Calculus: Parametric Equations and Polar Coordinates flashcards

Question 1

In a parametric curve, what is the parameter?

Accepted Answer

An independent variable on which both x and y depend, usually t

Question 2

Is there only one way to parameterize a curve?

Accepted Answer

No: there is always more than one way

Question 3

What is a cycloid?

Accepted Answer

The path of a point on the rim of a circle as it rolls, without slipping, along a straight line

Question 4

What is a space-filling curve?

Accepted Answer

A curve that fills up a two-dimensional region of the plane entirely

Question 5

dy/dx for a parametric curve x = x(t), y = y(t)?

Accepted Answer

dy/dx = y′(t)/x′(t)

Question 6

Second derivative d²y/dx² for a parametric curve?

Accepted Answer

(d/dt)(dy/dx) divided by dx/dt

Question 7

Arc length of a parametric curve from t₁ to t₂?

Accepted Answer

s = ∫ from t₁ to t₂ of √((dx/dt)² + (dy/dt)²) dt

Question 8

Area under a parametric curve?

Accepted Answer

A = ∫ y(t)x′(t) dt

Question 9

Surface area when a parametric curve is revolved about the x-axis?

Accepted Answer

S = 2π ∫ y(t)√((x′(t))² + (y′(t))²) dt

Question 10

What is the pole of the polar coordinate system?

Accepted Answer

Its central point, which plays the role of the origin in Cartesian coordinates

Question 11

What does the radial coordinate r measure?

Accepted Answer

How far a point in the plane is from the pole

Question 12

Polar to rectangular: x and y in terms of r and θ?

Accepted Answer

x = r cos θ, y = r sin θ

Question 13

Rectangular to polar: r² and tan θ in terms of x and y?

Accepted Answer

r² = x² + y², tan θ = y/x

Question 14

Area bounded by the polar curve r = f(θ), α ≤ θ ≤ β?

Accepted Answer

A = (1/2) ∫ from α to β of [f(θ)]² dθ

Question 15

Arc length of the polar curve r = f(θ), α ≤ θ ≤ β?

Accepted Answer

L = ∫ from α to β of √([f(θ)]² + [f′(θ)]²) dθ

Question 16

When is the limaçon r = a + b sin θ a cardioid?

Accepted Answer

When a = b

Question 17

What is a conic section?

Accepted Answer

A curve formed where a plane intersects a double cone (a cone of two nappes)

Question 18

What is a nappe?

Accepted Answer

One half of a double cone

Question 19

Eccentricity of a circle, an ellipse, a parabola and a hyperbola?

Accepted Answer

Circle 0; ellipse less than 1; parabola 1; hyperbola greater than 1

Question 20

For a horizontal ellipse (x − h)²/a² + (y − k)²/b² = 1, where are the foci?

Accepted Answer

At (h ± c, k), with c² = a² − b²

Question 21

For a horizontal hyperbola (x − h)²/a² − (y − k)²/b² = 1, where are the foci?

Accepted Answer

At (h ± c, k), with c² = a² + b²

Question 22

Polar equation of a conic with eccentricity e and focal parameter p?

Accepted Answer

r = ep/(1 ± e cos θ) or r = ep/(1 ± e sin θ)

Calculus: Parametric Equations and Polar Coordinates flashcards

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