AP 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 written t

Question 2

What is the orientation of a parametric curve?

Accepted Answer

The direction a point moves along the graph as the parameter increases

Question 3

What curve does a point on a wheel's rim trace as the wheel rolls without slipping along a line?

Accepted Answer

A cycloid

Question 4

What is a space-filling curve?

Accepted Answer

A curve that fills up an entire two-dimensional region of the plane

Question 5

What is the formula for dy/dx of a curve x = x(t), y = y(t)?

Accepted Answer

dy/dx = (dy/dt)/(dx/dt) = y′(t)/x′(t)

Question 6

What is the second derivative d²y/dx² for parametric equations?

Accepted Answer

d²y/dx² = [(d/dt)(dy/dx)] / (dx/dt)

Question 7

What is the area under a parametric curve x = x(t), y = y(t) for a ≤ t ≤ b?

Accepted Answer

A = ∫ from a to b of y(t) x′(t) dt

Question 8

What is the 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 9

What is the surface area when a parametric curve is revolved around the x-axis?

Accepted Answer

S = 2π ∫ from a to b of y(t) √((x′(t))² + (y′(t))²) dt

Question 10

What is the surface area when a parametric curve is revolved around the y-axis?

Accepted Answer

S = 2π ∫ from a to b of x(t) √((x′(t))² + (y′(t))²) dt

Question 11

What does the radial coordinate r measure?

Accepted Answer

How far a point is from the pole

Question 12

What does the angular coordinate θ measure?

Accepted Answer

The angle, measured counterclockwise, between the positive x-axis and the segment from the origin to the point

Question 13

What is the pole in the polar coordinate system?

Accepted Answer

The center of the polar coordinate system, matching the Cartesian origin

Question 14

How do you convert polar coordinates (r, θ) to x and y?

Accepted Answer

x = r cos θ and y = r sin θ

Question 15

How do you find r and tan θ from rectangular coordinates (x, y)?

Accepted Answer

r = √(x² + y²) and tan θ = y/x

Question 16

What is the area of the region 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 17

What is the 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 18

What is the graph of r = a + b sin θ (or a + b cos θ) called?

Accepted Answer

A limaçon (a cardioid when a = b)

Question 19

What is a conic section?

Accepted Answer

Any curve formed where a plane intersects a cone of two nappes

Question 20

What is a nappe?

Accepted Answer

One half of a double cone

Question 21

How many foci does a parabola have, and how many does an ellipse have?

Accepted Answer

A parabola has one; an ellipse has two.

Question 22

How many directrices does a hyperbola have?

Accepted Answer

Two

Question 23

What is the eccentricity of a parabola?

Accepted Answer

Exactly 1 (an ellipse's is less than 1, a hyperbola's is greater than 1)

Question 24

What is the eccentricity of a circle?

Accepted Answer

0

Question 25

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

Accepted Answer

At (h ± c, k), where c² = a² + b²

AP Calculus: Parametric Equations and Polar Coordinates flashcards

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