unit 9 Flashcards
1
Q
Parametric derivatives
A
dy/dx = y’(t)/x’(t)
2
Q
Parametric second derivatives
A
d^2y/dx^2 = (dy/dx)’/x’(t)
3
Q
Arc length
A
∫ √ (x’(t)^2 + y’(t)^2)
4
Q
Vectors
A
- x and y are independent of each other
- Take derivatives and integrals separately.
- to find direction of vector, use trig and right triangles
5
Q
X in a polar equation
A
r * cos(θ)
6
Q
Y in a polar equation
A
r * sin(θ)
7
Q
Polar derivatives
A
dy/dx = y’(θ)/x’(θ)
8
Q
Area of a polar equation
A
A = 0.5 * ∫ r^2 dθ
find alpha and beta by setting r=0
9
Q
Equation of a limacon
A
r = a+bcos(θ) horizontal
r = a+bsin(θ) vertical
10
Q
Graphing limacons
A
- amplitude = |a| + |b|
- amplitude of smaller loop: |a| - |b|
- if b is negative, graph is greatest when it is negative
11
Q
Shape of limacon
A
- a/b < 1 : Inner loop
- a/b = 1 : carotid
- a/b > 1 : dimple
- a/b > 2 : convex
12
Q
Equation of a rose curve
A
r = acos(nθ) : petal is split in half
r = asin(nθ) : petal is offset
13
Q
Amount of petals in a rose curve
A
- n petals if n is odd
- 2n petals if n is even
14
Q
Lemniscates equation (infinity sign)
A
r^2 = a^2 sin(2θ) : diagonal
r^2 = a^2 cos(2θ) : horizontal
15
Q
Area between 2 polars
A
A = 0.5 * ∫ (r2)^2 - (r1)^2 dθ
- find alpha and beta by setting the equations equal to each other
