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}\$$right)^2$$ + \left(\dfrac{dy}{d	heta}\$$right)^2$$} 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

Guida di studio - Domande di pratica

Study guide - Flashcard