91Ó°ÊÓ

\(36-37\) Plot the space curve and its curvature function \(\kappa(t)\) Comment on how the curvature reflects the shape of the curve. $$ \mathbf{r}(t)=\langle t-\sin t, 1-\cos t, 4 \cos (t / 2)\rangle, \quad 0 \leqslant t \leqslant 8 \pi $$

Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and view- points that reveal the true nature of the curve. $$ \mathbf{r}(t)=\langle\cos (8 \cos t) \sin t, \sin (8 \cos t) \sin t, \cos t\rangle $$

(a) If a particle moves along a straight line, what can you say about its acceleration vector? (b) If a particle moves with constant speed along a curve, what can you say about its acceleration vector?

Evaluate the integral. $$ \int_{1}^{4}\left(2 t^{3 / 2} \mathbf{i}+(t+1) \sqrt{t} \mathbf{k}\right) d t $$

Plot the space curve and its curvature function \(\kappa(t)\) Comment on how the curvature reflects the shape of the curve. $$ \mathbf{r}(t)=\left\langle t e^{t}, e^{-t}, \sqrt{2} t\right\rangle, \quad-5 \leqslant t \leqslant 5 $$

Find the tangential and normal components of the acceleration vector. $$ \mathbf{r}(t)=\left(t^{2}+1\right) \mathbf{i}+t^{3} \mathbf{j}, \quad t \geqslant 0 $$

Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and view- points that reveal the true nature of the curve. $$ \mathbf{r}(t)=\langle\cos 2 t, \cos 3 t, \cos 4 t\rangle $$

Evaluate the integral. $$ \int_{0}^{1}\left(\frac{1}{t+1} \mathbf{i}+\frac{1}{t^{2}+1} \mathbf{j}+\frac{t}{t^{2}+1} \mathbf{k}\right) d t $$

Evaluate the integral. $$ \int_{0}^{\pi / 4}\left(\sec t \tan t \mathbf{i}+t \cos 2 t \mathbf{j}+\sin ^{2} 2 t \cos 2 t \mathbf{k}\right) d t $$

Find the tangential and normal components of the acceleration vector. $$ \mathbf{r}(t)=2 t^{2} \mathbf{i}+\left(\frac{2}{3} t^{3}-2 t\right) \mathbf{j} $$