Chapter 13

In Exercises \(1-7,\) find the velocity and acceleration vectors in terms of \(\mathbf{u}_{r}\) and \(\mathbf{u}_{\theta} .\) $$ r=\theta \quad \text { and } \quad \frac{d \theta}{d t}=2 $$

Evaluate the integrals in Exercises \(1-10\) $$ \int_{0}^{1}\left[t^{3} \mathbf{i}+7 \mathbf{j}+(t+1) \mathbf{k}\right] d t $$

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the plane curves in Exercises \(1-4\) $$ \mathbf{r}(t)=t \mathbf{i}+(\ln \cos t) \mathbf{j}, \quad-\pi / 2 < t < \pi / 2 $$

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(2 \cos t) \mathbf{i}+(2 \sin t) \mathbf{j}+\sqrt{5} t \mathbf{k}, \quad 0 \leq t \leq \pi $$

In Exercises \(1-4,\) find the given limits. $$\lim _{t \rightarrow \pi}\left[\left(\sin \frac{t}{2}\right) \mathbf{i}+\left(\cos \frac{2}{3} t\right) \mathbf{j}+\left(\tan \frac{5}{4} t\right) \mathbf{k}\right]$$

In Exercises \(1-7,\) find the velocity and acceleration vectors in terms of \(\mathbf{u}_{r}\) and \(\mathbf{u}_{\theta} .\) $$ r=\frac{1}{\theta} \quad \text { and } \quad \frac{d \theta}{d t}=t^{2} $$

Evaluate the integrals in Exercises \(1-10\) $$ \int_{1}^{2}\left[(6-6 t) \mathbf{i}+3 \sqrt{t} \mathbf{j}+\left(\frac{4}{t^{2}}\right) \mathbf{k}\right] d t $$

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the plane curves in Exercises \(1-4\) $$ \mathbf{r}(t)=(\ln \sec t) \mathbf{i}+t \mathbf{j}, \quad-\pi / 2 < t <\pi / 2 $$

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(6 \sin 2 t) \mathbf{i}+(6 \cos 2 t) \mathbf{j}+5 t \mathbf{k}, \quad 0 \leq t \leq \pi $$

In Exercises \(1-4,\) find the given limits. $$ \lim _{t \rightarrow-1}\left[t^{3} \mathbf{i}+\left(\sin \frac{\pi}{2} t\right) \mathbf{j}+(\ln (t+2)) \mathbf{k}\right] $$

In Exercises \(1-7,\) find the velocity and acceleration vectors in terms of \(\mathbf{u}_{r}\) and \(\mathbf{u}_{\theta} .\) $$ r=a(1-\cos \theta) \quad \text { and } \quad \frac{d \theta}{d t}=3 $$

Evaluate the integrals in Exercises \(1-10\) $$ \int_{-\pi / 4}^{\pi / 4}\left[(\sin t) \mathbf{i}+(1+\cos t) \mathbf{j}+\left(\sec ^{2} t\right) \mathbf{k}\right] d t $$

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=t \mathbf{i}+(2 / 3) t^{3 / 2} \mathbf{k}, \quad 0 \leq t \leq 8 $$

In Exercises \(1-4,\) find the given limits. $$ \lim _{t \rightarrow 1}\left[\left(\frac{t^{2}-1}{\ln t}\right) \mathbf{i}-\left(\frac{\sqrt{t}-1}{1-t}\right) \mathbf{j}+\left(\tan ^{-1} t\right) \mathbf{k}\right] $$

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the plane curves in Exercises \(1-4\) $$ \mathbf{r}(t)=(2 t+3) \mathbf{i}+\left(5-t^{2}\right) \mathbf{j} $$

Write a in the form \(\mathbf{a}=a_{\mathrm{T}} \mathbf{T}+a_{\mathrm{N}} \mathbf{N}\) at the given value of \(t\) without finding \(\mathbf{T}\) and \(\mathbf{N} .\) \(\mathbf{r}(t)=(t+1) \mathbf{i}+2 t \mathbf{j}+t^{2} \mathbf{k}, \quad t=1\)

In Exercises \(1-7,\) find the velocity and acceleration vectors in terms of \(\mathbf{u}_{r}\) and \(\mathbf{u}_{\theta} .\) $$ r=a \sin 2 \theta \quad \text { and } \quad \frac{d \theta}{d t}=2 t $$

Evaluate the integrals in Exercises \(1-10\) $$\int_{0}^{\pi / 3}[(\sec t \tan t) \mathbf{i}+(\tan t) \mathbf{j}+(2 \sin t \cos t) \mathbf{k}] d t$$

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the plane curves in Exercises \(1-4\) $$ \mathbf{r}(t)=(\cos t+t \sin t) \mathbf{i}+(\sin t-t \cos t) \mathbf{j}, \quad t>0 $$

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(2+t) \mathbf{i}-(t+1) \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 3 $$

In Exercises \(1-4,\) find the given limits. $$ \lim _{t \rightarrow 0}\left[\left(\frac{\sin t}{t}\right) \mathbf{i}+\left(\frac{\tan ^{2} t}{\sin 2 t}\right) \mathbf{j}-\left(\frac{t^{3}-8}{t+2}\right) \mathbf{k}\right] $$

In Exercises \(1-7,\) find the velocity and acceleration vectors in terms of \(\mathbf{u}_{r}\) and \(\mathbf{u}_{\theta} .\) $$ r=e^{a \theta} \quad \text { and } \quad \frac{d \theta}{d t}=2 $$

Evaluate the integrals in Exercises \(1-10\) $$ \int_{0}^{\pi / 3}[(\sec t \tan t) \mathbf{i}+(\tan t) \mathbf{j}+(2 \sin t \cos t) \mathbf{k}] d t $$

Write a in the form \(\mathbf{a}=a_{\mathrm{T}} \mathbf{T}+a_{\mathrm{N}} \mathbf{N}\) at the given value of \(t\) without finding \(\mathbf{T}\) and \(\mathbf{N} .\) \(\mathbf{r}(t)=t^{2} \mathbf{i}+\left(t+(1 / 3) t^{3}\right) \mathbf{j}+\left(t-(1 / 3) t^{3}\right) \mathbf{k}, \quad t=0\)

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=\left(\cos ^{3} t\right) \mathbf{j}+\left(\sin ^{3} t\right) \mathbf{k}, \quad 0 \leq t \leq \pi / 2 $$

In Exercises \(5-8, \mathbf{r}(t)\) is the position of a particle in the \(x y\) -plane at time \(t .\) Find an equation in \(x\) and \(y\) whose graph is the path of the par- ticle. Then find the particle's velocity and acceleration vectors at the given value of \(t\) . $$ \mathbf{r}(t)=(t+1) \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}, \quad t=1 $$

In Exercises \(1-7,\) find the velocity and acceleration vectors in terms of \(\mathbf{u}_{r}\) and \(\mathbf{u}_{\theta} .\) $$ r=a(1+\sin t) \quad \text { and } \quad \theta=1-e^{-t} $$

Evaluate the integrals in Exercises \(1-10 .\) \(\int_{0}^{1}\left[\frac{2}{\sqrt{1-t^{2}}} \mathbf{i}+\frac{\sqrt{3}}{1+t^{2}} \mathbf{k}\right] d t\)

A formula for the curvature of a parametrized plane curve $$ \begin{array}{c}{\text { a. Show that the curvature of a smooth curve } \mathbf{r}(t)=f(t) \mathbf{i}+} \\ {g(t) \mathbf{j} \text { defined by twice- differentiable functions } x=f(t) \text { and }} \\ {y=g(t) \text { is given by the formula }} \\ {\kappa=\frac{|\ddot{x} \ddot{y}-\dot{y} \ddot{x}|}{\left(\dot{x}^{2}+\dot{y}^{2}\right)^{3 / 2}}}\end{array} $$The dots in the formula denote differentiation with respect to \(t,\) one derivative for each dot. Apply the formula to find the curvatures of the following curves. $$ \begin{array}{l}{\text { b. } \mathbf{r}(t)=t \mathbf{i}+(\ln \sin t) \mathbf{j}, \quad 0 < t<\pi} \\ {\text { c. } \mathbf{r}(t)=\left[\tan ^{-1}(\sinh t)\right] \mathbf{i}+(\ln \cosh t) \mathbf{j}}\end{array} $$

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=6 t^{3} \mathbf{i}-2 t^{3} \mathbf{j}-3 t^{3} \mathbf{k}, \quad 1 \leq t \leq 2 $$

In Exercises \(5-8, \mathbf{r}(t)\) is the position of a particle in the \(x y\) -plane at time \(t .\) Find an equation in \(x\) and \(y\) whose graph is the path of the par- ticle. Then find the particle's velocity and acceleration vectors at the given value of \(t\) . $$ \mathbf{r}(t)=\frac{t}{t+1} \mathbf{i}+\frac{1}{t} \mathbf{j}, \quad t=-\frac{1}{2} $$

In Exercises \(1-7,\) find the velocity and acceleration vectors in terms of \(\mathbf{u}_{r}\) and \(\mathbf{u}_{\theta} .\) $$ r=2 \cos 4 t \quad \text { and } \quad \theta=2 t $$

Evaluate the integrals in Exercises \(1-10\) $$ \int_{0}^{1}\left[\frac{2}{\sqrt{1-t^{2}}} \mathbf{i}+\frac{\sqrt{3}}{1+t^{2}} \mathbf{k}\right] d t $$

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(t \cos t) \mathbf{i}+(t \sin t) \mathbf{j}+(2 \sqrt{2} / 3) t^{3 / 2} \mathbf{k}, \quad 0 \leq t \leq \pi $$

In Exercises \(5-8, \mathbf{r}(t)\) is the position of a particle in the \(x y\) -plane at time \(t .\) Find an equation in \(x\) and \(y\) whose graph is the path of the par- ticle. Then find the particle's velocity and acceleration vectors at the given value of \(t\) . $$ \mathbf{r}(t)=e^{t} \mathbf{i}+\frac{2}{9} e^{2 t} \mathbf{j}, \quad t=\ln 3 $$

Find \(\mathbf{r}, \mathbf{T}, \mathbf{N},\) and \(\mathbf{B}\) at the given value of \(t\). Then find equations for the osculating, normal, and rectifying planes at that value of \(t\). \(\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}-\mathbf{k}, \quad t=\pi / 4\)

Evaluate the integrals in Exercises \(1-10\) $$ \int_{1}^{\ln 3}\left[t e^{t} \mathbf{i}+e^{t} \mathbf{j}+\ln t \mathbf{k}\right] d t $$

Find \(\mathbf{r}, \mathbf{T}, \mathbf{N},\) and \(\mathbf{B}\) at the given value of \(t\). Then find equations for the osculating, normal, and rectifying planes at that value of \(t\). \(\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+t \mathbf{k}, \quad t=0\)

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(t \sin t+\cos t) \mathbf{i}+(t \cos t-\sin t) \mathbf{j}, \quad \sqrt{2} \leq t \leq 2 $$

In Exercises \(5-8, \mathbf{r}(t)\) is the position of a particle in the \(x y\) -plane at time \(t .\) Find an equation in \(x\) and \(y\) whose graph is the path of the par- ticle. Then find the particle's velocity and acceleration vectors at the given value of \(t\) . $$ \mathbf{r}(t)=(\cos 2 t) \mathbf{i}+(3 \sin 2 t) \mathbf{j}, \quad t=0 $$

Circular orbits Show that a planet in a circular orbit moves with a constant speed. (Hint: This is a consequence of one of Kepler's laws.)

Evaluate the integrals in Exercises \(1-10\) $$ \int_{0}^{\pi / 2}\left[\cos t \mathbf{i}-\sin 2 t \mathbf{j}+\sin ^{2} t \mathbf{k}\right] d t $$

Find \(\mathbf{B}\) and \(\tau\) for these space curves. \(\mathbf{r}(t)=(3 \sin t) \mathbf{i}+(3 \cos t) \mathbf{j}+4 t \mathbf{k}\)

Find the point on the curve $$ \mathbf{r}(t)=(5 \sin t) \mathbf{i}+(5 \cos t) \mathbf{j}+12 t \mathbf{k} $$ at a distance 26\(\pi\) units along the curve from the point \((0,5,0)\) in the direction of increasing arc length.

Exercises \(9-12\) give the position vectors of particles moving along various curves in the \(x y-\) plane. In each case, find the particle's velocity and acceleration vectors at the stated times and sketch them as vectors on the curve. Motion on the circle \(x^{2}+y^{2}=1\) $$\mathbf{r}(t)=(\sin t) \mathbf{i}+(\cos t) \mathbf{j} ; \quad t=\pi / 4\( and \)\pi / 2$$

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the space curves in Exercises \(9-16\) $$ \mathbf{r}(t)=(3 \sin t) \mathbf{i}+(3 \cos t) \mathbf{j}+4 t \mathbf{k} $$

Suppose that \(\mathbf{r}\) is the position vector of a particle moving along a plane curve and \(d A / d t\) is the rate at which the vector sweeps out area. Without introducing coordinates, and assuming the necessary derivatives exist, give a geometric argument based on increments and limits for the validity of the equation $$ \frac{d A}{d t}=\frac{1}{2}|\mathbf{r} \times \dot{\mathbf{r}}| $$

Evaluate the integrals in Exercises \(1-10\) $$ \int_{0}^{\pi / 4}\left[\sec t \mathbf{i}+\tan ^{2} t \mathbf{j}-t \sin t \mathbf{k}\right] d t $$

Find \(\mathbf{B}\) and \(\tau\) for these space curves. \(\mathbf{r}(t)=(\cos t+t \sin t) \mathbf{i}+(\sin t-t \cos t) \mathbf{j}+3 \mathbf{k}\)

Find the point on the curve $$ \mathbf{r}(t)=(12 \sin t) \mathbf{i}-(12 \cos t) \mathbf{j}+5 t \mathbf{k} $$ at a distance 13\(\pi\) units along the curve from the point \((0,-12,0)\) in the direction opposite to the direction of increasing arc length.