Chapter 2

Given \(\mathbf{r}(t)=\left(3 t^{2}-2\right) \mathbf{i}+(2 t-\sin (t)) \mathbf{j}\), find the velocity of a particle moving along this curve.

Compute the derivatives of the vector-valued functions.\(\mathbf{r}(t)=t^{3} \mathbf{i}+3 t^{2} \mathbf{j}+\frac{t^{3}}{6} \mathbf{k}\)

Finding the Arc Length Calculate the arc length for each of the following vector-valued functions: $$ \begin{aligned} &\text { a. } \mathbf{r}(t)=(3 t-2) \mathbf{i}+(4 t+5) \mathbf{j}, 1 \leq t \leq 5 \\ &\text { b. } \mathbf{r}(t)=\langle t \cos t, t \sin t, 2 t\rangle, 0 \leq t \leq 2 \pi \end{aligned} $$

Give the component functions \(x=f(t)\) and \(y=g(t)\) for the vector-valued function \(\mathbf{r}(t)=3 \sec t \mathbf{i}+2 \tan t \mathbf{j}\)

Given \(\mathbf{r}(t)=\left(3 t^{2}-2\right) \mathbf{i}+(2 t-\sin (t)) \mathbf{j}\), find the acceleration vector of a particle moving along the curve in the preceding exercise.

Compute the derivatives of the vector-valued functions.\(\mathbf{r}(t)=\sin (t) \mathbf{i}+\cos (t) \mathbf{j}+e^{t} \mathbf{k}\)

Calculate the arc length of the parameterized curve $$ \mathbf{r}(t)=\left\langle 2 t^{2}+1,2 t^{2}-1, t^{3}\right\rangle, 0 \leq t \leq 3 $$

Given \(\mathbf{r}(t)=3 \sec t \mathbf{i}+2 \tan t \mathbf{j}\), find the following values (if possible). a. \(\mathbf{r}\left(\frac{\pi}{4}\right)\) b. \(\mathbf{r}(\pi)\) c. r \(\left(\frac{\pi}{2}\right)\)

Given the following position functions, find the velocity, acceleration, and speed in terms of the parameter t. $$ \mathbf{r}(t)=\left\langle 3 \cos t, 3 \sin t, t^{2}\right\rangle $$

Compute the derivatives of the vector-valued functions.\(\mathbf{r}(t)=e^{-t} \mathbf{i}+\sin (3 t) \mathbf{j}+10 \sqrt{t} \mathbf{k}\). A sketch of the graph is shown here. Notice the varying periodic nature of the graph.

Finding an Arc-Length Parameterization Find the arc-length parameterization for each of the following curves: a. \(\mathbf{r}(t)=4 \cos t \mathbf{i}+4 \sin t \mathbf{j}, t \geq 0\) b. \(\mathbf{r}(t)=\langle t+3,2 t-4,2 t\rangle, t \geq 3\)

Given the following position functions, find the velocity, acceleration, and speed in terms of the parameter t. $$ \mathbf{r}(t)=e^{-t} \mathbf{i}+t^{2} \mathbf{j}+\tan t \mathbf{k} $$

Compute the derivatives of the vector-valued functions.\(\mathbf{r}(t)=e^{t} \mathbf{i}+2 e^{t} \mathbf{j}+\mathbf{k}\)

Find the arc-length function for the helix $$ \mathbf{r}(t)=\langle 3 \cos t, 3 \sin t, 4 t\rangle, t \geq 0 $$ Then, use the relationship between the arc length and the parameter \(t\) to find an arc-length parameterization of \(\mathbf{r}(t)\).

Evaluate \(\lim _{t \rightarrow 0}\left\langle e^{t} \mathbf{i}+\frac{\sin t}{t} \mathbf{j}+e^{-t} \mathbf{k}\right\rangle\).

Given the following position functions, find the velocity, acceleration, and speed in terms of the parameter t. $$ \mathbf{r}(t)=2 \cos t \mathbf{j}+3 \sin t \mathbf{k} \text { . The graph is shown here: } $$

Compute the derivatives of the vector-valued functions.\(\mathbf{r}(t)=\mathbf{i}+\mathbf{j}+\mathbf{k}\)

Finding Curvature Find the curvature for each of the following curves at the given point a. \(\mathbf{r}(t)=4 \cos t \mathbf{i}+4 \sin t \mathbf{j}+3 t \mathbf{k}, t=\frac{4 \pi}{3}\) b. \(f(x)=\sqrt{4 x-x^{2}}, x=2\)

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

Compute the derivatives of the vector-valued functions.\(\mathbf{r}(t)=t e^{t} \mathbf{i}+t \ln (t) \mathbf{j}+\sin (3 t) \mathbf{k}\)

Find the curvature of the curve defined by the function $$ y=3 x^{2}-2 x+4 $$ at the point \(x=2\).

Given the vector-valued function \(\mathbf{r}(t)=\left\langle t, t^{2}+1\right\rangle\), find the following values: a. \(\lim _{t \rightarrow-3} \mathbf{r}(t)\) b. \(\mathbf{r}(-3)\) c. Is \(\mathbf{r}(t)\) continuous at \(x=-3 ?\) d. \(\mathbf{r}(t+2)-\mathbf{r}(t)\)

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

Finding the Principal Unit Normal Vector and Binormal Vector For each of the following vector-valued functions, find the principal unit normal vector. Then, if possible, find the binormal vector. $$ \begin{aligned} &\text { a. } \mathbf{r}(t)=4 \cos t \mathbf{i}-4 \sin t \mathbf{j} \\ &\text { b. } \mathbf{r}(t)=(6 t+2) \mathbf{i}+5 t^{2} \mathbf{j}-8 t \mathbf{k} \end{aligned} $$

Find the velocity, acceleration, and speed of a particle with the given position function. $$ \mathbf{r}(t)=\langle\sin t, t, \cos t\rangle . \text { The graph is shown here: } $$

Compute the derivatives of the vector-valued functions.\(\mathbf{r}(t)=\tan (2 t) \mathbf{i}+\sec (2 t) \mathbf{j}+\sin ^{2}(t) \mathbf{k}\)

Find the unit normal vector for the vector-valued function \(\mathbf{r}(t)=\left(t^{2}-3 t\right) \mathbf{i}+(4 t+1) \mathbf{j}\) and evaluate it at \(t=2\).

Find the limit of the following vector-valued functions at the indicated value of \(t\). $$ \lim _{t \rightarrow 4}\left\langle\sqrt{t-3}, \frac{\sqrt{t}-2}{t-4}, \tan \left(\frac{\pi}{t}\right)\right\rangle $$

The position function of an object is given by \(\mathbf{r}(t)=\left\langle t^{2}, 5 t, t^{2}-16 t\right\rangle .\) At what time is the speed a minimum?

Compute the derivatives of the vector-valued functions.\(\mathbf{r}(t)=3 \mathbf{i}+4 \sin (3 t) \mathbf{j}+t \cos (t) \mathbf{k}\)

Find the limit of the following vector-valued functions at the indicated value of \(t\). $$ \lim _{t \rightarrow \pi / 2} \mathbf{r}(t) \text { for } \mathbf{r}(t)=e^{t} \mathbf{i}+\sin t \mathbf{j}+\ln t \mathbf{k} $$

Let \(\mathbf{r}(t)=r \cosh (\omega t) \mathbf{i}+r \sinh (\omega t) \mathbf{j}\). Find the velocity and acceleration vectors and show that the acceleration is proportional to \(\mathbf{r}(t)\).

Compute the derivatives of the vector-valued functions.\(\mathbf{r}(t)=t^{2} \mathbf{i}+t e^{-2 t} \mathbf{j}-5 e^{-4 t} \mathbf{k}\)

Find the limit of the following vector-valued functions at the indicated value of \(t\). $$ \lim _{t \rightarrow \infty}\left\langle e^{-2 t}, \frac{2 t+3}{3 t-1}, \arctan (2 t)\right\rangle $$

For the following problems, find a tangent vector at the indicated value of \(t$$\mathbf{r}(t)=t \mathbf{i}+\sin (2 t) \mathbf{j}+\cos (3 t) \mathbf{k} ; t=\frac{\pi}{3}\)

Find the arc length of the curve on the given interval.\(\mathbf{r}(t)=t^{2} \mathbf{i}+14 t \mathbf{j}, 0 \leq t \leq 7\). This portion of the graph is shown here:

Find the limit of the following vector-valued functions at the indicated value of \(t\). $$ \lim _{t \rightarrow e^{2}}\left\langle t \ln (t), \frac{\ln t}{t^{2}}, \sqrt{\ln \left(t^{2}\right)}\right\rangle $$

For the following problems, find a tangent vector at the indicated value of \(t$$\mathbf{r}(t)=3 t^{3} \mathbf{i}+2 t^{2} \mathbf{j}+\frac{1}{t} \mathbf{k} ; t=1\)

Find the arc length of the curve on the given interval.\(\mathbf{r}(t)=t^{2} \mathbf{i}+\left(2 t^{2}+1\right) \mathbf{j}, 1 \leq t \leq 3\)

Find the limit of the following vector-valued functions at the indicated value of \(t\). $$ \lim _{t \rightarrow \pi / 6}\left\langle\cos ^{2} t, \sin ^{2} t, 1\right\rangle $$

A person on a hang glider is spiraling upward as a result of the rapidly rising air on a path having position vector \(\mathbf{r}(t)=(3 \cos t) \mathbf{i}+(3 \sin t) \mathbf{j}+t^{2} \mathbf{k}\). The path is similar to that of a helix, although it is not a helix. The graph is shown here:

For the following problems, find a tangent vector at the indicated value of \(t$$\mathbf{r}(t)=3 e^{t} \mathbf{i}+2 e^{-3 t} \mathbf{j}+4 e^{2 t} \mathbf{k} ; t=\ln (2)\)

Find the arc length of the curve on the given interval.\(\mathbf{r}(t)=\langle 2 \sin t, 5 t, 2 \cos t\rangle, 0 \leq t \leq \pi .\) This portion of the graph is shown here:

Find the limit of the following vector-valued functions at the indicated value of \(t\). $$ \lim _{t \rightarrow \infty} \mathbf{r}(t) \text { for } \mathbf{r}(t)=2 e^{-t} \mathbf{i}+e^{-t} \mathbf{j}+\ln (t-1) \mathbf{k} $$

For the following problems, find a tangent vector at the indicated value of \(t$$\mathbf{r}(t)=\cos (2 t) \mathbf{i}+2 \sin t \mathbf{j}+t^{2} \mathbf{k} ; t=\frac{\pi}{2}\)

Find the arc length of the curve on the given interval.\(\mathbf{r}(t)=\left\langle t^{2}+1,4 t^{3}+3\right\rangle,-1 \leq t \leq 0\)

Describe the curve defined by the vector-valued function \(\mathbf{r}(t)=(1+t) \mathbf{i}+(2+5 t) \mathbf{j}+(-1+6 t) \mathbf{k}\).

Find the unit tangent vector for the following parameterized curves.\(\mathbf{r}(t)=6 \mathbf{i}+\cos (3 t) \mathbf{j}+3 \sin (4 t) \mathbf{k}, 0 \leq t<2 \pi\)

Find the arc length of the curve on the given interval.\(\mathbf{r}(t)=\left\langle e^{-t} \cos t, e^{-t} \sin t\right\rangle\) over the interval \(\left[0, \frac{\pi}{2}\right] .\) Here is the portion of the graph on the indicated interval:

Find the domain of the vector-valued functions. $$ \text { Domain: } \mathbf{r}(t)=\left\langle t^{2}, \tan t, \ln t\right\rangle $$