Chapter 5: Problem 12
\(P(t)=60 t^{2}-t^{3}\) \(P^{\prime}(t)=120 t-3 t^{2}=900\) \(\Rightarrow t^{2}-40 t+300=0\) \(t=10,30\)
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slope of normal \(\Rightarrow 3 x-y+3=0\) $$ x=0 \& y=3 $$ Pt of normal \(=(0,3)\) \(\frac{d y}{d x}=\frac{-1}{3}=f^{\prime}(0)\) $\lim _{x \rightarrow 0} \frac{x^{2}}{f\left(x^{2}\right)+4 f\left(7 x^{2}\right)-5 f\left(4 x^{2}\right)}$ $\lim _{x \rightarrow 0} \frac{2 x}{x\left[2 f^{\prime}\left(x^{2}\right)+56 f^{\prime}\left(7 x^{2}\right)-40 f^{\prime}\left(4 x^{2}\right)\right]}$ \(=\frac{2}{-6}=\frac{-1}{3}\)
\(x y^{2}=1\) \(y^{2}+2 x y \frac{d y}{d x}=0\) \(\frac{d y}{d x}=\frac{-y}{2 x}\) \(-\frac{d x}{d y}=\frac{2 x}{y}=\frac{2}{y^{3}}\) \(y-y_{1}=\frac{2}{y_{1}^{3}}\left(x-x_{1}\right)\) \(+y_{1}^{4}=2 x_{1}\) \(y_{1}^{6}=2\) \(y_{1}=\pm 2^{1 / 6}\) \(x_{1}=\pm 2^{-\sqrt{3}}\)
\(y=\frac{x^{2}}{4}-3 x+10\) For \(y=2-\frac{x^{2}}{4}\) eqn of tangent \(y-2=m x+m^{2}\) Solving (II) \& (I) \(m x+2+m^{2}=\frac{x^{2}}{4}-3 x+10\) \(\Rightarrow \frac{x^{2}}{4}-(m+3) x+\left(8-m^{2}\right)=0\) \(\mathrm{D}=0\) \((m+3)^{2}-\left(8-m^{2}\right)=0\) \(2 m^{2}+6 m+1=0\) \(\mathrm{m}_{1}+\mathrm{m}_{2}=-3\)
\(x=t^{2}+t+1\) \(\frac{d x}{d t}=2 t+1\) \(y=t^{2}-t+1\) \(\frac{d y}{d t}=2 t-1\) \(\frac{d y}{d x}=\frac{2 t-1}{2 t+1}\) $\Rightarrow y-\left(t^{2}-t+1\right)=\frac{2 t-1}{2 t+1}\left(x-\left(t^{2}+t+1\right)\right)$ $\Rightarrow\left(t-t^{2}\right)=\left(\frac{2 t-1}{2 t+1}\right)\left(-\left(t^{2}+t\right)\right)$ \(\Rightarrow 2 t^{2}+t-2 t^{3}-t^{2}=t^{2}+t-2 t^{3}-2 t^{2}\) \(\Rightarrow 2 t^{2}=0\) \(\Rightarrow t=0\)
Eqn of tangent $\cos \frac{\mathrm{t}}{2} \mathrm{y}-\mathrm{a}\left[2 \cos \frac{\mathrm{t}}{2} \sin \mathrm{t}-\sin 2 \operatorname{t} \cos \frac{\mathrm{t}}{2}\right]=$ $-\sin \frac{\mathrm{t}}{2} \mathrm{~s}+\mathrm{a}\left[2 \sin \frac{\mathrm{t}}{2} \cos \mathrm{t}+\cos 2 \mathrm{t} \sin \frac{\mathrm{t}}{2}\right]$ $\left(\cos \frac{t}{2}\right) y+\left(\sin \frac{t}{2}\right) x=a\left[\begin{array}{l}2\left[\sin \frac{t}{2} \cos t+\cos \frac{t}{2} \sin t\right] \\ +\cos 2 \operatorname{t} \sin \frac{t}{2}-\sin 2 t \operatorname{t} 0 s \frac{t}{2}\end{array}\right]$ $$ =a\left[2 \sin \frac{3 t}{2}-\sin \frac{3 t}{2}\right] $$ \(=a \sin \frac{3 t}{2} .\) so, \(p=a \sin \frac{3 t}{2}, p_{1}=3 a \cos \frac{3 t}{2}\) \(9 p^{2}+p_{1}^{2}=9 a^{2}\)
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