Chapter 10 Review Flashcards Preview

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Flashcards in Chapter 10 Review Deck (21):
1

Polar Coordinates

used to plot points on a sphere

2

(r,θ)

r - distance from the pole
θ - angle measured counterclockwise from the polar axis

3

How to name 4 polar representations

Take first coordinate
Add or subtract 2pi
Add or subtract pi from both the original and the new

4

changing polar coordinates to cartesian coordinates

*plug in (r,θ) into each of these equations to find (x,y)
x = rcosθ
y = rsinθ

r^2 = x^2 + y^2
θ = inverse tan (y/x)

5

Changing cartesian coordinates to polar coordinates

*find r by plugging into r^2 = x^2 + y^2 and θ by plugging into inverse tan
r^2 = x^2 + y^2
θ = inverse tan (y/x)

6

Horizontal Line

y = b
r = b / sin θ
OR
r = b * cscθ

7

Vertical Line

x = a
r = a / cosθ
oR
r = a * secθ

8

Line Through the Pole

θ = k

9

Sin Coordinate values

θ sin θ

0 0
π/6 1/2
π/2 1
3π/2 -1
11π/6 -1/2

*symmetrical over y-axis

10

Cos Coordinate Values

θ cos θ

0 1
π/3 1/2
π/2 0
2π/3 -1/2
π -1

*symmetrical over x-axis

11

Circles

r = ± 2a cosθ
circle on the x-axis
through pole , radius = a

r = ± 2a sinθ
circle on the y-axis, through pole, radius = a

12

Cardioid

r = a ± acosθ
on x axis

r = a ± asinθ
on y axis

13

Limacon

r = a ± bcosθ
r = a ± bsinθ

If abs. value of....
a/b is greater than or equal to 2, then: convex limacon
1 < abs value a/b < 2: dimpled limacon
a/b < 1 then: limacon with inner loop
a/b = 1 : cardioid

14

Rose

r = acos(nθ) r = asin(nθ)
# petals: n if n is odd, 2n if n is even
length of petal: a
first petal: set r = absvalue a and solve for θ
next petal: +2π/# petals

15

Spiral of Archimedes

r = aθ (spiral out) r = a/θ (spiral in)
Use: θ±π

θ r

0
π/4
π/2
3π/4
π
5π/4
3π/2
7π/4

16

Polar Area

one half the integral of theta 1 to theta 2 of r squared dθ

17

Slope of a polar curve

dy/dx = (d/dθ)(rsinθ) / (d/dθ)(rcosθ)

18

Tangent lines of Polar Curves

horizontal when dy/dθ = 0 and dx/dθ does not = 0
Vertical (vice versa)
If BOTH equal 0, use limits to determine it is is horizontal or vertical

19

Steps to finding both horizontal and vertical tangent lines

1. derivative
2. set top equal to zero for horiz, set bottom equal to zero for vert.
3. if the same answer for both, use a limit to find out
4. plug each value into either y=rsinθ or x=rcosθ`

20

Finding tangent lines at the pole

1. set function equal to zero
2. plug in values to tanθ

21

Length of a curve

L = integral of theta 1 to theta 2 of root r squared plus (dr/dθ) squared dθ