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

$$x = r \cos \theta, \ y = r \sin \theta$$

Question 1

Which of the following best describes a conic section?

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) = 2t$, what is the Cartesian equation of the curve?

Accepted Answer

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

Question 3

Which of the following is the correct transformation from Cartesian to polar coordinates for a point $$(x, y)$$?

Accepted Answer

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

Question 4

A particle moves in the plane according to $$x(t) = \cos t$$, $$y(t) = \sin t. $$What is the geometric path traced by the particle?

Accepted Answer

A circle of radius 1 centered at the origin.

Question 5

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

Accepted Answer

Spiral

Question 6

Given the polar coordinates $$(r, 	heta)$$, which equations correctly convert to Cartesian coordinates $$(x, y)$$?

Accepted Answer

$$x = r \cos 	heta, \ y = r \sin 	heta$$

Question 7

If a curve is given by $$y = f(x)$$, what does its graph represent?

Accepted Answer

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

Question 8

Which of the following statements about parametric curves is TRUE?

Accepted Answer

A parametric curve in the plane can be described by $$x(t)$$ and $$y(t)$$, where $$t is a $$parameter.

Question 9

Suppose a particle's position at time $$t is $$given by $$\gamma(t) = (x(t), y(t)). $$What does $$t$$ typically represent in this context?

Accepted Answer

Time

Question 10

Given the Cartesian coordinates $$(x, y)$$, what is the formula for $$r in $$polar coordinates?

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

Which of the following real-world phenomena can be modeled by a parametric curve?

Accepted Answer

The trajectory of a projectile in two dimensions.

Question 13

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

Accepted Answer

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

Question 14

Which of the following is a correct interpretation of the parametric equations $$x(t) = t$$, $$y(t) = t^2$?

Accepted Answer

A particle moving along a parabola as $$t$$ increases.

Question 15

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

Accepted Answer

$$y = r \sin 	heta$$

Curves, Parametric Equations, and Coordinate Systems in Calculus III

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