Q: Given $$x = 5\cos(\theta)$$ and $$y = 3\sin(\theta)$$ for $$-\pi \leq \theta \leq \pi$$, which integral gives the area under the curve?

$$\int_{-\pi}^{\pi} y \frac{dx}{d\theta} d\theta$$

Q: Which of the following integrals correctly computes the area under one arch of the cycloid $$x = 5(\theta - \sin(\theta))$$, $$y = 5(1 - \cos(\theta))$$ for $$\theta in$$ $$[0, 2\pi]$$?

$$\int_{0}^{2\pi} y \frac{dx}{d\theta} d\theta$$

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 $$x = 5\cos(	heta)$$ and $$y = 3\sin(	heta)$$ for $$-\pi \leq 	heta \leq \pi$$, which integral gives the area under the curve?

Accepted Answer

$$\int_{-\pi}^{\pi} y \frac{dx}{d	heta} d	heta$$

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 the cycloid?

Accepted Answer

$$25\pi$$

Question 5

If $$x = t^2$ + 1$$ and $$y = t^2$ - 4t$$, what is the equation of the tangent line at $$t = 3$$?

Accepted Answer

$$y + 3 = 2(x - 10)$$

Question 6

Which of the following best describes the geometric meaning of $$\frac{dy}{dx}$$ for parametric equations?

Accepted Answer

It gives the rate of change of $$y$$ with respect to $$x$$ along the curve.

Question 7

Suppose a particle moves along the curve $$x = 5\cos(t)$$, $$y = 3\sin(t)$$ for $$t in$$ $$[-\pi, \pi]. $$What is the total area swept out by the curve?

Accepted Answer

$$15\pi$$

Question 8

For the parametric equations $$x = f(t)$$ and $$y = g(t)$$, which of the following is NOT a necessary condition for finding $$\frac{dy}{dx}$$?

Accepted Answer

$$f(t)$$ must be a linear function$

Question 9

A car travels along a path described by $$x = t^2$ + 1$$, $$y = t^2$ - 4t. At $$what value of $$t$$ does the tangent line to the path have zero slope?

Accepted Answer

$$t = 2$$

Question 10

Given $$x = 5\cos(t)$$ and $$y = 3\sin(t)$$, what is $$\frac{dy}{dx} at$$ $$t = 0$$?

Accepted Answer

$$\frac{3}{-5}$$

Question 11

Which real-world scenario can be modeled using parametric equations?

Accepted Answer

The trajectory of a projectile in two dimensions.

Question 12

If $$x = 5(	heta - \sin(	heta))$$ and $$y = 5(1 - \cos(	heta))$$, what is $$\frac{dx}{d	heta} at$$ $$	heta = 0$$?

Accepted Answer

$$5$$

Question 13

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

Accepted Answer

$$(10, -3)$$

Question 14

Which of the following integrals correctly computes the area under one arch of the cycloid $$x = 5(	heta - \sin(	heta))$$, $$y = 5(1 - \cos(	heta))$$ for $$	heta in$$ $$[0, 2\pi]$$?

Accepted Answer

$$\int_{0}^{2\pi} y \frac{dx}{d	heta} d	heta$$

Question 15

A medical imaging device traces a path given by $$x = 5\cos(t)$$, $$y = 3\sin(t) as$$ $$t$$ varies. What is the instantaneous rate of change of $$y$$ with respect to $$x at$$ $$t = \frac{\pi}{2}$$?

Accepted Answer

Undefined

Question 16

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

Accepted Answer

$$1$$

Question 17

Which of the following is a correct step in finding the area under a parametric curve $$x = f(t)$$, $$y = g(t)$$ from $$t = a to$$ $$t = b$$?

Accepted Answer

Compute $$\int_{a}^{b} y \frac{dx}{dt} dt$$

Question 18

A roller coaster follows the path $$x = 5\cos(t)$$, $$y = 3\sin(t). At $$what value of $$t$$ does the tangent line become vertical?

Accepted Answer

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

Tangent Lines and Area for Parametric Equations

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