Question 1

Find a parametrization for the curve.
-The line segment with endpoints (-4, -7) and ( , -15)

Accepted Answer

$\text { Answers will vary. Possible answer: } x = - 4 + 4 t , y = - 7 - 8 t , 0 \leq t \leq 1$

Question 2

Find an equation for the line tangent to the curve at the point defined by the given value of t.
-$$x = 4 \sin t , y = 4 \cos t , t = \frac { 3 \pi } { 4 }$$
A) $y = x - 4 \sqrt { 2 }$
B) $y = 4 x + 4 \sqrt { 2 }$
C) $y = - x + 4 \sqrt { 2 }$
D) $y = 4 \sqrt { 2 } x + 1$

Accepted Answer

To find the equation of the tangent line, we first need to find the slope of the tangent line at the given point. The slope of the tangent line can be found by differentiating $y$ with respect to $x$ using the chain rule, which gives us $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$. Differentiating $x = 4\sin(t)$ with respect to $t$ gives $dx/dt = 4\cos(t)$, and differentiating $y = 4\cos(t)$ with respect to $t$ gives $dy/dt = -4\sin(t)$. Thus, $\frac{dy}{dx} = \frac{-4\sin(t)}{4\cos(t)} = -\tan(t)$. At $t = \frac{3\pi}{4}$, $\tan\left(\frac{3\pi}{4}\right) = -1$, so the slope of the tangent line is $1$. The point on the curve at $t = \frac{3\pi}{4}$ is $x = 4\sin\left(\frac{3\pi}{4}\right) = 4\left(\frac{\sqrt{2}}{2}\right) = 2\sqrt{2}$ and $y = 4\cos\left(\frac{3\pi}{4}\right) = 4\left(-\frac{\sqrt{2}}{2}\right) = -2\sqrt{2}$. Using the point-slope form of a line, $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is the point on the line, we get $y - (-2\sqrt{2}) = 1(x - 2\sqrt{2})$, which simplifies to $y = x - 4\sqrt{2}$.

Question 3

Graph the pair of parametric equations with the aid of a graphing calculator.
-$x = 2 \cos t + \cos 2 t , y = 2 \sin t - \sin 2 t , 0 \leq t \leq 2 \pi$

A)

B)

C)

D)

Accepted Answer

B

Question 4

Graph the pair of parametric equations with the aid of a graphing calculator.
-$x=6 \cos t+2 \cos 3 t, y=6 \sin t-2 \sin 3 t, 0 \leq t \leq 2 \pi$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the pair of parametric equations with...

Question 5

Graph the pair of parametric equations with the aid of a graphing calculator.
-$x=4 \sin t-\sin 4 t, y=4 \cos t-\cos 4 t, 0 \leq t \leq 2 \pi$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the pair of parametric equations with...

Question 6

Graph the pair of parametric equations with the aid of a graphing calculator.
-$x=\cos t+4 \cos 3 t, y=5 \cos t-4 \sin 3 t, 0 \leq t \leq 2 \pi$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the pair of parametric equations with...

Question 7

Parametric equations and and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph
traced by the particle and the direction of motion.
-$x=5 t+5, y=15 t+9,-\leq t \leq \infty$

A) $y = 3 x - 6$; Entire line, from right to left

B) $y = 3 x + 6$; Entire line, from right to left

C) $y = 3 x - 6$; Entire line, from left to right

D) $y = 3 x + 6$; Entire line, from left to right

Accepted Answer

To find a Cartesian equation, we need to eliminate the parameter t. Solving for t in terms of y from the second equation gives us $t=\frac{y-9}{15}$. Substitution into the first equation gives us $x=5 \left(\frac{y-9}{15}\right)+5$ or $x=\frac{y}{3}$. This is the equation of a line with slope 3 passing through the origin. Therefore, the particle's path is a line passing through the origin with a slope of 3. Since the parameter interval is $-\leq t \leq \infty$, the particle covers the entire line from left to right.

The correct choice is C.

Question 8

Parametric equations and and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph
traced by the particle and the direction of motion.
-$x=4 t^{2}, y=2 t, x \leq t \leq \infty$

A) $x = y ^ { 2 }$; Entire parabola, top to bottom (from1st quadrant to origin to 2nd quadrant)

B) $y = x ^ { 2 }$; Entire parabola, right to left (from 1st quadrant to origin to 4 th quadrant)

C) $x = y ^ { 2 }$; Entire parabola, bottom to top (from 2nd quadrant to origin to 1st quadrant)

D) $y = x ^ { 2 }$; Entire parabola, left to right (from 4th quadrant to origin to 1st quadrant)

Accepted Answer

The answer of Parametric equations and and a parameter interval...

Question 9

Parametric equations and and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph
traced by the particle and the direction of motion.
-$x=2 \sin t, y=5 \cos t, 0 \leq t \leq 2 \pi$

A) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 25 } = 1$; Counterclockwise from$( 2,0 )$ to $( 2,0 )$ one rotation

B) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 4 } = 1$; Counterclockwise from ,  $( 5,0 )$ to $( 5,0 )$, one rotation

C) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 4 } = 1$; Counterclockwise from  $( 0,2 )$ to $( 0,2 )$, one rotation

D) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 25 } = 1$; Counterclockwise from $( 0,5 )$ to $( 0,5 )$, one rotation

Accepted Answer

The answer of Parametric equations and and a parameter interval...

Question 10

Parametric equations and and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph
traced by the particle and the direction of motion.
-$x=2 \cos t, y=2 \sin t, \pi \leq t \leq 2 \pi$

A) $x ^ { 2 } + y ^ { 2 } = 4$; Counterclockwise from
 $( 2,0 )$ to $( 2,0 )$, one rotation

B) $x ^ { 2 } + y ^ { 2 } = 4 ;$ Clockwise from $( 2,0 )$ to $( - 2,0 )$

C) $x ^ { 2 } + y ^ { 2 } = 1$; Clockwise from$( 1,0 )$ to $( 1,0 )$

D) $x ^ { 2 } + y ^ { 2 } = 4$; Counterclockwise from , one rotation $( - 2,0 )$ to $( 2,0 )$

Accepted Answer

The answer of Parametric equations and and a parameter interval...

Question 11

Find a parametrization for the curve.
-The ray (half line) with initial point (-3, ) that passes through the point (-10, -3)

Accepted Answer

The answer of Find a parametrization for the curve.
-The ray...

Question 12

Graph the pair of parametric equations with the aid of a graphing calculator.
-$x=t-\sin t, y=1-\cos t,-4 \pi \leq t \leq 4 \pi$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the pair of parametric equations with...

Question 13

Graph the pair of parametric equations with the aid of a graphing calculator.
-$x = 5 \cos 2 t + 4 \cos 6 t , y = 5 \sin 2 t - 4 \sin 6 t , 0 \leq t \leq \pi$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the pair of parametric equations with...

Question 14

Graph the pair of parametric equations with the aid of a graphing calculator.
-$x=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pi$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the pair of parametric equations with...

Question 15

Graph the pair of parametric equations with the aid of a graphing calculator.
-$x=7 t-7 \sin t, y=7-7 \cos t,-2 \pi \leq t \leq 2 \pi$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the pair of parametric equations with...

Question 16

Solve the problem.
-Find the point on the curve $x = 3 \sin t , y = \cos t , - \frac { \pi } { 2 } \leq t \leq \frac { \pi } { 2 }$, closest to the point $\left( \frac { 4 } { 3 } , 0 \right)$. (Hint: Minimize the square of the distance as a function of $t$.)
A) $\left( \frac { 3 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right)$
B) $( - 3,0 )$
C) $\left( \frac { \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right)$
D) $( 3,0 )$

Accepted Answer

The distance squared between a point on the curve and the point $\left( \frac { 4 } { 3 } , 0 \right)$ is given by $d^2 = \left( x - \frac { 4 } { 3 } \right)^2 + y^2$. Substituting $x = 3 \sin t$ and $y = \cos t$, we get $d^2 = \left( 3 \sin t - \frac { 4 } { 3 } \right)^2 + \cos^2 t$. To find the minimum distance, we differentiate $d^2$ with respect to $t$ and set the derivative equal to zero. Solving this will give the value of $t$ that minimizes the distance. The point corresponding to this value of $t$ is $\left( \frac { 3 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right)$, which is choice A.

Question 17

Find a parametrization for the curve.
-$\text { The upper half of the parabola } x - 1 = y ^ { 2 }$

Accepted Answer

The answer of Find a parametrization for the curve.
-\(\text {...

Question 18

Graph the pair of parametric equations with the aid of a graphing calculator.
-$x = 4 ( t - \sin t ) , y = 4 ( 1 - \cos t ) , 0 \leq t \leq 4 \pi$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the pair of parametric equations with...

Question 19

Solve the problem.
-Find the point on the parabola $x = t ^ { 2 } , y = t , * < t < \infty$, closest to the point $\left( \frac { 1 } { 2 } , 54 \right)$, (Hint: Minimize the square of the distance as a function of $\mathrm { t }$.)
A) $( 0,0 )$
B) $( 36,6 )$
C) $( 9,3 )$
D) $\left( \frac { 9 } { 4 } , \frac { 3 } { 2 } \right)$

Accepted Answer

To find the point on the parabola closest to $\left( \frac{1}{2}, 54 \right)$, minimize the square of the distance: $D(t) = (t^2 - \frac{1}{2})^2 + (t - 54)^2$. Solving $\frac{dD}{dt} = 0$ gives $t = 3$, leading to the point $(9, 3)$.

Question 20

Graph the pair of parametric equations with the aid of a graphing calculator.
-$x=5 \tan t, y=2 \sec t ;-\pi / 2 \leq t \leq \pi / 2$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the pair of parametric equations with...

Quiz 12: Parametric Equations and Polar Coordinates

Find a parametrization for the curve. -The line segment with endpoints (-4, -7) and ( , -15)

Find an equation for the line tangent to the curve at the point defined by the given value of t. - $x = 4 \sin t , y = 4 \cos t , t = \frac { 3 \pi } { 4 }$

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=4 \sin t-\sin 4 t, y=4 \cos t-\cos 4 t, 0 \leq t \leq 2 \pi$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=4 \sin t-\sin 4 t, y=4 \cos t-\cos 4 t, 0 \leq t \leq 2 \pi A) B) C) D)$

Find a parametrization for the curve. -The ray (half line) with initial point (-3, ) that passes through the point (-10, -3)

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=t-\sin t, y=1-\cos t,-4 \pi \leq t \leq 4 \pi$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=t-\sin t, y=1-\cos t,-4 \pi \leq t \leq 4 \pi A) B) C) D)$

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pi$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pi A) B) C) D)$

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=7 t-7 \sin t, y=7-7 \cos t,-2 \pi \leq t \leq 2 \pi$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=7 t-7 \sin t, y=7-7 \cos t,-2 \pi \leq t \leq 2 \pi A) B) C) D)$

Solve the problem. -Find the point on the curve $x = 3 \sin t , y = \cos t , - \frac { \pi } { 2 } \leq t \leq \frac { \pi } { 2 }$ , closest to the point $\left( \frac { 4 } { 3 } , 0 \right)$ . (Hint: Minimize the square of the distance as a function of $t$ .)

Find a parametrization for the curve. - $\text { The upper half of the parabola } x - 1 = y ^ { 2 }$

Solve the problem. -Find the point on the parabola $x = t ^ { 2 } , y = t , * < t < \infty$ , closest to the point $\left( \frac { 1 } { 2 } , 54 \right)$ , (Hint: Minimize the square of the distance as a function of $\mathrm { t }$ .)

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=5 \tan t, y=2 \sec t ;-\pi / 2 \leq t \leq \pi / 2$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=5 \tan t, y=2 \sec t ;-\pi / 2 \leq t \leq \pi / 2 A) B) C) D)$

Quiz 12: Parametric Equations and Polar Coordinates

Find a parametrization for the curve. -The line segment with endpoints (-4, -7) and ( , -15)

Find an equation for the line tangent to the curve at the point defined by the given value of t. - x=4sin⁡t,y=4cos⁡t,t=3π4x = 4 \sin t , y = 4 \cos t , t = \frac { 3 \pi } { 4 }x=4sint,y=4cost,t=43π​

Graph the pair of parametric equations with the aid of a graphing calculator. - x=2cos⁡t+cos⁡2t,y=2sin⁡t−sin⁡2t,0≤t≤2πx = 2 \cos t + \cos 2 t , y = 2 \sin t - \sin 2 t , 0 \leq t \leq 2 \pix=2cost+cos2t,y=2sint−sin2t,0≤t≤2π

Graph the pair of parametric equations with the aid of a graphing calculator. - x=6cos⁡t+2cos⁡3t,y=6sin⁡t−2sin⁡3t,0≤t≤2πx=6 \cos t+2 \cos 3 t, y=6 \sin t-2 \sin 3 t, 0 \leq t \leq 2 \pix=6cost+2cos3t,y=6sint−2sin3t,0≤t≤2π

Graph the pair of parametric equations with the aid of a graphing calculator. - x=4sin⁡t−sin⁡4t,y=4cos⁡t−cos⁡4t,0≤t≤2πx=4 \sin t-\sin 4 t, y=4 \cos t-\cos 4 t, 0 \leq t \leq 2 \pix=4sint−sin4t,y=4cost−cos4t,0≤t≤2π

Graph the pair of parametric equations with the aid of a graphing calculator. - x=cos⁡t+4cos⁡3t,y=5cos⁡t−4sin⁡3t,0≤t≤2πx=\cos t+4 \cos 3 t, y=5 \cos t-4 \sin 3 t, 0 \leq t \leq 2 \pix=cost+4cos3t,y=5cost−4sin3t,0≤t≤2π

Find a parametrization for the curve. -The ray (half line) with initial point (-3, ) that passes through the point (-10, -3)

Graph the pair of parametric equations with the aid of a graphing calculator. - x=t−sin⁡t,y=1−cos⁡t,−4π≤t≤4πx=t-\sin t, y=1-\cos t,-4 \pi \leq t \leq 4 \pix=t−sint,y=1−cost,−4π≤t≤4π

Graph the pair of parametric equations with the aid of a graphing calculator. - x=5cos⁡2t+4cos⁡6t,y=5sin⁡2t−4sin⁡6t,0≤t≤πx = 5 \cos 2 t + 4 \cos 6 t , y = 5 \sin 2 t - 4 \sin 6 t , 0 \leq t \leq \pix=5cos2t+4cos6t,y=5sin2t−4sin6t,0≤t≤π

Graph the pair of parametric equations with the aid of a graphing calculator. - x=cos⁡5t,y=sin⁡4t,0≤t≤2πx=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pix=cos5t,y=sin4t,0≤t≤2π

Graph the pair of parametric equations with the aid of a graphing calculator. - x=7t−7sin⁡t,y=7−7cos⁡t,−2π≤t≤2πx=7 t-7 \sin t, y=7-7 \cos t,-2 \pi \leq t \leq 2 \pix=7t−7sint,y=7−7cost,−2π≤t≤2π

Find a parametrization for the curve. - The upper half of the parabola x−1=y2\text { The upper half of the parabola } x - 1 = y ^ { 2 } The upper half of the parabola x−1=y2

Graph the pair of parametric equations with the aid of a graphing calculator. - x=4(t−sin⁡t) ,y=4(1−cos⁡t) ,0≤t≤4πx = 4 ( t - \sin t ) , y = 4 ( 1 - \cos t ) , 0 \leq t \leq 4 \pix=4(t−sint) ,y=4(1−cost) ,0≤t≤4π

Solve the problem. -Find the point on the parabola x=t2,y=t,∗<t<∞x = t ^ { 2 } , y = t , * < t < \inftyx=t2,y=t,∗<t<∞ , closest to the point (12,54) \left( \frac { 1 } { 2 } , 54 \right) (21​,54) , (Hint: Minimize the square of the distance as a function of t\mathrm { t }t .)

Graph the pair of parametric equations with the aid of a graphing calculator. - x=5tan⁡t,y=2sec⁡t;−π/2≤t≤π/2x=5 \tan t, y=2 \sec t ;-\pi / 2 \leq t \leq \pi / 2x=5tant,y=2sect;−π/2≤t≤π/2

Find an equation for the line tangent to the curve at the point defined by the given value of t. - $x = 4 \sin t , y = 4 \cos t , t = \frac { 3 \pi } { 4 }$

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=4 \sin t-\sin 4 t, y=4 \cos t-\cos 4 t, 0 \leq t \leq 2 \pi$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=4 \sin t-\sin 4 t, y=4 \cos t-\cos 4 t, 0 \leq t \leq 2 \pi A) B) C) D)$

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=t-\sin t, y=1-\cos t,-4 \pi \leq t \leq 4 \pi$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=t-\sin t, y=1-\cos t,-4 \pi \leq t \leq 4 \pi A) B) C) D)$

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pi$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pi A) B) C) D)$

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=7 t-7 \sin t, y=7-7 \cos t,-2 \pi \leq t \leq 2 \pi$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=7 t-7 \sin t, y=7-7 \cos t,-2 \pi \leq t \leq 2 \pi A) B) C) D)$

Find a parametrization for the curve. - $\text { The upper half of the parabola } x - 1 = y ^ { 2 }$

Solve the problem. -Find the point on the parabola $x = t ^ { 2 } , y = t , * < t < \infty$ , closest to the point $\left( \frac { 1 } { 2 } , 54 \right)$ , (Hint: Minimize the square of the distance as a function of $\mathrm { t }$ .)

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=5 \tan t, y=2 \sec t ;-\pi / 2 \leq t \leq \pi / 2$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=5 \tan t, y=2 \sec t ;-\pi / 2 \leq t \leq \pi / 2 A) B) C) D)$