# Thomas Calculus

## Educators

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 $t .$

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Problem 2

${r}(t)$ 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 $t .$

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Problem 3

${r}(t)$ 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}{9} e^{2 t} \mathbf{j}, \quad t=\ln 3

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Problem 4

${r}(t)$ 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.

\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

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

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

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

${r}(t)$ 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{t^{3}}{3} \mathbf{k}, \quad t=1

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Problem 11

${r}(t)$ 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

${r}(t)$ 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

${r}(t)$ 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}+\frac{t^{2}}{2} \mathbf{k}, \quad t=1

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Problem 14

${r}(t)$ 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^{-}\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

${r}(t)$ 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

${r}(t)$ 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) \mathbf{i}+\left(\tan ^{-1} t\right) \mathbf{j}+\sqrt{t^{2}+1} \mathbf{k}\right.

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Problem 18

${r}(t)$ 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^{z} \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 $t_{0}$ 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 $t_{0}$ 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 $t_{0}$ 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.
$\begin{array}{l}{\text { 1) Does the particle have constant speed? If so, what is its contant }} \\ {\text {speed? }} \\ {\text { ii) Is the particle's acceleration vector always orthogonal to its }} \\ {\text { velocity vector? }}\end{array}$
$\begin{array}{l}{\text { iii) Does the particle move clockwise or counterclockwise }} \\ {\text { around the circle? }} \\ {\text { iv) Does the particle begin at the point }(1,0) ?}\end{array}$
$\begin{array}{l}{\text { a. } \mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}, \quad t \geqq 0} \\ {\text { b. } \mathbf{r}(t)=\cos (2 t) \mathbf{i}+\sin (2 t) \mathbf{j}, \quad t \geq 0} \\ {\text { c. } \mathbf{r}(t)=\cos (t-\pi / 2) \mathbf{i}+\sin (t-\pi / 2) \mathbf{j}, \quad t \geq 0} \\ {\text { d. } \mathbf{r}(t)=(\cos t) \mathbf{i}-(\sin t) \mathbf{j}, \quad t \geq 0} \\ {\text { e. } \mathbf{r}(t)=\cos \left(t^{2}\right) \mathbf{i}+\sin \left(t^{2}\right) \mathbf{j}, \quad t \geq 0}\end{array}$

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Problem 24

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)

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 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}

$\begin{array}{l}{\text { a. Graph } \mathbf{r}(t) . \text { The resulting curve is a cycloid. }} \\ {\text { b. Find the maximum and minimum values of }|\mathbf{v}| \text { and }|\mathbf{a}| \text { . }} \\ {\text { (Hint: Find the extreme values of }|\mathbf{v}|^{2} \text { and }|\mathbf{a}|^{2} \text { first and take }} \\ {\text { square roots later.) }}\end{array}$

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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

\begin{array}{l}{\text { a. Show that if } \mathbf{u}, \mathbf{v}, \text { and w are differentiable vector functions of }} \\ {t, \text { then }}\end{array}

$$\begin{array}{l}{\frac{d}{d t}(\mathbf{u} \cdot \mathbf{v} \times \mathbf{w})=\frac{d \mathbf{u}}{d t} \cdot \mathbf{v} \times \mathbf{w}+\mathbf{u} \cdot \frac{d \mathbf{v}}{d t} \times \mathbf{w}+\mathbf{u} \cdot \mathbf{v} \times \frac{d \mathbf{w}}{d t}} \\ {\text { b. Show that }}\end{array}$$
$$\frac{d}{d t}\left(\mathbf{r} \cdot \frac{d \mathbf{r}}{d t} \times \frac{d^{2} \mathbf{r}}{d t^{2}}\right)=\mathbf{r} \cdot\left(\frac{d \mathbf{r}}{d t} \times \frac{d^{3} \mathbf{r}}{d t^{3}}\right)$$
(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 $\mathbf{r}$ 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 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 determinant for-
mula for cross products and the Limit Product Rule for scalar
functions 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 $C,$ then $d \mathbf{u} / d t=0 .$

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Problem 35

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

\begin{array}{l}{\text { a. Plot the space curve traced out by the position vector } \mathbf{r} \text { . }} \\ {\text { b. Find the components of the velocity vector } d \mathbf{r} / d t \text { . }} \\ {\text { c. Evaluate } d \mathbf{r} / d t \text { at the given point } t_{0} \text { and determine the equation of }} \\ {\text { the tangent line to the curve at } \mathbf{r}\left(t_{0}\right) .} \\ {\text { d. Plot the tangent line together with the curve over the given interval. }}\end{array}

\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 36

Use a CAS to perform the following steps.

\begin{array}{l}{\text { a. Plot the space curve traced out by the position vector } \mathbf{r} \text { . }} \\ {\text { b. Find the components of the velocity vector } d \mathbf{r} / d t \text { . }} \\ {\text { c. Evaluate } d \mathbf{r} / d t \text { at the given point } t_{0} \text { and determine the equation of }} \\ {\text { the tangent line to the curve at } \mathbf{r}\left(t_{0}\right) .} \\ {\text { d. Plot the tangent line together with the curve over the given interval. }}\end{array}

\mathbf{r}(t)=\sqrt{2} t \mathbf{i}+e^{t} \mathbf{j}+e^{-i} \mathbf{k}, \quad-2 \leq t \leq 3, \quad t_{0}=1

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Problem 37

Use a CAS to perform the following steps.

\begin{array}{l}{\text { a. Plot the space curve traced out by the position vector } \mathbf{r} \text { . }} \\ {\text { b. Find the components of the velocity vector } d \mathbf{r} / d t \text { . }} \\ {\text { c. Evaluate } d \mathbf{r} / d t \text { at the given point } t_{0} \text { and determine the equation of }} \\ {\text { the tangent line to the curve at } \mathbf{r}\left(t_{0}\right) .} \\ {\text { d. Plot the tangent line together with the curve over the given interval. }}\end{array}

\begin{array}{l}{\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{array}

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Problem 38

Use a CAS to perform the following steps.

\begin{array}{l}{\text { a. Plot the space curve traced out by the position vector } \mathbf{r} \text { . }} \\ {\text { b. Find the components of the velocity vector } d \mathbf{r} / d t \text { . }} \\ {\text { c. Evaluate } d \mathbf{r} / d t \text { at the given point } t_{0} \text { and determine the equation of }} \\ {\text { the tangent line to the curve at } \mathbf{r}\left(t_{0}\right) .} \\ {\text { d. Plot the tangent line together with the curve over the given interval. }}\end{array}

\begin{array}{l}{\mathbf{r}(t)=\left(\ln \left(t^{2}+2\right) \mathbf{i}+\left(\tan ^{-1} 3 t\right) \mathbf{j}+\sqrt{t^{2}+1} \mathbf{k}\right.} \\ {-3 \leq t \leq 5, \quad t_{0}=3}\end{array}

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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.

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

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Problem 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.

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

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