)}{% +\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*}} + +\newproblem{diffeasy:arcsin}{% +\(y = \arcsin(x)\)}{% +\[\sin(y) = x\] +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*}} + +\newproblem{diffeasy:arccos}{% +$y = \arccos x$.}{% +\(\cos y = x\) +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*}} + +\newproblem{diffeasy:tan}{% +\(y = \tan x\)}{% +\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*}} + +\newproblem{diffeasy:arctan}{% +\(y = \arctan x = \tan^{-1}x\)}{% +\[\tan y = x\] +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*}} + +\newproblem{diffeasy:cot}{% +\(y = (\tan x)^{-1} = \cot x\)}{% +\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*}} + +\newproblem{diffeasy:cosxsqsinx}{% +$y = \cos(x^2)\sin x$.}{% +\[\frac{dy}{dx} = -\sin(x^2)2x\sin x + \cos(x^2)\cos x\]} + +\newproblem{diffeasy:xlnx}{% +$y = (x+1)\ln(x+1)$.}{% +\begin{eqnarray*} +\frac{dy}{dx} & = & \ln(x+1) + \frac{x+1}{x+1}\\ + & = & 1 + \ln(x+1). +\end{eqnarray*}} + +\newproblem{diffeasy:glng}{% +$f(x) = g(x)\ln(g(x))$.}{% +\begin{eqnarray*} +f'(x) & = & g'(x)\ln(g(x)) + \frac{g(x)}{g(x)}g'(x)\\ + & = & g'(x)(1+\ln(g(x))). +\end{eqnarray*}} + +\newproblem{diffeasy:sinx/x}{% +$y = \frac{\sin x}{x}$.}{% +\[\frac{dy}{dx} = \frac{\cos x}{x} - \frac{\sin x}{x^2}\]} + +\newproblem{diffeasy:exp4x}{% + $y = \exp(4x)$ +}% +{% + \[\frac{dy}{dx} = 4\exp(4x)\] +} + +\newproblem{diffeasy:exp3x+2}{% + $y = \exp(3x+2)$ +}% +{% + \[\frac{dy}{dx} = 3\exp(3x+2)\] +} + +\newproblem{diffeasy:cubic}{% + $y=x^3 + 4x^2 - x + 3$ +}% +{% + \[\frac{dy}{dx} = 3x^2 + 8x - 1\] +} + +\newproblem{diffeasy:quad}{% + $y=2x^3 + 6x -1$ +}% +{% + \[\frac{dy}{dx} = 6x + 6 = 6(x+1)\] +} |