Chapter Nine: Parametric Equations and Polar Coordinates

Question 1

what parametric equations describe a cycloid?

Answer 1

x(𝛉) = 𝛉 - sin(𝛉) y(𝛉) = 1 - cos(𝛉)

Question 2

what is dy/dx of a parametric curve?

Answer 2

dy/dx = dy/dt ÷ dx/dt, where dx/dt ≠ 0

Question 3

what is the arc length of a parametric curve?

Answer 3

L = ∫_a^b √(dx/dt)² + (dy/dt)² dt

Question 4

how do you convert from cartesian to polar coordinates?

Answer 4

r = √(x² + y²)

𝛉 = tan^-1(y/x)

Question 5

how do you convert from polar to cartesian coordinates?

Answer 5

x = rcos𝛉

y = rsin𝛉

Question 6

what is r = asin𝛉?

Answer 6

a circle centered at (0, a/2) with a radius a/2

Question 7

what is r = acos𝛉?

Answer 7

a circle centered at (a/2, 0) with radius a/2

Question 8

what is r = cos(n𝛉)?

Answer 8

a rose curve

if n is odd there are n petals

if n is even there are 2n petals

the first petal is always centered on the positive x-axis

Question 9

what is r = a + bcos𝛉 when a = b?

Answer 9

a cardioid

Question 10

what is r = a + bcos𝛉 when a < b?

Answer 10

a limaçon

if a < b = limaçon with inner loop

if b < a < 2b = dimpled limaçon

if a ≥ 2b = convex limaçon

Question 11

what is r² = a²sin(2𝛉)?

Answer 11

a lemniscate in quadrants I and III

Question 12

what is r² = a²cos(2𝛉)?

Answer 12

a lemniscate centered on the x-axis

Question 13

what is dy/dx of a polar curve?

Answer 13

dy/dx = (f(𝛉)cos𝛉 + f’(𝛉)sin𝛉) ÷ (-f(𝛉)sin𝛉 + f’(𝛉)cos𝛉)

or

dy/d𝛉 ÷ dx/d𝛉

Question 14

what is the area under a polar curve?

Answer 14

A = ∫_a^b 1/2(f(𝛉))² d𝛉

a and b are angles!

Question 15

what is the area between two polar curves?

Answer 15

A = 1/2∫_a^b (f(𝛉)² - g(𝛉)²) d𝛉

f is the farther curve

g is the nearer curve

Question 16

what is the distance between the petals of a rose curve?

Answer 16

2π ÷ number of petals

remember in n𝛉:

if n is odd there are n number of petals

if n is even there are 2n number of petals