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Thomas Calculus 12

George B. Thomas, Jr. Maurice D. Weir, Joel Hass

Chapter 13

Vector-Valued Functions and Motion in Space

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Problem 1

In Exercises $1-4, \mathbf{r}(t)$ 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 particle. Then find the particle's velocity and acceleration vectors at the given value of $t .$
$$\mathbf{r}(t)=(t+1) \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}, \quad t=1$$

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Problem 2

In Exercises $1-4, \mathbf{r}(t)$ 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 particle. Then find the particle's velocity and acceleration vectors at the given value of $t .$
$$\mathbf{r}(t)=\frac{t}{t+1} \mathbf{i}+\frac{1}{t} \mathbf{j}, \quad t=-1 / 2$$

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Problem 3

In Exercises $1-4, \mathbf{r}(t)$ 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 particle. Then find the particle's velocity and acceleration vectors at the given value of $t .$
$$\mathbf{r}(t)=e^{t} \mathbf{i}+\frac{2}{\overline{0}} e^{2 t} \mathbf{j}, \quad t=\ln 3$$

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Problem 4

In Exercises $1-4, \mathbf{r}(t)$ 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 particle. Then find the particle's velocity and acceleration vectors at the given value of $t .$
$$\mathbf{r}(t)=(\cos 2 t) \mathbf{i}+(3 \sin 2 t) \mathbf{j}, \quad t=0$$

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Problem 5

Exercises 5-8 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 $x^{2}+y^{2}=1$
$$
\mathbf{r}(t)=(\sin t) \mathbf{i}+(\cos t) \mathbf{j} ; \quad t=\pi / 4 \text { and } \pi / 2
$$

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Problem 6

Exercises 5-8 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 $x^{2}+y^{2}=16$
$$
\mathbf{r}(t)=\left(4 \cos \frac{t}{2}\right) \mathbf{i}+\left(4 \sin \frac{t}{2}\right) \mathbf{j} ; \quad t=\pi \text { and } 3 \pi / 2
$$

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Problem 7

Exercises 5-8 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 $x=t-\sin t, \quad y=1-\cos t$
$$
\mathbf{r}(t)=(t-\sin t) \mathbf{i}+(1-\cos t) \mathbf{j} ; \quad t=\pi \text { and } 3 \pi / 2
$$

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Problem 8

Exercises 5-8 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 $y=x^{2}+1$
$$
\mathbf{r}(t)=t \mathbf{i}+\left(t^{2}+1\right) \mathbf{j} ; \quad t=-1,0, \text { and } 1
$$

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Problem 9

In Exercises $9-14, \mathbf{r}(t)$ 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.
$$\mathbf{r}(t)=(t+1) \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}+2 t \mathbf{k}, \quad t=1$$

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Problem 10

In Exercises $9-14, \mathbf{r}(t)$ 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.
$$\mathbf{r}(t)=(1+t) \mathbf{i}+\frac{t^{2}}{\sqrt{2}} \mathbf{j}+\frac{l^{3}}{3} \mathbf{k}, \quad t=1$$

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Problem 11

In Exercises $9-14, \mathbf{r}(t)$ 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.
$$\mathbf{r}(t)=(2 \cos t) \mathbf{i}+(3 \sin t) \mathbf{j}+4 t \mathbf{k}, \quad t=\pi / 2$$

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Problem 12

In Exercises $9-14, \mathbf{r}(t)$ 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.
$$\mathbf{r}(t)=(\sec t) \mathbf{i}+(\tan t) \mathbf{j}+\frac{4}{3} t \mathbf{k}, \quad t=\pi / 6$$

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Problem 13

In Exercises $9-14, \mathbf{r}(t)$ 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.
$$\mathbf{r}(t)=(2 \ln (t+1)) \mathbf{i}+t^{2} \mathbf{j}+\left\|\frac{t^{2}}{2} \mathbf{k}, \quad t=1\right.$$

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Problem 14

In Exercises $9-14, \mathbf{r}(t)$ 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.
$$\mathbf{r}(t)=\left(e^{-t}\right) \mathbf{i}+(2 \cos 3 t) \mathbf{j}+(2 \sin 3 t) \mathbf{k}, \quad t=0$$

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Problem 15

In Exercises $15-18, \mathbf{r}(t)$ 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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Problem 16

In Exercises $15-18, \mathbf{r}(t)$ is the position of a particle in space at time $t .$ Find the angle between the velocity and acceleration vectors at time $t=0 .$
$$\mathbf{r}(t)=\left(\frac{\sqrt{2}}{2} t\right) \mathbf{i}+\left(\frac{\sqrt{2}}{2} t-16 t^{2}\right) \mathbf{j}$$

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Problem 17

In Exercises $15-18, \mathbf{r}(t)$ is the position of a particle in space at time $t .$ Find the angle between the velocity and acceleration vectors at time $t=0 .$
$$\mathbf{r}(t)=\left(\ln \left(t^{2}+1\right)\right) \mathbf{i}+\left(\tan ^{-1} t\right) \mathbf{j}+\sqrt{t^{2}+1} \mathbf{k}$$

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Problem 18

In Exercises $15-18, \mathbf{r}(t)$ is the position of a particle in space at time $t .$ Find the angle between the velocity and acceleration vectors at time $t=0 .$
$$\mathbf{r}(t)=\frac{4}{9}(1+t)^{3 / 2} \mathbf{i}+\frac{4}{9}(1-t)^{3 / 2} \mathbf{j}+\frac{1}{3} t \mathbf{k}$$

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Problem 19

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 $19-22,$ find parametric equations for the line that is tangent to the given curve at the given parameter value $t=t_{0}$ .
$$\mathbf{r}(t)=(\sin t) \mathbf{i}+\left(t^{2}-\cos t\right) \mathbf{j}+e^{\prime} \mathbf{k}, \quad t_{0}=0$$

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Problem 20

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 $19-22,$ find parametric equations for the line that is tangent to the given curve at the given parameter value $t=t_{0}$ .
$$\mathbf{r}(t)=t^{2} \mathbf{i}+(2 t-1) \mathbf{j}+t^{3} \mathbf{k}, \quad t_{0}=2$$

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Problem 21

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 $19-22,$ find parametric equations for the line that is tangent to the given curve at the given parameter value $t=t_{0}$ .
$$\mathbf{r}(t)=\ln t \mathbf{i}+\frac{t-1}{t+2} \mathbf{j}+t \ln t \mathbf{k}, \quad t_{0}=1$$

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Problem 22

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 $19-22,$ find parametric equations for the line that is tangent to the given curve at the given parameter value $t=t_{0}$ .
$$\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+(\sin 2 t) \mathbf{k}, \quad t_{0}=\frac{\pi}{2}$$

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Problem 23

Motion along a circle 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) Does the particle begin at the point $(1,0) ?$
a. $\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}, \quad t \geq 0$
b. $\mathbf{r}(t)=\cos (2 t) \mathbf{i}+\sin (2 t) \mathbf{j}, \quad t \geq 0$ $\mathbf{c} \cdot \mathbf{r}(t)=\cos (t-\pi / 2) \mathbf{i}+\sin (t-\pi / 2) \mathbf{j}, \quad t \geq 0$
d. $\mathbf{r}(t)=(\cos t) \mathbf{i}-(\sin t) \mathbf{j}, \quad t \geq 0$
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 24

$$\begin{array}{l}{\text { Motion along a circle Show that the vector-valued function }} \\ {\mathbf{r}(t)=(2 \mathbf{i}+2 \mathbf{j}+\mathbf{k})} \\ {+\cos t\left(\frac{1}{\sqrt{2}} \mathbf{i}-\frac{1}{\sqrt{2}} \mathbf{j}\right)+\sin t\left(\frac{1}{\sqrt{3}} \mathbf{i}+\frac{1}{\sqrt{3}} \mathbf{j}+\frac{1}{\sqrt{3}} \mathbf{k}\right)}\end{array}$$
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 25

Motion along a parabola 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 26

Motion along a cycloid 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 27

Let $\mathbf{r}$ 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 28

Derivatives of triple scalar products
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 29

Prove the two Scalar Multiple Rules for vector functions.

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Problem 30

Prove the Sum and Difference Rules for vector functions.

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Problem 31

Component Test for Continuity at a Point Show that the vector function $r$ 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 32

Limits of cross products of vector functions Suppose
that $\mathbf{r}_{1}(t)=f_{1}(t) \mathbf{i}+f_{2}(t) \mathbf{j}+f_{3}(t) \mathbf{k}, \mathbf{r}_{2}(t)=g_{1}(t) \mathbf{i}+g_{2}(t) \mathbf{j}+$
$g_{3}(t) \mathbf{k}, \lim _{t \rightarrow t_{0}} \mathbf{r}_{1}(t)=\mathbf{A},$ and $\lim _{t \rightarrow t_{0}} \mathbf{r}_{2}(t)=\mathbf{B} .$ Use the determi-
nant formula for cross products and the Limit Product Rule for
scalar finctions to show that
$$\lim _{t \rightarrow t_{0}}\left(\mathbf{r}_{1}(t) \times \mathbf{r}_{2}(t)\right)=\mathbf{A} \times \mathbf{B}$$

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Problem 33

Differentiable vector functions are continuous 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 it is
continuous at $t_{0}$ as well.

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Problem 34

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

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Problem 35

Use a CAS to perform the following steps in Exercises $35-38$ .
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.

$$\mathbf{r}(t)=(\sin t-t \cos t) \mathbf{i}+(\cos t+t \sin t) \mathbf{j}+t^{2} \mathbf{k}$$
$$0 \leq t \leq 6 \pi, \quad t_{0}=3 \pi / 2$$

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Problem 36

Use a CAS to perform the following steps in Exercises $35-38$ .
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.
$$\mathbf{r}(t)=\sqrt{2 t \mathbf{i}+e^{t} \mathbf{j}+e^{-} \mathbf{k},}-2 \leq t \leq 3, \quad t_{0}=1$$

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Problem 37

Use a CAS to perform the following steps in Exercises $35-38$ .
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.
$$\mathbf{r}(t)=(\sin 2 t) \mathbf{i}+(\ln (1+t)) \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 4 \pi$$t_{0}=\pi / 4$$

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Problem 38

Use a CAS to perform the following steps in Exercises $35-38$ .
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.
$$\mathbf{r}(t)=\left(\ln \left(t^{2}+2\right)\right) \mathbf{i}+\left(\tan ^{-1} 3 t\right) \mathbf{j}+\sqrt{t^{2}+1} \mathbf{k}$$
$$-3 \leq t \leq 5, \quad t_{0}=3$$

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Problem 39

In Exercises 39 and $40,$ 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 $a$ it creases through these positive values.

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Problem 40

In Exercises 39 and $40,$ 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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