Chapter 3

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

Sketch the curve of the vector-valued function \(\mathbf{r}(t)=3 \sec t \mathbf{i}+2 \tan t \mathbf{j}\) and give the orientation of the curve. Sketch asymptotes as a guide to the graph.

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 vector-valued function \(\mathbf{r}(t)=\langle\cos t, \sin t\rangle,\) find the following values: a. \(\lim _{t \rightarrow \frac{\pi}{4}} \mathbf{r}(t)\) b. \(\quad \mathbf{r}\left(\frac{\pi}{3}\right)\) c. Is \(\mathbf{r}(t)\) continuous at \(t=\frac{\pi}{3} ?\) d. Graph \(\mathbf{r}(t)\).

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

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

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

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

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

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

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

Find the domain of the vector-valued functions. Domain: \(\mathbf{r}(t)=\left\langle t^{2}, \sqrt{t-3}, \frac{3}{2 t+1}\right\rangle\)

Find the domain of the vector-valued functions. Domain: \(\mathbf{r}(t)=\left\langle\csc (t), \frac{1}{\sqrt{t-3}}, \ln (t-2)\right\rangle\)

Let \(\mathbf{r}(t)=\langle\cos t, t, \sin t\rangle\) and use it to answer the following questions. For what values of \(t\) is \(\mathbf{r}(t)\) continuous?

Let \(\mathbf{r}(t)=\langle\cos t, t, \sin t\rangle\) and use it to answer the following questions. Sketch the graph of \(\mathbf{r}(t)\).

Find the domain of \(\mathbf{r}(t)=2 e^{-t} \mathbf{i}+e^{-t} \mathbf{j}+\ln (t-1) \mathbf{k}\) .

Eliminate the parameter \(t\), write the equation in Cartesian coordinates, then sketch the graphs of the vector-valued functions. \(\mathbf{r}(t)=2 t \mathbf{i}+t^{2} \mathbf{j}\)

Use a graphing utility to sketch each of the following vector-valued functions: \(\mathbf{r}(t)=2 \cos t^{2} \mathbf{i}+(2-\sqrt{t}) \mathbf{j}\)

Use a graphing utility to sketch each of the following vector-valued functions: \([\mathrm{T}] \mathbf{r}(t)=\left\langle e^{\cos (3 t)}, e^{-\sin (t)}\right\rangle\)

Use a graphing utility to sketch each of the following vector-valued functions: \(\mathbf{r}(t)=\langle 2-\sin (2 t), 3+2 \cos t\rangle\)

Use a graphing utility to sketch each of the following vector-valued functions: \(\mathbf{r}(t)=\left\langle t, t^{2}\right\rangle ;\) from left to right

Use a graphing utility to sketch each of the following vector-valued functions: The line through \(P\) and \(Q\) where \(P\) is (1,4,-2) and \(Q\) is (3,9,6)

Consider the curve described by the vector-valued function $$\mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k}$$ What is the initial point of the path corresponding to \(\mathbf{r}(0) ?\)

Consider the curve described by the vector-valued function $$\mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k}$$ What is \(\lim _{t \rightarrow \infty} \mathbf{r}(t) ?\)

Consider the curve described by the vector-valued function $$\mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k}$$ Use technology to sketch the curve.

Consider the curve described by the vector-valued function $$\mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k}$$ Eliminate the parameter \(t\) to show that \(z=5-\frac{r}{10}\) where \(r^{2}=x^{2}+y^{2}\)

Let \(r(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+0.3 \sin (2 t) \mathbf{k}\). Use technology to graph the curve (called the roller-coaster curve) over the interval \([0,2 \pi)\). Choose at least two views to determine the peaks and valleys.

a. Graph the curve \(\mathbf{r}(t)=(4+\cos (18 t)) \cos (t) \mathbf{i}+(4+\cos (18 t) \sin (t)) \mathbf{j}+0.3 \sin (18 t) \mathbf{k}\) using two viewing angles of your choice to see the overall shape of the curve. b. Does the curve resemble a "slinky"? c. What changes to the equation should be made to increase the number of coils of the slinky?

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

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

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.

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

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

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

Compute the derivatives of the vector-valued functions. \(\mathbf{r}(t)=\frac{1}{t+1} \mathbf{i}+\arctan (t) \mathbf{j}+\ln t^{3} \mathbf{k}\)

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

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

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 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 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 a tangent vector at the indicated value of \(t\). \(\mathbf{r}(t)=3 e^{t} \mathbf{i}+2 e^{-3 t} \mathbf{j}+4 e^{2 t} \mathbf{k} ; \quad t=\ln (2)\)

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

Find the unit tangent vector for the following parameterized curves. \(\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+\sin t \mathbf{k}, \quad 0 \leq t<2 \pi\). Two views of this curve are presented here:

Find the unit tangent vector for the following parameterized curves. \(\mathbf{r}(t)=3 \cos (4 t) \mathbf{i}+3 \sin (4 t) \mathbf{j}+5 t \mathbf{k}, 1 \leq t \leq 2\)

Find the unit tangent vector for the following parameterized curves. \(\mathbf{r}(t)=t \mathbf{i}+3 t \mathbf{j}+t^{2} \mathbf{k}\)

Let \(\quad \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}-t^{4} \mathbf{k} \quad\) and \(s(t)=\sin (t) \mathbf{i}+e^{t} \mathbf{j}+\cos (t) \mathbf{k}\). Here is the graph of the function: Find the following. \(\frac{d}{d t}\left[r\left(t^{2}\right)\right]\)

Let \(\quad \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}-t^{4} \mathbf{k} \quad\) and \(s(t)=\sin (t) \mathbf{i}+e^{t} \mathbf{j}+\cos (t) \mathbf{k}\). Here is the graph of the function: Find the following. \(\frac{d}{d t}[r(t) \cdot s(t)]\)