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authorKarl Berry <karl@freefriends.org>2017-07-13 23:06:59 +0000
committerKarl Berry <karl@freefriends.org>2017-07-13 23:06:59 +0000
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treeb11e221ce28f24eee3995a8efb51d2e76993472c /Master/texmf-dist/doc/latex/probsoln/samples/prob-implicit.tex
parent04f93eb1202717fcb407f1271d28f4153aaa0ca0 (diff)
probsoln (10jul17)
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+ % This file is public domain
+ %
+ % These are all implicit differentiation problems
+
+\newproblem{imd:circ}{%
+Find the gradient of the unit circle ($x^2 + y^2 = 1$).}{%
+Differentiating with respect to $x$ gives:
+\begin{eqnarray*}
+2x + 2y\frac{dy}{dx} & = & 0\\
+\frac{dy}{dx} & = & \frac{-2x}{2y}\\
+ & = & \frac{-x}{\sqrt{1-x^2}}.
+\end{eqnarray*}}
+
+\newproblem{imd:ellipse}{%
+Find the gradient of the ellipse given by $4x^2 + 3y^2 = 25$.}{%
+Differentiating with respect to $x$ gives:
+\begin{eqnarray*}
+8x + 6y\frac{dy}{dx} & = & 0\\
+\frac{dy}{dx} & = & \frac{-8x}{6y}\\
+ & = & \frac{-4x}{\sqrt{25-4x^2}}.
+\end{eqnarray*}}
+
+\newproblem{imd:ysq:xcuov2mx}{%
+Find $\frac{dy}{dx}$, given
+\begin{displaymath}
+y^2 = \frac{x^3}{2-x}
+\end{displaymath}}{%
+Differentiating both sides w.r.t.\ $x$:
+\begin{eqnarray*}
+2y\frac{dy}{dx} & = & \frac{(2-x)3x^2 - x^3(-1)}{(2-x)^2}\\
+ & = & \frac{3x^2(2-x) + x^3}{(2-x)^2}\\
+ & = & \frac{6x^2 - 3x^3 + x^3}{(2-x)^2}\\
+ & = & \frac{6x^2-2x^3}{(2-x)^2}\\
+ & = & 2x^2\frac{3-x}{(2-x)^2}
+\end{eqnarray*}
+Therefore
+\begin{displaymath}
+y\frac{dy}{dx} = x^2\frac{3-x}{(2-x)^2}
+\end{displaymath}}
+
+\newproblem{imd:exy:IIxay}{%
+Differentiate w.r.t.\ $x$:
+\begin{displaymath}
+e^{xy} = 2x + y
+\end{displaymath}}{%
+Differentiating both sides w.r.t.\ $x$:
+\begin{eqnarray*}
+e^{xy}(1y + x\frac{dy}{dx}) & = & 2 + \frac{dy}{dx}\\
+xe^{xy}\frac{dy}{dx} - \frac{dy}{dx} & = & 2 - ye^{xy} \\
+\frac{dy}{dx}(xe^{xy}-1) & = & 2 - ye^{xy}\\
+\frac{dy}{dx} & = & \frac{2-ye^{xy}}{xe^{xy}-1}
+\end{eqnarray*}}