Chapter 13

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

Find the derivative of the vector function. $$ \mathbf{r}(t)=a t \cos 3 t \mathbf{i}+b \sin ^{3} t \mathbf{j}+c \cos ^{3} t \mathbf{k} $$

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

Suppose you start at the point \((0,0,3)\) and move 5 units along the curve \(x=3 \sin t, y=4 t, z=3 \cos t\) in the positive direction. Where are you now?

\(15-16\) Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position. $$\mathbf{a}(t)=\mathbf{i}+2 \mathbf{j}, \quad \mathbf{v}(0)=\mathbf{k}, \quad \mathbf{r}(0)=\mathbf{i}$$

Find the derivative of the vector function. $$ \mathbf{r}(t)=\mathbf{a}+t \mathbf{b}+t^{2} \mathbf{c} $$

\(15-18\) Find a vector equation and parametric equations for the line segment that joins \(P\) to \(Q .\) $$ P(0,0,0), \quad Q(1,2,3) $$

Reparametrize the curve $$\mathbf{r}(t)=\left(\frac{2}{t^{2}+1}-1\right) \mathbf{i}+\frac{2 t}{t^{2}+1} \mathbf{j}$$ with respect to arc length measured from the point \((1,0)\) in the direction of increasing \(t\) . Express the reparametrization in its simplest form. What can you conclude about the curve?

15-16 Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position. $$\mathbf{a}(t)=2 \mathbf{i}+6 t \mathbf{j}+12 t^{2} \mathbf{k}, \quad \mathbf{v}(0)=\mathbf{i}, \quad \mathbf{r}(0)=\mathbf{j}-\mathbf{k}$$

Find the derivative of the vector function. $$ \mathbf{r}(t)=t \mathbf{a} \times(\mathbf{b}+t \mathbf{c}) $$

\(15-18\) Find a vector equation and parametric equations for the line segment that joins \(P\) to \(Q .\) $$ P(1,0,1), \quad Q(2,3,1) $$

(a) Find the unit tangent and unit normal vectors \(\mathbf{T}(t)\) and \(\mathbf{N}(t)\) (b) Use Formula 9 to find the curvature. \(\mathbf{r}(t)=\langle 2 \sin t, 5 t, 2 \cos t\rangle\)

\(17-18\) (a) Find the position vector of a particle that has the given acceleration and the specified initial velocity and position. (b) Use a computer to graph the path of the particle. $$\mathbf{a}(t)=2 t \mathbf{i}+\sin t \mathbf{j}+\cos 2 t \mathbf{k}, \quad \mathbf{v}(0)=\mathbf{i}, \quad \mathbf{r}(0)=\mathbf{j}$$

Find the unit tangent vector \(\mathbf{T}(t)\) at the point with the given value of the parameter \(t .\) $$ \mathbf{r}(t)=\left\langle t e^{-t}, 2 \text { arctan } t, 2 e^{t}\right\rangle, \quad t=0 $$

\(15-18\) Find a vector equation and parametric equations for the line segment that joins \(P\) to \(Q .\) $$ P(1,-1,2), \quad Q(4,1,7) $$

(a) Find the unit tangent and unit normal vectors \(\mathbf{T}(t)\) and \(\mathbf{N}(t)\) (b) Use Formula 9 to find the curvature. \(\mathbf{r}(t)=\left\langle t^{2}, \sin t-t \cos t, \cos t+t \sin t\right\rangle, \quad t>0\)

\(17-18\) (a) Find the position vector of a particle that has the given acceleration and the specified initial velocity and position. (b) Use a computer to graph the path of the particle. $$\mathbf{a}(t)=t \mathbf{i}+e^{t} \mathbf{j}+e^{-l} \mathbf{k}, \quad \mathbf{v}(0)=\mathbf{k}, \quad \mathbf{r}(0)=\mathbf{j}+\mathbf{k}$$

Find the unit tangent vector \(\mathbf{T}(t)\) at the point with the given value of the parameter \(t .\) $$ \mathbf{r}(t)=4 \sqrt{t} \mathbf{i}+t^{2} \mathbf{j}+t \mathbf{k}, \quad t=1 $$

\(15-18\) Find a vector equation and parametric equations for the line segment that joins \(P\) to \(Q .\) $$ P(-2,4,0), \quad Q(6,-1,2) $$

(a) Find the unit tangent and unit normal vectors \(\mathbf{T}(t)\) and \(\mathbf{N}(t)\) (b) Use Formula 9 to find the curvature. \(\mathbf{r}(t)=\left\langle\sqrt{2} t, e^{t}, e^{-t}\right\rangle\)

Find the unit tangent vector \(\mathbf{T}(t)\) at the point with the given value of the parameter \(t .\) $$ \mathbf{r}(t)=\cos t \mathbf{i}+3 t \mathbf{j}+2 \sin 2 t \mathbf{k}, \quad t=0 $$

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

(a) Find the unit tangent and unit normal vectors \(\mathbf{T}(t)\) and \(\mathbf{N}(t)\) (b) Use Formula 9 to find the curvature. \(\mathbf{r}(t)=\left\langle t, \frac{1}{2} t^{2}, t^{2}\right\rangle\)

Find the unit tangent vector \(\mathbf{T}(t)\) at the point with the given value of the parameter \(t .\) $$ \mathbf{r}(t)=2 \sin t \mathbf{i}+2 \cos t \mathbf{j}+\tan t \mathbf{k}, \quad t=\pi / 4 $$

What force is required so that a particle of mass \(m\) has the position function \(\mathbf{r}(t)=t^{3} \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k} ?\)

If $$ \mathbf{r}(t)=\left\langle t, t^{2}, t^{3}\right\rangle, \text { find } \mathbf{r}^{\prime}(t), \mathbf{T}(1), \mathbf{r}^{\prime \prime}(t), \text { and } \mathbf{r}^{\prime}(t) \times \mathbf{r}^{\prime \prime}(t) $$

Show that if a particle moves with constant speed, then the velocity and acceleration vectors are orthogonal.

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$ x=1+2 \sqrt{t}, \quad y=t^{3}-t, z=t^{3}+t ; \quad(3,0,2) $$

A projectile is fired with an initial speed of 500 \(\mathrm{m} / \mathrm{s}\) and angle of elevation \(30^{\circ} .\) Find (a) the range of the projectile, (b) the maximum height reached, and (c) the speed at impact.

Find the curvature of \(\mathbf{r}(t)=\left\langle e^{t} \cos t, e^{t} \sin t, t\right\rangle\) at the point \((1,0,0) .\)

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$ \boldsymbol{x}=e^{t}, \quad y=t e^{t}, \quad z=t e^{t^{2}} ; \quad(1,0,0) $$

Find the curvature of \(\mathbf{r}(t)=\left\langle t, t^{2}, t^{3}\right\rangle\) at the point \((1,1,1)\)

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$ x=e^{-t} \cos t, \quad y=e^{-t} \sin t, \quad z=e^{-t} ; \quad(1,0,1) $$

Show that the curve with parametric equations \(x=t \cos t,\) \(y=t \sin t, z=t\) lies on the cone \(z^{2}=x^{2}+y^{2}\) , and use this fact to help sketch the curve.

Graph the curve with parametric equations $$x=t \quad y=4 t^{3 / 2} \quad z=-t^{2}$$ and find the curvature at the point \((1,4,-1).\)

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$ x=\ln t, \quad y=2 \sqrt{t}, \quad z=t^{2} ; \quad(0,2,1) $$

A gun is fired with angle of elevation \(30^{\circ} .\) What is the muzzle speed if the maximum height of the shell is 500 \(\mathrm{m} ?\)

Show that the curve with parametric equations \(x=\sin t\) \(y=\cos t, z=\sin ^{2} t\) is the curve of intersection of the surfaces \(z=x^{2}\) and \(x^{2}+y^{2}=1 .\) Use this fact to help sketch the curve.

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Illustrate by graphing both the curve and the tangent line on a common screen. $$ x=t, y=e^{-t}, z=2 t-t^{2} ; \quad(0,1,0) $$

A gun has muzzle speed 150 \(\mathrm{m} / \mathrm{s}\) . Find two angles of elevation that can be used to hit a target 800 \(\mathrm{m}\) away.

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Illustrate by graphing both the curve and the tangent line on a common screen. $$ x=2 \cos t, y=2 \sin t, z=4 \cos 2 t ; \quad(\sqrt{3}, 1,2) $$

At what points does the helix \(\mathbf{r}(t)=\langle\sin t, \cos t, t\rangle\) intersect the sphere \( x^{2}+y^{2}+z^{2}=5 ?\)

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Illustrate by graphing both the curve and the tangent line on a common screen. $$ x=t \cos t, y=t, z=t \sin t ; \quad(-\pi, \pi, 0) $$

\(29-32\) Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve. $$ \mathbf{r}(t)=\langle\cos t \sin 2 t, \sin t \sin 2 t, \cos 2 t\rangle $$

A medieval city has the shape of a square and is protected by walls with length 500 \(\mathrm{m}\) and height 15 \(\mathrm{m} .\) You are the commander of an attacking army and the closest you can get to the wall is 100 \(\mathrm{m} .\) Your plan is to set fire to the city by catapulting heated rocks over the wall (with an initial speed of 80 \(\mathrm{m} / \mathrm{s} ) .\) At what range of angles should you tell your men to set the catapult? (Assume the path of the rocks is perpendicular to the wall.)

(a) Find the point of intersection of the tangent lines to the curve \(\mathbf{r}(t)=\langle\sin \pi t, 2 \sin \pi t, \cos \pi t\rangle\) at the points where \( t=0\) and \(t=0.5 .\) (b) Illustrate by graphing the curve and both tangent lines.

At what point does the curve have maximum curvature? What happens to the curvature as \(x \rightarrow \infty ?\) \(y=\ln x\)

\(29-32\) Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve. $$ \mathbf{r}(t)=\left\langle t^{2}, \ln t, t\right\rangle $$

The curves \(\mathbf{r}_{1}(t)=\left\langle t, t^{2}, t^{3}\right\rangle\) and \(\mathbf{r}_{2}(t)=\langle\sin t, \sin 2 t, t\rangle\) intersect at the origin. Find their angle of intersection correct to the nearest degree.

At what point does the curve have maximum curvature? What happens to the curvature as \(x \rightarrow \infty ?\) \(y=e^{x}\)