$$r = \sqrt{x^2$ + y^2$}$$, $$\theta = \arccos\left(\frac{x}{r}\right)$$

$$\theta = \arctan\left(\frac{y}{x}\right)$$

Question 1

Which of the following best describes a conic section in the context of Calculus III?

Accepted Answer

A curve obtained by the intersection of a plane with a cone.

Question 2

Given the parametric equations $$x(t) = t^2$ and y(t$$) = $$t^3$$, what is the Cartesian equation relating $$x$$ and $$y$$?

Accepted Answer

$$y = x$$^{3/2}$$

Question 3

If a curve is parameterized as $$\gamma(t) = (x(t), y(t))$$, what does $$t$$ typically represent in physical applications?

Accepted Answer

Time

Question 4

Which of the following is the correct formula for converting Cartesian coordinates $$(x, y) to $$polar coordinates $$(r, 	heta)$$?

Accepted Answer

$$r = \sqrt{x^2$ + y^2$}$$, $$	heta = \arccos\left(\frac{x}{r}ight)$$

Question 5

A particle moves along a curve given by $$x(t) = t \cos 	heta$$, $$y(t) = t \sin 	heta. $$What is the distance from the origin at time $$t$$?

Accepted Answer

$$r = t$$

Question 6

Which of the following is NOT a type of conic section?

Accepted Answer

Rectangle

Question 7

Given the polar coordinates $$(r, 	heta)$$, which equation correctly expresses $$x in $$terms of $$r$$ and $$	heta$$?

Accepted Answer

$$x = r \cos 	heta$$

Question 8

If $$y = f(x) is $$the graph of a function, which of the following statements is true?

Accepted Answer

The graph is the set of points $$(x, f(x)) in $$the plane.

Question 9

Which formula gives the radius $$R in $$terms of Cartesian coordinates $$x$$ and $$y$$?

Accepted Answer

$$R = \sqrt{x^2$ + y^2$}$$

Question 10

A curve is parameterized by $$x(t) = t^2$, y(t) = 2t$. What is the equation of the curve in Cartesian coordinates?

Accepted Answer

$$y = 2\sqrt{x}$$

Question 11

Which of the following is the correct expression for $$\cos 	heta in $$terms of $$x$$, $$y$$, and $$R$$?

Accepted Answer

$$\cos 	heta = \frac{x}{R}$$

Question 12

Which of the following is the correct expression for $$\sin 	heta in $$terms of $$x$$, $$y$$, and $$R$$?

Accepted Answer

$$\sin 	heta = \frac{y}{R}$$

Question 13

If a curve is given by $$y = f(x)$$, which of the following best describes its graph?

Accepted Answer

The set of all points $$(x, f(x))$$ for $$x in $$the domain of $$f$$

Question 14

Which of the following is a correct parametric representation for a circle of radius $$R$$ centered at the origin?

Accepted Answer

$$x = R \cos t$$, $$y = R \sin t$$

Question 15

Which of the following is a real-world application of parametric curves?

Accepted Answer

Describing the trajectory of a particle in physics.

Question 16

If $$x = r \cos 	heta$$ and $$y = r \sin 	heta$$, what is the value of $$r in $$terms of $$x$$ and $$y$$?

Accepted Answer

$$r = \sqrt{x^2$ + y^2$}$$

Question 17

Which of the following best describes the meaning of $$\gamma(t) in $$the context of parametric curves?

Accepted Answer

$$\gamma(t) is $$the position of a particle at time $$t$$

Question 18

Which of the following is the correct formula for $$	heta in $$terms of $$x$$ and $$y$$?

Accepted Answer

$$	heta = \arctan\left(\frac{y}{x}ight)$$

Curves in the Plane, Conic Sections, and Coordinate Systems

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