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• Notes

Q1: Find the lenght of curve

$\vec{r}(t)=\langle t, 3 \cos t, 3 \sin t\rangle,\quad - 5 \leq t \leq 5$

(1) $\vec r^{\prime}(t)?$

$\vec r^{\prime}(t)=\langle 1,-3 \sin t, 3 \cos t)$

(2) $L=\int_{-5}^{5}\left|\vec r^{\prime}(t)\right| d t$

$L=\int_{-5}^{5} \sqrt{1+9 \sin ^{2} t+9 \cos ^{2} t} dt$

$=\int_{-5}^{5} \sqrt{1+9} dt=\int_{-5}^{5} \sqrt{10} dt$

$L=[t \sqrt{10}]_{-5}^{5}$

$L=(\sqrt{10}\times 5)-(\sqrt{10} \times -5)$

$L=5 \sqrt{10}+5 \sqrt{10}$

$L=10 \sqrt{10}$

Q2: Req. find The Curviture

$\overrightarrow{r(t)}=\left\langle t^{2}, \ln t, t(\ln t\rangle,(1,0,10)\right.$

sol.

(1) $\vec{r^{\prime}(t)}=\left\langle 2 t, \frac{1}{t}, 1+\ln t\right\rangle$

$\left|\vec r^{\prime}(t)\right|= \sqrt{4 t^{2}+\frac{1}{t^{2}}+(1+\ln t)^{2}}$

(2) $\overrightarrow{r^{\prime \prime}(t)}=\left\langle 2, \frac{-1}{t^{2}}, \frac{1}{t}\right\rangle$

$k(t)=\frac {\left|\vec r^{\prime}(t)\times \vec r^{\prime \prime}(t)\right|}{\left|\vec r^{\prime}(t)\right|^{3}}$

$\overrightarrow{r^{\prime}}(t) \times \overrightarrow{r^{\prime\prime}}(t)=\left|\begin{array}{ccc}{i} & {j} & {k} \\ {2 t} & {\frac{1}{t}} & {1+\ln t} \\ {2} & {\frac{-1}{t^{2}}} & {\frac{1}{t}}\end{array}\right|$

$=\frac{-1}{t^{2}}(2+\ln t) \hat{i}+(2 \cdot \ln t) \hat{j}-\frac{4}{t} \hat{k}$

$k(t)=\frac{\left|\frac{1}{t^{2}} (2+\ln t)\hat i+2 \ln t \hat{j}-\frac{4}{t} \hat{k}\right|}{\left(\sqrt{4 t^{2}+\frac{1}{t^{2}}+(1+\ln t)^{2}}\right)^{3}}$

$k(1)=\frac{|2\hat i+0\hat j-4 \hat k|}{6^{\frac{3}{2}}}=\frac{\sqrt{4+16}}{\sqrt{2^{3} \cdot 3^{3}}}$

$k(1)=\frac{\sqrt{2} 0}{6 \sqrt{6}}=\frac{\sqrt{30}}{18}$

Q3: Max. Curviture

$y=\ln x$

(1) $y^{\prime}=\frac{1}{x}, y^{\prime \prime}=\frac{-1}{x^{2}}$

(2) $k(x)=\frac{\left|f^{\prime \prime}(x)\right|}{\left[1+\left(f^{\prime}(x)\right)^{2}\right]^{\frac{3}{2}}}=\frac{\left|\frac{-1}{x^{2}}\right|}{\left[1+\left(\frac{1}{x}\right)^{2}\right]^{\frac{3}{2}}}$

$k(x)=\frac{1}{x^{2}\left[1+\frac{1}{x^{2}}\right]^{\frac{3}{2}}}=x^{-2}\left[1+\frac{1}{x^{2}}\right]^{-\frac{3}{2}}$

(3) $k^{\prime}(x)=\frac{-2 x^{2}\left(1+\frac{1}{x^{2}}\right)+3}{x^{5}\left(1+\frac{1}{x^{2}}\right)^{\frac{5}{2}}}=\frac{1-2 x^{2}}{x^{3}\left(1+\frac{1}{x^{2}}\right)^{\frac{5}{2}}}$

(4) $k^{\prime}(x)=0$

$1-2 x^{2}=0$

\$x=\frac{1}{\sqrt{2}}\quad , \quad x=-\frac{1}{\sqrt{2}}$

Max cur. $\left(\frac{1}{\sqrt{2}}, \ln \frac{1}{\sqrt{2}}\right)$