Question 1

Which of the following best describes a parametric curve in the plane?

Accepted Answer

A set of points $$(x, y)$$ where $$x$$ and $$y$$ are both functions of a third variable $$t.$$

Question 2

Given the parametric equations $$x = t + 1$$, $$y = t^2$ for -1$$ \leq t \leq 3$$, what is the Cartesian equation of the curve?

Accepted Answer

$$y = (x - 1)^2$

Question 3

For the parametric equations $$x = 4 \cos t$$, $$y = 4 \sin t$$, $$0 \leq t \leq 2\pi$$, what is the geometric shape of the curve?

Accepted Answer

A circle of radius 4 centered at the origin

Question 4

How do you find the slope of the tangent line to a parametric curve $$x = f(t)$$, $$y = g(t)$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}$$

Question 5

Given $$x = a( t - \sin t )$$, $$y = a( 1 - \cos t )$$, what is the name of the curve traced by these equations?

Accepted Answer

Cycloid

Question 6

Suppose a particle moves along the curve $$x = \ln t$$, $$y = t^2$ for 1$$ \leq t \leq 4$$. What is the orientation of the curve as t$ increases?

Accepted Answer

From left to right and upward

Question 7

For the parametric equations $$x = 3 \cos^3$ t$$, $$y = 3 \sin^3$ t$$, $$0 \leq t \leq 2\pi$$, what is the name of the curve?

Accepted Answer

Astroid

Question 8

Given $$x = 4 \cos t$$, $$y = 4 \sin t$$, for which values of $$t$$ does the curve have a horizontal tangent?

Accepted Answer

$$t = \frac{\pi}{2}, \frac{3\pi}{2}$$

Question 9

If $$x = t^2$, y$$ = $$t^3 - t$, find $$\frac{dy}{dx}$$ in terms of t$.

Accepted Answer

$$\frac{dy}{dx} = \frac{3t^2$ - 1}{2t}$$

Question 10

A circle of radius $$r is $$rolling along a straight line. The path traced by a point on the circumference is given by $$x = r(t - \sin t)$$, $$y = r(1 - \cos t). $$What real-world phenomenon does this describe?

Accepted Answer

The path of a reflector on a bicycle wheel

Question 11

Given the parametric equations $$x = t$$, $$y = e^t$$, for $$-\infty \u003c t \u003c \infty$$, what is the Cartesian equation?

Accepted Answer

$$y = e^x$$

Question 12

For the parametric equations $$x = 2 \cos t$$, $$y = 3 \sin t$$, $$0 \leq t \leq 2\pi$$, what is the Cartesian equation?

Accepted Answer

$$\frac{x^2$}{4} + \frac{y^2$}{9} = 1$$

Question 13

Given $$x = t^2$, y$$ = $$t^3 - t$, at what value(s) of t$ does the curve have a vertical tangent?

Accepted Answer

$$t = 0$$

Question 14

Which of the following is a valid set of parametric equations for the line $$y = 3x + 2$$?

Accepted Answer

$$x = t, y = 3t + 2$$

Question 15

If $$x = \cos t$$, $$y = \sin t$$, what is the orientation of the curve as $$t$$ increases from $$0 to$$ $$2\pi$$?

Accepted Answer

Counterclockwise

Question 16

Given $$x = t^2$, y$$ = $$t^2 - 2t$, for 0$$ \leq t \leq 2$$, what is the range of y$?

Accepted Answer

$$-2 \leq y \leq 0$$

Question 17

For the parametric equations $$x = t$$, $$y = t^2$, what is the orientation of the curve as t$ increases?

Accepted Answer

From left to right and upward

Parametric Curves and Their Derivatives in Calculus III

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