Q: For the parametric equations $x = \cos^3(\theta)$, $y = \sin^3(\theta)$, $0 \leq \theta \leq \dfrac{\pi}{2}$, what is the surface area formed by rotating the curve about the $x$-axis?

$\int_0^{\frac{\pi}{2}} 2\pi y \sqrt{\left(\dfrac{dx}{d\theta}\right)^2 + \left(\dfrac{dy}{d\theta}\right)^2} d\theta$

Question 1

Given the parametric equations $x = t^2 + t$, $y = 2t - 1$, which of the following is the correct Cartesian equation relating $x$ and $y$?

Accepted Answer

$x = \dfrac{(y+1)^2}{4} - \dfrac{y+1}{2}$

Question 2

Which of the following parametric equations describes a circle of radius $7$ centered at $(2, -3)$?

Accepted Answer

$x = 2 + 7\cos(t),\ y = -3 + 7\sin(t)$

Question 3

Given $x = 5\cos(3t)$ and $y = 2\sin(3t)$ for $t \in [0, 2\pi]$, which of the following is the correct Cartesian equation for the curve?

Accepted Answer

$\dfrac{x^2}{25} + \dfrac{y^2}{4} = 1$

Question 4

For the parametric curve $x = 3\sin(t)$, $y = 3\cos(t)$, $0 \leq t \leq 2\pi$, what is the total arc length of the curve?

Accepted Answer

$6\pi$

Question 5

Given $x = f(t) = 6(t - \sin(t))$, $y = g(t) = 6(1 - \cos(t))$, what is the area enclosed by the parametric curve and the $x$-axis for $t$ from $0$ to $2\pi$?

Accepted Answer

$36\pi$

Question 6

Find the equation of the tangent line to the parametric curve $x = t^3 - 3t$, $y = 3t^2 - 9$ at $t = -2$.

Accepted Answer

$y = -12x - 21$

Question 7

For the parametric equations $x = \cos^3(	heta)$, $y = \sin^3(	heta)$, $0 \leq 	heta \leq \dfrac{\pi}{2}$, what is the surface area formed by rotating the curve about the $x$-axis?

Accepted Answer

$\int_0^{\frac{\pi}{2}} 2\pi y \sqrt{\left(\dfrac{dx}{d	heta}ight)^2 + \left(\dfrac{dy}{d	heta}ight)^2} d	heta$

Question 8

At which value of $t$ does the tangent to the curve $x = t^2 + t$, $y = 2t - 1$ become vertical?

Accepted Answer

$t = -\dfrac{1}{2}$

Question 9

For the parametric curve $x = 5\cos(3t)$, $y = 2\sin(3t)$, $t \in [0, 2\pi]$, what is the maximum value of $y$?

Accepted Answer

$2$

Question 10

Given the parametric equations $x = 3\sin(t)$, $y = 3\cos(t)$, what is the geometric shape described by the curve?

Accepted Answer

Circle

Question 11

For the parametric curve $x = t^2 + t$, $y = 2t - 1$, what is the slope of the tangent line at $t = 1$?

Accepted Answer

$\dfrac{2}{3}$

Question 12

Which of the following is the correct formula for the arc length $L$ of a parametric curve $x = f(t)$, $y = g(t)$ from $t = a$ to $t = b$?

Accepted Answer

$L = \int_a^b \sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2} dt$

Question 13

For the parametric curve $x = 3\sin(t)$, $y = 3\cos(t)$, at which value of $t$ is the tangent line horizontal?

Accepted Answer

$t = 0$

Question 14

Given $x = 2 + 7\cos(t)$, $y = -3 + 7\sin(t)$, what is the center and radius of the circle described by these parametric equations?

Accepted Answer

Center $(2, -3)$, radius $7$

Question 15

For the parametric curve $x = t^2 + t$, $y = 2t - 1$, what is the $y$-coordinate when $x = 6$?

Accepted Answer

$y = 5$

Question 16

Which of the following best describes the geometric locus of the parametric equations $x = 5\cos(3t)$, $y = 2\sin(3t)$?

Accepted Answer

Ellipse centered at the origin

Question 17

For the parametric curve $x = 3\sin(t)$, $y = 3\cos(t)$, what is the value of $x$ when $y = 0$?

Accepted Answer

$x = 0$

Question 18

Given $x = t^2 + t$, $y = 2t - 1$, what is the equation of the tangent line at $t = 0$?

Accepted Answer

$y = 2x - 1$

Question 19

For the parametric curve $x = 3\sin(t)$, $y = 3\cos(t)$, what is the value of $t$ where the tangent line is vertical?

Accepted Answer

$t = \dfrac{\pi}{2}$

Final Exam: 50-57 Parametric Equations

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