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author | Karl Berry <karl@freefriends.org> | 2008-09-08 00:42:04 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2008-09-08 00:42:04 +0000 |
commit | 036469499e69e21110bdbbd77af3e90de86a382a (patch) | |
tree | 34d517f0b3490d8eed24f45e7bd1ce7ab63ab825 /Master/texmf-dist/doc/latex/probsoln/samples/1stprncp.tex | |
parent | 2f2e79372e149e87d92bdd0d33d596793e6b1252 (diff) |
probsoln 3.0 (27aug08)
git-svn-id: svn://tug.org/texlive/trunk@10474 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Master/texmf-dist/doc/latex/probsoln/samples/1stprncp.tex')
-rw-r--r-- | Master/texmf-dist/doc/latex/probsoln/samples/1stprncp.tex | 61 |
1 files changed, 61 insertions, 0 deletions
diff --git a/Master/texmf-dist/doc/latex/probsoln/samples/1stprncp.tex b/Master/texmf-dist/doc/latex/probsoln/samples/1stprncp.tex new file mode 100644 index 00000000000..514abf57763 --- /dev/null +++ b/Master/texmf-dist/doc/latex/probsoln/samples/1stprncp.tex @@ -0,0 +1,61 @@ + % These all involve differentiating from 1st principles + +\newproblem{dfp:xcube}{% +Differentiate $f(x) = x^3$ with respect to $x$ by first principles.}{% +\begin{eqnarray*} +\frac{dy}{dx} & = & \lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x) - f(x)}{\Delta x}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{(x+\Delta x)^3-x^3}{\Delta x}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{(x+\Delta x)(x^2+2x\Delta x+(\Delta x)^2)-x^3}{\Delta x}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{x^3+3x^2\Delta x+3x(\Delta x)^2+(\Delta x)^3-x^3}{\Delta x}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{3x^2\Delta x+3x(\Delta x)^2+(\Delta x)^3}{\Delta x}\\ + & = & \lim_{\Delta x\rightarrow 0}3x^2+3x\Delta x + (\Delta x)^2\\ + & = & 3x^2 +\end{eqnarray*}} + +\newproblem{dfp:Ioverxsq}{% +Differentiate $\displaystyle f(x) = \frac{1}{x^2}$ with respect to $x$ by first principles.}{% +\begin{eqnarray*} +\frac{df}{dx} & = & \lim_{\Delta x\rightarrow 0}\frac{\frac{1}{(x+\Delta x)^2}-\frac{1}{x^2}}{\Delta x}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{\frac{x^2-(x+\Delta x)^2}{x^2(x+\Delta x)^2}}{\Delta x}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{x^2-(x^2+2x\Delta x+(\Delta x)^2)}{x^2\Delta x(x+\Delta x)^2}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{-2x\Delta x-(\Delta x)^2}{x^2\Delta x(x+\Delta x)^2}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{-2x-\Delta x}{x^2(x+\Delta x)^2}\\ + & = & \frac{-2x}{x^2x^2}\\ + & = & -\frac{2}{x^3} +\end{eqnarray*}} + +\newproblem{dfp:sqrtx}{% +Differentiate from first principles $f(x) = \surd x$}{% +\begin{eqnarray*} +\frac{df}{dx} & = & \lim_{\Delta x\rightarrow 0}\frac{\sqrt{x+\Delta x}-\surd x}{\Delta x}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{(\sqrt{x+\Delta x}-\surd x)(\sqrt{x+\delta x}+\surd x)}{\Delta x(\sqrt{x+\Delta x}+\surd x)}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{x+\Delta x - x}{\Delta x(\sqrt{x+\Delta x}+\surd x)}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{\Delta x}{\Delta x(\sqrt{x+\Delta x}+\Delta x)}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{1}{\sqrt{x+\Delta x}+\surd x}\\ + & = & \frac{1}{2\surd x} +\end{eqnarray*}} + +\newproblem{dfp:cons}{% +Differentiate from first principles $f(x) = c$ where $c$ is a constant.}{% +\begin{eqnarray*} +\frac{df}{dx} & = & \lim_{\Delta x\rightarrow 0}\frac{c-c}{\Delta x}\\ + & = & \lim_{\Delta x\rightarrow 0}0\\ + & = & 0 +\end{eqnarray*}} + +\newproblem{dfp:cosx}{% +Given +\begin{eqnarray*} +\lim_{x \rightarrow 0} \frac{\cos x - 1}{x} & = & 0\\ +\lim_{x \rightarrow 0} \frac{\sin x}{x} & = & 1 +\end{eqnarray*} +differentiate from first principles $f(x) = \cos x$.}{% +\begin{eqnarray*} +\frac{df}{dx} & = & \lim_{\Delta x \rightarrow 0}\frac{f(x + \Delta x) - f(x)}{\Delta x}\\ + & = & \lim_{\Delta x \rightarrow 0} \frac{\cos(x + \Delta x) - \cos(x)}{\Delta x}\\ + & = & \lim_{\Delta x \rightarrow 0} \frac{\cos x\cos\Delta x - \sin x\sin\Delta x - \cos x}{\Delta x}\\ + & = & \lim_{\Delta x \rightarrow 0} \frac{\cos x(\cos\Delta x - 1) - \sin x\sin\Delta x}{\Delta x}\\ + & = & \cos x\lim_{\Delta x \rightarrow 0}\frac{\cos\Delta x - 1}{\Delta x} + - \sin x\lim_{\Delta x \rightarrow 0}\frac{\sin\Delta x}{\Delta x}\\ + & = & -1 \qquad\mbox{(using given results)} +\end{eqnarray*}} |