Need Help?

  • Notes
  • Comments & Questions

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

No comments yet

Join the conversation

Join Notatee Today!