Chapter 13

Find the length of the curve. \(\mathbf{r}(t)=\langle 2 \sin t, 5 t, 2 \cos t\rangle, \quad-10 \leqslant t \leqslant 10\)

The table gives coordinates of a particle moving through space along a smooth curve. (a) Find the average velocities over the time intervals \([0,1]\) \([0.5,1],[1,2],\) and \([1,1.5] .\) (b) Estimate the velocity and speed of the particle at \(t=1\) $$\begin{array}{|c|c|c|c|}\hline t & {x} & {y} & {z} \\ \hline 0 & {2.7} & {9.8} & {3.7} \\ {0.5} & {3.5} & {7.2} & {3.3} \\ {1.0} & {4.5} & {6.0} & {3.0} \\ {1.5} & {5.9} & {6.4} & {2.8} \\ {2.0} & {7.3} & {7.8} & {2.7} \\\ \hline\end{array}$$

I-2 Find the domain of the vector function. $$\mathbf{r}(t)=\left\langle\sqrt{4-t^{2}}, e^{-3 t}, \ln (t+1)\right\rangle$$

Find the length of the curve. \(\mathbf{r}(t)=\left\langle 2 t, t^{2}, \frac{1}{3} t^{3}\right\rangle, \quad 0 \leqslant t \leqslant 1\)

(a) Make a large sketch of the curve described by the vector function \(\mathbf{r}(t)=\left\langle t^{2}, t\right\rangle, 0 \leqslant t \leqslant 2,\) and draw the vectors \(\mathbf{r}(1), \mathbf{r}(1.1),\) and \(\mathbf{r}(1.1)-\mathbf{r}(1)\) (b) Draw the vector \(\mathbf{r}^{\prime}(1)\) starting at \((1,1)\) and compare it with the vector $$ \frac{\mathbf{r}(1.1)-\mathbf{r}(1)}{0.1} $$ Explain why these vectors are so close to each other in length and direction.

I-2 Find the domain of the vector function. $$\mathbf{r}(t)=\frac{t-2}{t+2} \mathbf{i}+\sin t \mathbf{j}+\ln \left(9-t^{2}\right) \mathbf{k}$$

Find the length of the curve. \(\mathbf{r}(t)=\sqrt{2} t \mathbf{i}+e^{t} \mathbf{j}+e^{-t} \mathbf{k}, \quad 0 \leqslant t \leqslant 1\)

\(3-8\) Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of \(t .\) $$\mathbf{r}(t)=\left\langle-\frac{1}{2} t^{2}, t\right\rangle, \quad t=2$$

(a) Sketch the plane curve with the given vector equation. (b) Find \(\mathbf{r}^{\prime}(t) .\) (c) Sketch the position vector \(\mathbf{r}(t)\) and the tangent vector \(\mathbf{r}^{\prime}(t)\) for the given value of \(t\) . $$ \mathbf{r}(t)=\left\langle t-2, t^{2}+1\right\rangle, \quad t=-1 $$

Find the length of the curve. \(\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+\ln \cos t \mathbf{k}, \quad 0 \leqslant t \leqslant \pi / 4\)

\(3-8\) Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of \(t .\) $$\mathbf{r}(t)=\langle 2-t, 4 \sqrt{t}\rangle, \quad t=1$$

(a) Sketch the plane curve with the given vector equation. (b) Find \(\mathbf{r}^{\prime}(t) .\) (c) Sketch the position vector \(\mathbf{r}(t)\) and the tangent vector \(\mathbf{r}^{\prime}(t)\) for the given value of \(t\) . $$ \mathbf{r}(t)=\langle 1+t, \sqrt{t}\rangle, \quad t=1 $$

\(3-6\) Find the limit. $$ \lim _{t \rightarrow 0}\left\langle\frac{e^{t}-1}{t}, \frac{\sqrt{1+t}-1}{t}, \frac{3}{1+t}\right\rangle $$

Find the length of the curve. \(\mathbf{r}(t)=\mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k}, \quad 0 \leqslant t \leqslant 1\)

\(3-8\) Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of \(t .\) $$\mathbf{r}(t)=3 \cos t \mathbf{i}+2 \sin t \mathbf{j}, \quad t=\pi / 3$$

(a) Sketch the plane curve with the given vector equation. (b) Find \(\mathbf{r}^{\prime}(t) .\) (c) Sketch the position vector \(\mathbf{r}(t)\) and the tangent vector \(\mathbf{r}^{\prime}(t)\) for the given value of \(t\) . $$ \mathbf{r}(t)=\sin t \mathbf{i}+2 \cos t \mathbf{j}, \quad t=\pi / 4 $$

\(3-6\) Find the limit. $$ \lim _{t \rightarrow 0}\left(e^{-3 t} \mathbf{i}+\frac{t^{2}}{\sin ^{2} t} \mathbf{j}+\cos 2 t \mathbf{k}\right) $$

Find the length of the curve. \(\mathbf{r}(t)=12 t \mathbf{i}+8 t^{3 / 2} \mathbf{j}+3 t^{2} \mathbf{k}, \quad 0 \leqslant t \leqslant 1\)

\(3-8\) Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of \(t .\) $$\mathbf{r}(t)=e^{t} \mathbf{i}+e^{2 t} \mathbf{j}, \quad t=0$$

(a) Sketch the plane curve with the given vector equation. (b) Find \(\mathbf{r}^{\prime}(t) .\) (c) Sketch the position vector \(\mathbf{r}(t)\) and the tangent vector \(\mathbf{r}^{\prime}(t)\) for the given value of \(t\) . $$ \mathbf{r}(t)=e^{t} \mathbf{i}+e^{-t} \mathbf{j}, \quad t=0 $$

\(3-6\) Find the limit. $$ \lim _{t \rightarrow \infty}\left\langle\arctan t, e^{-2 t}, \frac{\ln t}{t}\right\rangle $$

Find the length of the curve correct to four decimal places. (Use your calculator to approximate the integral.) \(\mathbf{r}(t)=\left\langle\sqrt{t}, t, t^{2}\right\rangle, \quad 1 \leqslant t \leqslant 4\)

\(3-8\) Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of \(t .\) $$\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+2 \mathbf{k}, \quad t=1$$

\(7-14\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=\langle\sin t, t\rangle $$

(a) Sketch the plane curve with the given vector equation. (b) Find \(\mathbf{r}^{\prime}(t) .\) (c) Sketch the position vector \(\mathbf{r}(t)\) and the tangent vector \(\mathbf{r}^{\prime}(t)\) for the given value of \(t\) . $$ \mathbf{r}(t)=e^{t} \mathbf{i}+e^{3 t} \mathbf{j}, \quad t=0 $$

Find the length of the curve correct to four decimal places. (Use your calculator to approximate the integral.) \(\mathbf{r}(t)=\langle t, \ln t, t \ln t\rangle, \quad 1 \leqslant t \leqslant 2\)

\(3-8\) Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of \(t .\) $$\mathbf{r}(t)=t \mathbf{i}+2 \cos t \mathbf{j}+\sin t \mathbf{k}, \quad t=0$$

(a) Sketch the plane curve with the given vector equation. (b) Find \(\mathbf{r}^{\prime}(t) .\) (c) Sketch the position vector \(\mathbf{r}(t)\) and the tangent vector \(\mathbf{r}^{\prime}(t)\) for the given value of \(t\) . $$ \mathbf{r}(t)=(1+\cos t) \mathbf{i}+(2+\sin t) \mathbf{j}, \quad t=\pi / 6 $$

\(7-14\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=\left\langle t^{3}, t^{2}\right\rangle $$

Find the length of the curve correct to four decimal places. (Use your calculator to approximate the integral.) \(\mathbf{r}(t)=\langle\sin t, \cos t, \tan t\rangle, \quad 0 \leqslant t \leqslant \pi / 4\)

\(9-14\) Find the velocity, acceleration, and speed of a particle with the given position function. $$\mathbf{r}(t)=\left\langle t^{2}+1, t^{3}, t^{2}-1\right\rangle$$

Find the derivative of the vector function. $$ \mathbf{r}(t)=\left\langle t \sin t, t^{2}, t \cos 2 t\right\rangle $$

\(7-14\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=\langle t, \cos 2 t, \sin 2 t\rangle $$

Graph the curve with parametric equations \(x=\sin t\) \(y=\sin 2 t, z=\sin 3 t\) . Find the total length of this curve correct to four decimal places.

\(9-14\) Find the velocity, acceleration, and speed of a particle with the given position function. $$\mathbf{r}(t)=\langle 2 \cos t, 3 t, 2 \sin t\rangle$$

Find the derivative of the vector function. $$ \mathbf{r}(t)=\left\langle\tan t, \sec t, 1 / t^{2}\right\rangle $$

\(7-14\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=\langle 1+t, 3 t,-t\rangle $$

Let \(C\) be the curve of intersection of the parabolic cylinder \(x^{2}=2 y\) and the surface \(3 z=x y .\) Find the exact length of \(C\) from the origin to the point \((6,18,36) .\)

\(9-14\) Find the velocity, acceleration, and speed of a particle with the given position function. $$\mathbf{r}(t)=\sqrt{2} t \mathbf{i}+e^{t} \mathbf{j}+e^{-t} \mathbf{k}$$

Find the derivative of the vector function. $$ \mathbf{r}(t)=\mathbf{i}-\mathbf{j}+e^{4 t} \mathbf{k} $$

\(7-14\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=\langle 1, \cos t, 2 \sin t\rangle $$

Find, correct to four decimal places, the length of the curve of intersection of the cylinder \(4 x^{2}+y^{2}=4\) and the plane \(x+y+z=2.\)

\(9-14\) Find the velocity, acceleration, and speed of a particle with the given position function. $$\mathbf{r}(t)=t^{2} \mathbf{i}+\ln t \mathbf{j}+t \mathbf{k}$$

Find the derivative of the vector function. $$ \mathbf{r}(t)=\sin ^{-1} t \mathbf{i}+\sqrt{1-t^{2}} \mathbf{j}+\mathbf{k} $$

\(7-14\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=t^{2} \mathbf{i}+t \mathbf{j}+2 \mathbf{k} $$

Reparametrize the curve with respect to arc length measured from the point where \(t=0\) in the direction of increasing \(t.\) \(\mathbf{r}(t)=2 t \mathbf{i}+(1-3 t) \mathbf{j}+(5+4 t) \mathbf{k}\)

\(9-14\) Find the velocity, acceleration, and speed of a particle with the given position function. $$\mathbf{r}(t)=e^{t}(\cos t \mathbf{i}+\sin t \mathbf{j}+t \mathbf{k})$$

Find the derivative of the vector function. $$ \mathbf{r}(t)=e^{t^{\prime}} \mathbf{i}-\mathbf{j}+\ln (1+3 t) \mathbf{k} $$

\(7-14\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=t^{2} \mathbf{i}+t^{4} \mathbf{j}+t^{6} \mathbf{k} $$

Reparametrize the curve with respect to arc length measured from the point where \(t=0\) in the direction of increasing \(t.\) \(\mathbf{r}(t)=e^{2 t} \cos 2 t \mathbf{i}+2 \mathbf{j}+e^{2 t} \sin 2 t \mathbf{k}\)