Chapter 10 Review Flashcards
Polar Coordinates
used to plot points on a sphere
(r,θ)
r - distance from the pole
θ - angle measured counterclockwise from the polar axis
How to name 4 polar representations
Take first coordinate
Add or subtract 2pi
Add or subtract pi from both the original and the new
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)
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)
Horizontal Line
y = b
r = b / sin θ
OR
r = b * cscθ
Vertical Line
x = a
r = a / cosθ
oR
r = a * secθ
Line Through the Pole
θ = k
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
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
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
Cardioid
r = a ± acosθ
on x axis
r = a ± asinθ
on y axis
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
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
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 2π
Polar Area
one half the integral of theta 1 to theta 2 of r squared dθ
Slope of a polar curve
dy/dx = (d/dθ)(rsinθ) / (d/dθ)(rcosθ)
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
Steps to finding both horizontal and vertical tangent lines
- derivative
- set top equal to zero for horiz, set bottom equal to zero for vert.
- if the same answer for both, use a limit to find out
- plug each value into either y=rsinθ or x=rcosθ`
Finding tangent lines at the pole
- set function equal to zero
2. plug in values to tanθ
Length of a curve
L = integral of theta 1 to theta 2 of root r squared plus (dr/dθ) squared dθ
