Q: For the cycloid $$x = 5(\theta - \sin(\theta))$$, $$y = 5(1 - \cos(\theta))$$, what is the correct integral to find the area under one arch ($$0 \leq \theta \leq 2\pi$$)?

$$\$$int_0$$^{2\pi} 5(1 - \cos(\theta)) \cdot \frac{d}{d\theta}[5(\theta - \sin(\theta))] \, d\theta$$

Question 1

Given the parametric equations $$x = f(t)$$ and $$y = g(t)$$, which formula correctly expresses $$\frac{dy}{dx} in $$terms of $$t$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{g'(t)}{f'(t)}$$

Question 2

For the parametric curve $$c(t) = (t^2$ + 1, t^2$ - 4t)$$, what is the slope of the tangent line at $$t = 3$$?

Accepted Answer

$$-2$$

Question 3

Given $$x = 5\cos(t)$$ and $$y = 3\sin(t)$$ for $$-\pi \leq t \leq \pi$$, which integral gives the area under the curve traced by these parametric equations?

Accepted Answer

$$\int_{-\pi}^{\pi} 3\sin(t) \cdot \frac{d}{dt}[5\cos(t)]\,dt$$

Question 4

A cycloid is defined by $$x = 5(	heta - \sin(	heta))$$, $$y = 5(1 - \cos(	heta)). $$What is the area under one arch of this cycloid?

Accepted Answer

Question 5

Which of the following best describes the process for finding the equation of the tangent line to a parametric curve at $$t = t_0$?

Accepted Answer

Find $$x(t_0$)$$ and $$y(t_0$)$$, then use $$\frac{dy}{dx} as $$the slope.

Question 6

Suppose a particle moves along the curve $$x = t^2$, y$$ = $$t^3. At$$ $$t = 2$$, what is the equation of the tangent line in point-slope form?

Accepted Answer

$$y - 8 = 12(x - 4)$$

Question 7

For the parametric equations $$x = t^2$ + 1$$, $$y = t^2$ - 4t$$, what is the point on the curve at $$t = 3$$?

Accepted Answer

$$(10, -6)$$

Question 8

If $$x = f(t)$$ and $$y = g(t)$$, which of the following is the correct formula for the area under the parametric curve from $$t = a to$$ $$t = b$$?

Accepted Answer

$$\int_a^b y(t) \frac{dx}{dt}\,dt$$

Question 9

A medical imaging device traces a path given by $$x = 5\cos(t)$$, $$y = 3\sin(t)$$ for $$0 \leq t \leq \pi. $$What is the total area enclosed by this path?

Accepted Answer

$$15\pi$$

Question 10

Which of the following is a common mistake when finding $$\frac{dy}{dx}$$ for parametric equations?

Accepted Answer

Using $$\frac{g(t)}{f(t)}$$ instead of $$\frac{g'(t)}{f'(t)}$$

Question 11

A car's position is modeled by $$x = 2t$$, $$y = 3t^2$. What is the slope of the tangent to its path at t = 1$?

Accepted Answer

$$6$$

Question 12

For the cycloid $$x = 5(	heta - \sin(	heta))$$, $$y = 5(1 - \cos(	heta))$$, what is the correct integral to find the area under one arch ($$0 \leq 	heta \leq 2\pi$$)?

Accepted Answer

$$\$$int_0$$^{2\pi} 5(1 - \cos(	heta)) \cdot \frac{d}{d	heta}[5(	heta - \sin(	heta))] \, d	heta$$

Question 13

A roller coaster follows $$x = t^2$, y$$ = $$t^3 - t$. At t = 2$, what is the equation of the tangent line?

Accepted Answer

$$y - 6 = 11(x - 4)$$

Question 14

Which real-world scenario best illustrates the use of parametric equations for area calculation?

Accepted Answer

Calculating the area of a lake with irregular boundaries using GPS data

Question 15

If $$x = t^2$, y$$ = $$t^3$$, what is $$\frac{dy}{dx} at$$ $$t = 2$$?

Accepted Answer

$$6$$

Question 16

A drone traces $$x = 5\cos(t)$$, $$y = 3\sin(t)$$ for $$0 \leq t \leq \pi. $$What is the value of $$\frac{dy}{dx} at$$ $$t = \frac{\pi}{2}$$?

Accepted Answer

$$\infty$$

Question 17

Which of the following is NOT a valid application of parametric equations?

Accepted Answer

Calculating the mean of a data set

Question 18

For $$x = t^2$ + 1$$, $$y = t^2$ - 4t$$, what is $$\frac{dy}{dx} at$$ $$t = 0$$?

Accepted Answer

$$-4$$

Question 19

A parametric curve is given by $$x = 3t$$, $$y = 2t^2$. What is the equation of the tangent line at t = 1$?

Accepted Answer

$$y - 2 = \frac{4}{3}(x - 3)$$

Tangent Lines and Area for Parametric Equations

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