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

$$x = R \cdot \cos \theta, y = R \cdot \sin \theta$$

Q: What is the polar angle $$\theta$$ for a point $$(x, y) in $$the plane?

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

Question 1

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

Accepted Answer

A function $$\gamma(t) = (x(t), y(t))$$ where $$t$$ belongs to an interval $$I \subset \mathbb{R}.$$

Question 2

What is the geometric interpretation of $$\gamma(t) in $$the context of parametrized curves?

Accepted Answer

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

Question 3

Given the parametric equations $$x = t \cdot \cos 	heta$$ and $$y = t \cdot \sin 	heta$$, what type of coordinate system is being used?

Accepted Answer

Polar coordinates

Question 4

Which equation correctly converts Cartesian coordinates $$(x, y) to $$the polar radius $$R$$?

Accepted Answer

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

Question 5

If a curve is defined by $$y = f(x)$$, what is the image of the function?

Accepted Answer

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

Question 6

Which of the following is NOT a conic section obtained by the intersection of a plane with a cone?

Accepted Answer

Rectangle

Question 7

Given $$\cos 	heta = \frac{x}{R}$$ and $$\sin 	heta = \frac{y}{R}$$, what is $$R$$?

Accepted Answer

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

Question 8

Which formula expresses $$y in $$terms of $$x$$ and $$y in $$polar coordinates?

Accepted Answer

$$y = R \cdot \sin 	heta$$

Question 9

A particle moves along a curve given by $$\gamma(t) = (x(t), y(t)). If$$ $$x(t) = t^2$ and y(t$$) = $$t^3$$, what is the position of the particle at $$t = 2$$?

Accepted Answer

$$(4, 8)$$

Question 10

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

Accepted Answer

$$x = R \cdot \cos 	heta, y = R \cdot \sin 	heta$$

Question 11

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

Accepted Answer

The graph of $$y = f(x) is a $$curve in the plane.

Question 12

Which coordinate system is best suited for describing the position of a point $$P$$ given by $$(x, y)$$?

Accepted Answer

Cartesian coordinates

Question 13

What is the polar angle $$	heta$$ for a point $$(x, y) in $$the plane?

Accepted Answer

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

Question 14

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

Accepted Answer

Describing the trajectory of a moving particle in physics.

Question 15

If $$x = R \cdot \cos 	heta$$ and $$y = R \cdot \sin 	heta$$, what is the value of $$R$$ for the point $$(3, 4)$$?

Accepted Answer

$$R = 5$$

Question 16

Which conic section is formed when a plane intersects a cone parallel to its base?

Accepted Answer

Circle

Question 17

What is the correct formula for $$\cos 	heta in $$terms of $$x$$ and $$R$$?

Accepted Answer

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

Question 18

Which equation represents the distance from the origin to a point $$(x, y) in $$the plane?

Accepted Answer

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

Question 19

If a curve is defined parametrically by $$x(t) = t$$ and $$y(t) = t^2$, what is the Cartesian equation of the curve?

Accepted Answer

$$y = x^2$

Curves in the Plane, Conics, and Coordinate Systems – Calculus III Study Notes

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