University Calculus: Early Transcendentals 4th

Joel Hass, Christopher Heil, Przemyslaw Bogacki

Chapter 12

Vector-Valued Functions and Motion in Space

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Problem 1

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

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Will E.
Numerade Educator

Problem 2

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

WE
Will E.
Numerade Educator

Problem 3

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

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Will E.
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Problem 4

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

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Will E.
Numerade Educator

Problem 5

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 particle. Then find the particle's velocity and acceleration vectors at the given value of $t$.
$$\mathbf{r}(t)=(t+1) \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}, \quad t=1$$

WE
Will E.
Numerade Educator

Problem 6

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 particle. Then find the particle's velocity and acceleration vectors at the given value of $t$.
$$\mathbf{r}(t)=\frac{t}{t+1} \mathbf{i}+\frac{1}{t} \mathbf{j}, \quad t=-\frac{1}{2}$$

WE
Will E.
Numerade Educator

Problem 7

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 particle. Then find the particle's velocity and acceleration vectors at the given value of $t$.
$$\mathbf{r}(t)=e^{t} \mathbf{i}+\frac{2}{9} e^{2 t} \mathbf{j}, \quad t=\ln 3$$

WE
Will E.
Numerade Educator

Problem 8

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 particle. Then find the particle's velocity and acceleration vectors at the given value of $t$.
$$\mathbf{r}(t)=(\cos 2 t) \mathbf{i}+(3 \sin 2 t) \mathbf{j}, \quad t=0$$

WE
Will E.
Numerade Educator

Problem 9

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 $x^{2}+y^{2}=1$
$$\mathbf{r}(t)=(\sin t) \mathbf{i}+(\cos t) \mathbf{j} ; \quad t=\pi / 4 \text { and } \pi / 2$$

WE
Will E.
Numerade Educator

Problem 10

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 $x^{2}+y^{2}=16$
$$\mathbf{r}(t)=\left(4 \cos \frac{t}{2}\right) \mathbf{i}+\left(4 \sin \frac{t}{2}\right) \mathbf{j} ; \quad t=\pi \text { and } 3 \pi / 2$$

WE
Will E.
Numerade Educator

Problem 11

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 cycloid $x=t-\sin t, \quad y=1-\cos t$
$$\mathbf{r}(t)=(t-\sin t) \mathbf{i}+(1-\cos t) \mathbf{j} ; \quad t=\pi \text { and } 3 \pi / 2$$

WE
Will E.
Numerade Educator

Problem 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 parabola $y=x^{2}+1$
$$\mathbf{r}(t)=t \mathbf{i}+\left(t^{2}+1\right) \mathbf{j} ; \quad t=-1,0, \text { and } 1$$

WE
Will E.
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Problem 13

Is the position of a particle in space at time $t$ Find the particle's velocity and acceleration vectors. Then find the particle's speed and direction of motion at the given value of $t .$ Write the particle's velocity at that time as the product of its speed and direction.
$$\mathbf{r}(t)=(t+1) \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}+2 t \mathbf{k}, \quad t=1$$

WE
Will E.
Numerade Educator

Problem 14

Is the position of a particle in space at time $t$ Find the particle's velocity and acceleration vectors. Then find the particle's speed and direction of motion at the given value of $t .$ Write the particle's velocity at that time as the product of its speed and direction.
$$\mathbf{r}(t)=(1+t) \mathbf{i}+\frac{t^{2}}{\sqrt{2}} \mathbf{j}+\frac{t^{3}}{3} \mathbf{k}, \quad t=1$$

WE
Will E.
Numerade Educator

Problem 15

Is the position of a particle in space at time $t$ Find the particle's velocity and acceleration vectors. Then find the particle's speed and direction of motion at the given value of $t .$ Write the particle's velocity at that time as the product of its speed and direction.
$$\mathbf{r}(t)=(2 \cos t) \mathbf{i}+(3 \sin t) \mathbf{j}+4 t \mathbf{k}, \quad t=\pi / 2$$

WE
Will E.
Numerade Educator

Problem 16

Is the position of a particle in space at time $t$ Find the particle's velocity and acceleration vectors. Then find the particle's speed and direction of motion at the given value of $t .$ Write the particle's velocity at that time as the product of its speed and direction.
$$\mathbf{r}(t)=(\sec t) \mathbf{i}+(\tan t) \mathbf{j}+\frac{4}{3} t \mathbf{k}, \quad t=\pi / 6$$

WE
Will E.
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Problem 17

Is the position of a particle in space at time $t$ Find the particle's velocity and acceleration vectors. Then find the particle's speed and direction of motion at the given value of $t .$ Write the particle's velocity at that time as the product of its speed and direction.
$$\mathbf{r}(t)=(2 \ln (t+1)) \mathbf{i}+t^{2} \mathbf{j}+\frac{t^{2}}{2} \mathbf{k}, \quad t=1$$

WE
Will E.
Numerade Educator

Problem 18

Is the position of a particle in space at time $t$ Find the particle's velocity and acceleration vectors. Then find the particle's speed and direction of motion at the given value of $t .$ Write the particle's velocity at that time as the product of its speed and direction.
$$\mathbf{r}(t)=e^{-t} \mathbf{i}+(2 \cos 3 t) \mathbf{j}+(2 \sin 3 t) \mathbf{k}, \quad t=0$$

WE
Will E.
Numerade Educator

Problem 19

Is the position of a particle in space at time $t$ Find the angle between the velocity and acceleration vectors at time $t=0$.
$$\mathbf{r}(t)=(3 t+1) \mathbf{i}+\sqrt{3} t \mathbf{j}+t^{2} \mathbf{k}$$

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Will E.
Numerade Educator

Problem 20

Is the position of a particle in space at time $t$ Find the angle between the velocity and acceleration vectors at time $t=0$.
$$\mathbf{r}(t)=\left(\frac{\sqrt{2}}{2} t\right) \mathbf{i}+\left(\frac{\sqrt{2}}{2} t-16 t^{2}\right) \mathbf{j}$$

WE
Will E.
Numerade Educator

Problem 21

Is the position of a particle in space at time $t$ Find the angle between the velocity and acceleration vectors at time $t=0$.
$$\mathbf{r}(t)=\left(\ln \left(t^{2}+1\right)\right) \mathbf{i}+\left(\tan ^{-1} t\right) \mathbf{j}+\sqrt{t^{2}+1} \mathbf{k}$$

WE
Will E.
Numerade Educator

Problem 22

Is the position of a particle in space at time $t$ Find the angle between the velocity and acceleration vectors at time $t=0$.
$$\mathbf{r}(t)=\frac{4}{9}(1+t)^{3 / 2} \mathbf{i}+\frac{4}{9}(1-t)^{3 / 2} \mathbf{j}+\frac{1}{3} t \mathbf{k}$$

WE
Will E.
Numerade Educator

Problem 23

As mentioned in the text, the tangent line to a smooth curve $\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}$ at $t=t_{0}$ is the line that passes through
the point $\left(f\left(t_{0}\right), g\left(t_{0}\right), h\left(t_{0}\right)\right)$ parallel to $\mathbf{v}\left(t_{0}\right),$ the curve's velocity vector at $t_{0} .$ In Exercises $23-26,$ find parametric equations for the line that is tangent to the given curve at the given parameter value $t=t_{0}$.
$$\mathbf{r}(t)=(\sin t) \mathbf{i}+\left(t^{2}-\cos t\right) \mathbf{j}+e^{t} \mathbf{k}, \quad t_{0}=0$$

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Will E.
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Problem 24

the point $\left(f\left(t_{0}\right), g\left(t_{0}\right), h\left(t_{0}\right)\right)$ parallel to $\mathbf{v}\left(t_{0}\right),$ the curve's velocity vector at $t_{0} .$ In Exercises $23-26,$ find parametric equations for the line that is tangent to the given curve at the given parameter value $t=t_{0}$.
$$\mathbf{r}(t)=t^{2} \mathbf{i}+(2 t-1) \mathbf{j}+t^{3} \mathbf{k}, \quad t_{0}=2$$

WE
Will E.
Numerade Educator

Problem 25

the point $\left(f\left(t_{0}\right), g\left(t_{0}\right), h\left(t_{0}\right)\right)$ parallel to $\mathbf{v}\left(t_{0}\right),$ the curve's velocity vector at $t_{0} .$ In Exercises $23-26,$ find parametric equations for the line that is tangent to the given curve at the given parameter value $t=t_{0}$.
$$\mathbf{r}(t)=\ln t \mathbf{i}+\frac{t-1}{t+2} \mathbf{j}+t \ln t \mathbf{k}, \quad t_{0}=1$$

WE
Will E.
Numerade Educator

Problem 26

the point $\left(f\left(t_{0}\right), g\left(t_{0}\right), h\left(t_{0}\right)\right)$ parallel to $\mathbf{v}\left(t_{0}\right),$ the curve's velocity vector at $t_{0} .$ In Exercises $23-26,$ find parametric equations for the line that is tangent to the given curve at the given parameter value $t=t_{0}$.
$$\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+(\sin 2 t) \mathbf{k}, \quad t_{0}=\frac{\pi}{2}$$

WE
Will E.
Numerade Educator

Problem 27

Find the value(s) of $t$ so that the tangent line to the given curve contains the given point.
$$\mathbf{r}(t)=t^{2} \mathbf{i}+(1+t) \mathbf{j}+(2 t-3) \mathbf{k} ; \quad(-8,2,-1)$$

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Problem 28

Find the value(s) of $t$ so that the tangent line to the given curve contains the given point.
$$\mathbf{r}(t)=t \mathbf{i}+3 \mathbf{j}+\left(\frac{2}{3} t^{3 / 2}\right) \mathbf{k} ; \quad(0,3,-8 / 3)$$

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Problem 29

Find the value(s) of $t$ so that the tangent line to the given curve contains the given point.
$$\mathbf{r}(t)=2 t \mathbf{i}+t^{2} \mathbf{j}-t^{2} \mathbf{k} ; \quad(0,-4,4)$$

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Problem 30

Find the value(s) of $t$ so that the tangent line to the given curve contains the given point.
$$\mathbf{r}(t)=-t \mathbf{i}+t^{2} \mathbf{j}+(\ln t) \mathbf{k} ; \quad(2,-5,-3)$$

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Problem 31

Is the position of a particle in space at time $t$ Match each position function with one of the graphs A-F. (GRAPH CAN"T COPY)
$$\mathbf{r}(t)=(t \cos t) \mathbf{i}+(t \sin t) \mathbf{j}+t \mathbf{k}$$

WE
Will E.
Numerade Educator

Problem 32

Is the position of a particle in space at time $t$ Match each position function with one of the graphs A-F. (GRAPH CAN"T COPY)
$$\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+(\sin 2 t) \mathbf{k}$$

WE
Will E.
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Problem 33

Is the position of a particle in space at time $t$ Match each position function with one of the graphs A-F. (GRAPH CAN"T COPY)
$$\mathbf{r}(t)=t^{2} \mathbf{i}+\left(t^{2}+1\right) \mathbf{j}+t^{4} \mathbf{k}$$

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Will E.
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Problem 34

Is the position of a particle in space at time $t$ Match each position function with one of the graphs $\mathrm{A}-\mathrm{F}$. (GRAPH CANT COPY)
$$\mathbf{r}(t)=t \mathbf{i}+(\ln t) \mathbf{j}+(\sin t) \mathbf{k}$$

WE
Will E.
Numerade Educator

Problem 35

Is the position of a particle in space at time $t$ Match each position function with one of the graphs $\mathrm{A}-\mathrm{F}$. (GRAPH CANT COPY)
$$\mathbf{r}(t)=t \mathbf{i}+(\cos t) \mathbf{j}+(\sin t) \mathbf{k}$$

WE
Will E.
Numerade Educator

Problem 36

Is the position of a particle in space at time $t$ Match each position function with one of the graphs A-F. (GRAPH CANT COPY)
$$\mathbf{r}(t)=(t \sin t) \mathbf{i}+(t \cos t) \mathbf{j}+\left(\frac{t}{t^{2}+1}\right) \mathbf{k}$$

WE
Will E.
Numerade Educator

Problem 37

Each of the following equations in parts
(a)-(e) describes the motion of a particle having the same path, namely the unit circle $x^{2}+y^{2}=1 .$ Although the path of each particle in parts (a)-(e) is the same, the behavior, or "dynamics," of each particle is different. For each particle, answer the following questions.
i) Does the particle have constant speed? If so, what is its constant speed?
ii) Is the particle's acceleration vector always orthogonal to its velocity vector?
iii) Does the particle move clockwise or counterclockwise around the circle?
iv) Is the particle initially located at the point (1,0)$?$
a. $\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}, \quad t \geq 0$
b. $\mathbf{r}(t)=\cos (2 t) \mathbf{i}+\sin (2 t) \mathbf{j}, \quad t \geq 0$
c. $r(t)=\cos (t-\pi / 2) \mathbf{i}+\sin (t-\pi / 2) \mathbf{j}, \quad t \geq 0$
d. $\mathbf{r}(t)=(\cos t) \mathbf{i}-(\sin t) \mathbf{j}, \quad t \geq 0$
$\mathbf{e}_{.} \mathbf{r}(t)=\cos \left(t^{2}\right) \mathbf{i}+\sin \left(t^{2}\right) \mathbf{j}, \quad t \geq 0$

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Problem 38

Show that the vector-valued function
$\mathbf{r}(t)=(2 \mathbf{i}+2 \mathbf{j}+\mathbf{k})$
$$+\cos t\left(\frac{1}{\sqrt{2}} \mathbf{i}-\frac{1}{\sqrt{2}} \mathbf{j}\right)+\sin t\left(\frac{1}{\sqrt{3}} \mathbf{i}+\frac{1}{\sqrt{3}} \mathbf{j}+\frac{1}{\sqrt{3}} \mathbf{k}\right)$$
describes the motion of a particle moving in the circle of radius 1 centered at the point (2,2,1) and lying in the plane $x+y-2 z=2$

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Problem 39

A particle moves along the top of the parabola $y^{2}=2 x$ from left to right at a constant speed of 5 units per second. Find the velocity of the particle as it moves through the point (2,2).

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Problem 40

A particle moves in the $x y$ -plane in such a way that its position at time $t$ is
$$\mathbf{r}(t)=(t-\sin t) \mathbf{i}+(1-\cos t) \mathbf{j}$$
a. Graph $\mathbf{r}(t) .$ The resulting curve is a cycloid.
b. Find the maximum and minimum values of $|\mathbf{v}|$ and $|\mathbf{a}| .$ (Hint: Find the extreme values of $|\mathbf{v}|^{2}$ and $|\mathbf{a}|^{2}$ first, and take square roots later.)

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Problem 41

Let $r$ be a differentiable vector function of $t .$ Show that if $\mathbf{r} \cdot(d \mathbf{r} / d t)=0$ for all $t,$ then $|\mathbf{r}|$ is constant.

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Problem 42

a. Show that if $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ are differentiable vector functions of
$t,$ then
$$
\frac{d}{d t}(\mathbf{u} \cdot \mathbf{v} \times \mathbf{w})=\frac{d \mathbf{u}}{d t} \cdot \mathbf{v} \times \mathbf{w}+\mathbf{u} \cdot \frac{d \mathbf{v}}{d t} \times \mathbf{w}+\mathbf{u} \cdot \mathbf{v} \times \frac{d \mathbf{w}}{d t}
$$
b. Show that
$$
\frac{d}{d t}\left(\mathbf{r} \cdot \frac{d \mathbf{r}}{d t} \times \frac{d^{2} \mathbf{r}}{d t^{2}}\right)=\mathbf{r} \cdot\left(\frac{d \mathbf{r}}{d t} \times \frac{d^{3} \mathbf{r}}{d t^{3}}\right)
$$
(Hint: Differentiate on the left and look for vectors whose products are zero.)

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Problem 43

Prove the two Scalar Multiple Rules for vector functions.

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Problem 44

Prove the Sum and Difference Rules for vector functions.

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Problem 45

Show that the vector function $\mathbf{r}$ defined by $\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}$ is continuous at $t=t_{0}$ if and only if $f, g,$ and $h$ are continuous at $t_{0}$.

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Problem 46

Suppose that
$$\mathbf{r}_{1}(t)=f_{1}(t) \mathbf{i}+f_{2}(t) \mathbf{j}+f_{3}(t) \mathbf{k}, \quad \mathbf{r}_{2}(t)=g_{1}(t) \mathbf{i}+g_{2}(t) \mathbf{j}+g_{3}(t) \mathbf{k}$$
$\lim _{t \rightarrow t_{b}} \mathbf{r}_{1}(t)=\mathbf{A},$ and $\lim _{t \rightarrow t_{b}} \mathbf{r}_{2}(t)=\mathbf{B} .$ Use the determinant
formula for cross products and the Limit Product Rule for scalar functions to show that
$$\lim _{t \rightarrow t_{b}}\left(\mathbf{r}_{1}(t) \times \mathbf{r}_{2}(t)\right)=\mathbf{A} \times \mathbf{B}$$.

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Problem 47

Show that if $\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}$ is differentiable at $t=t_{0},$ then $\mathrm{it}$ is
continuous at $t_{0}$ as well.

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Problem 48

Prove that if $\mathbf{u}$ is the vector function with the constant value $\mathbf{C},$ then $d \mathbf{u} / d t=\mathbf{0}$.

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Problem 49

Use a CAS to perform the following steps.
a. Plot the space curve traced out by the position vector $\mathbf{r}$.
b. Find the components of the velocity vector $d \mathbf{r} / d t$
c. Evaluate $d \mathbf{r} / d t$ at the given point $t_{0}$ and determine the equation of the tangent line to the curve at $\mathbf{r}\left(t_{0}\right)$
d. Plot the tangent line together with the curve over the given interval.
$$\begin{array}{l}
\mathbf{r}(t)=(\sin t-t \cos t) \mathbf{i}+(\cos t+t \sin t) \mathbf{j}+t^{2} \mathbf{k} \\
0 \leq t \leq 6 \pi, \quad t_{0}=3 \pi / 2
\end{array}$$

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Problem 50

Use a CAS to perform the following steps.
a. Plot the space curve traced out by the position vector $\mathbf{r}$.
b. Find the components of the velocity vector $d \mathbf{r} / d t$
c. Evaluate $d \mathbf{r} / d t$ at the given point $t_{0}$ and determine the equation of the tangent line to the curve at $\mathbf{r}\left(t_{0}\right)$
d. Plot the tangent line together with the curve over the given interval.
$$\mathbf{r}(t)=\sqrt{2} t \mathbf{i}+e^{t} \mathbf{j}+e^{-t} \mathbf{k}, \quad-2 \leq t \leq 3, \quad t_{0}=1$$

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Problem 51

Use a CAS to perform the following steps.
a. Plot the space curve traced out by the position vector $\mathbf{r}$.
b. Find the components of the velocity vector $d \mathbf{r} / d t$
c. Evaluate $d \mathbf{r} / d t$ at the given point $t_{0}$ and determine the equation of the tangent line to the curve at $\mathbf{r}\left(t_{0}\right)$
d. Plot the tangent line together with the curve over the given interval.
$$\begin{aligned}
&\mathbf{r}(t)=(\sin 2 t) \mathbf{i}+(\ln (1+t)) \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 4 \pi\\
&t_{0}=\pi / 4
\end{aligned}$$

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Problem 52

Use a CAS to perform the following steps.
a. Plot the space curve traced out by the position vector $\mathbf{r}$.
b. Find the components of the velocity vector $d \mathbf{r} / d t$
c. Evaluate $d \mathbf{r} / d t$ at the given point $t_{0}$ and determine the equation of the tangent line to the curve at $\mathbf{r}\left(t_{0}\right)$
d. Plot the tangent line together with the curve over the given interval.
$$\begin{aligned}
&\mathbf{r}(t)=\left(\ln \left(t^{2}+2\right)\right) \mathbf{i}+\left(\tan ^{-1} 3 t\right) \mathbf{j}+\sqrt{t^{2}+1} \mathbf{k}\\
&-3 \leq t \leq 5, \quad t_{0}=3
\end{aligned}$$

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Problem 53

You will explore graphically the behavior of the helix
$$\mathbf{r}(t)=(\cos a t) \mathbf{i}+(\sin a t) \mathbf{j}+b t \mathbf{k}$$
as you change the values of the constants $a$ and $b$. Use a CAS to perform the steps in each exercise.

Set $b=1 .$ Plot the helix $\mathbf{r}(t)$ together with the tangent line to the curve at $t=3 \pi / 2$ for $a=1,2,4,$ and 6 over the interval $0 \leq t \leq 4 \pi .$ Describe in your own words what happens to the graph of the helix and the position of the tangent line as $a$ increases through these positive values.

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Problem 54

You will explore graphically the behavior of the helix
$$\mathbf{r}(t)=(\cos a t) \mathbf{i}+(\sin a t) \mathbf{j}+b t \mathbf{k}$$
as you change the values of the constants $a$ and $b$. Use a CAS to perform the steps in each exercise.
Set $a=1 .$ Plot the helix $\mathbf{r}(t)$ together with the tangent line to the curve at $t=3 \pi / 2$ for $b=1 / 4,1 / 2,2,$ and 4 over the interval $0 \leq t \leq 4 \pi .$ Describe in your own words what happens to the graph of the helix and the position of the tangent line as $b$ increases through these positive values.

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