Calculus Midterm 4 Flashcards
Polar Coordinate System
The location of a point described by its distance from a fixed point and angular direction
What is the fixed point called?
Pole
Polar Axis
Ray emanating from pole
Rectangular to coordinate equations
x = rcos(ϴ)
y = rsin(ϴ)
Polar to rectangular coordinates
r^2 = x^2 + y^2
tan(ϴ) = y/x
Equation for line in a polar function
ϴ = k
Equation for circles in a polar function
Centered at
Origin: r = k
x-axis: r = kcos(ϴ)
y-axis: r = ksin(ϴ)
Calculator Function for graphing points of intersections polar
Polar
Simul
Graph functions
Adjust window settings
When do polar functions intersect?
ONLY if the ϴ and r are the same at that point
How to find points of intersection in polar functions
Set two equations equal to each other and solve for ϴ
sin(ϴ) = 1/2
(1/2, pi/6), (1/2, 5pi/6)
Derivative of Polar Functions: dr/dϴ
The rate of change of radius as the angle changes
Derivative of Polar Functions: dy/dx
Slope of the polar curve in the xy-plane
Derivative of a polar function
How to find dy/dx of r = f(ϴ)
- find rectangular coordinates (x and y)
- Replace r with f(ϴ)
- find dx/dϴ and dy/dϴ
4/ find dy/dx
Arc Length
dL = sqrt((dx/dt)^2 + (dy/dt)^2) dt
Arc Length if ϴ is a parametric
r^2 + (dr/dϴ)^2
Area of a polar curve
dA = 1/2r^2dϴ
Distance vs. Displacement (definition)
Distance: total path covered by the objection at a time frame
Displacement: The total change in initial and final position
Distance vs. Displacement (use)
Distance:
- d
- positive always
- No direction
- Scalar
Displacement:
- s
- +, -, or zero (left or right of initial point)
- has direction vector
Distance formula
d = ∫b->a |v(t)| dt
Displacement formula
s = ∫b->a v(t) dt
Derivative of parametric functions
dy/dx = dy/dt / dx/dt
Find the parametric equations of the direct line segment from (1,2) to (3,5)
first coordinate of one pair * (1-t) + first coordinate of other pair (t)
Find Parametric equations of the line y - 4 = 1/2 (x-3)
x(t) = at + b
y(t) = ct + d
Slope = (c/a)
Point (b, d)
Find the parametric functions of a circle with the radius (r)
Origin is the center
x(t) = rcos(t)
y(t) = rsin(t)
(h,k) is the center
x(t) = rcos(t) + h
y(t) = rsin(t) + k
Find the parametric functions of an ellipse with the radius (r)
x(t) = acos(t)
y(t) = bsin(t)
Thinking about an ellipse like…
A circle with different y and x radii
Ellipse equation
(x/a)^2 + (x/b)^2 = 1
Component Form
v = ai + bj
<a, b>
i = x-axis
j = y-axis
Length aka norm aka magnitude of a vector
||v|| = sqrt(a^2 + b^2)
Unit Vector
v = <a,b>
u = 1/||v|| * <a, b>
Dot Product
“multiply vectors”
v = <v1, v2>
w = <w1, w2>
v * w = ||V||||w||cos(ϴ)
gives a visual
OR
v * w =
v1w1 + v2w2
gives a way to calculate it
what is ϴ in the dot product?
The angle between the two vectors
Orthogonal Projections
Decomposes vectors into the sum of the two orthogonal components
||b|| (length of b in orthogonal vector)
(dot product of v and w / ||v||^2) * v
Parametric equations for r(t) = <f(t), g(t)>
x(t) = f(t)
y(t) = g(t)
How to graph parametric equations in the calculator
- Parametric mode
- Plug in
- Choose a good window and t-step and t-interval
*it will show the path but not the vectors
Find the limit of r(t) if r(t) = <x(t), y(t)>
find the limits of each
r(t) is continuous at t=a if
lim r(t) = r(a)
t->a
Sin(2ϴ)
2sin(ϴ)cos(ϴ)
dy/dx with r and ϴ
dy/dϴ over dx/dϴ = dr/dϴ sin(ϴ) + rcos(ϴ) over dr/dϴ cos(ϴ) - rsin(ϴ)
