Multivariable 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

A wheel rolls without slipping along a straight line; what curve does a fixed point on its rim trace?

Accepted Answer

A cycloid

Question 3

What is a space-filling curve?

Accepted Answer

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

Question 4

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 5

In polar coordinates, what are x and y in terms of r and θ?

Accepted Answer

x = r cos θ and y = r sin θ

Question 6

What is the pole of the polar coordinate system?

Accepted Answer

Its center point, which plays the role of the Cartesian origin

Question 7

What does the radial coordinate r measure?

Accepted Answer

How far the point lies from the pole

Question 8

From which axis, and in which direction, is the angular coordinate θ measured?

Accepted Answer

From the positive x-axis (polar axis), counterclockwise

Question 9

Which curve has the polar equation r = a(1 + sin θ) or r = a(1 + cos θ)?

Accepted Answer

A cardioid

Question 10

What graph is r = a + b sin θ or r = a + b cos θ?

Accepted Answer

A limaçon (a cardioid when a = b)

Question 11

What is the graph of r = a cos 2θ or r = a sin 2θ, for a positive constant a?

Accepted Answer

A rose

Question 12

What integral gives the area of the polar region r = f(θ), α ≤ θ ≤ β?

Accepted Answer

A = (1/2) ∫ from α to β of [f(θ)]² dθ

Question 13

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 14

What is a conic section?

Accepted Answer

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

Question 15

What is a nappe?

Accepted Answer

One half of a double cone

Question 16

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

Accepted Answer

A parabola has one focus; an ellipse has two

Question 17

What is the eccentricity of a parabola?

Accepted Answer

Exactly 1

Question 18

What is the eccentricity of a circle?

Accepted Answer

0

Question 19

Is the eccentricity of a hyperbola less than, equal to, or greater than 1?

Accepted Answer

Greater than 1

Question 20

For the horizontal ellipse (x − h)²/a² + (y − k)²/b² = 1, how is c (the focal distance) found?

Accepted Answer

c² = a² − b², with foci at (h ± c, k)

Question 21

For the horizontal hyperbola (x − h)²/a² − (y − k)²/b² = 1, how is c (the focal distance) found?

Accepted Answer

c² = a² + b², with foci at (h ± c, k)

Question 22

If the discriminant 4AC − B² of a conic is negative, which conic is it?

Accepted Answer

A hyperbola (positive gives an ellipse, zero a parabola)

Multivariable Calculus: Parametric Equations and Polar Coordinates flashcards

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