• Shuffle
    Toggle On
    Toggle Off
  • Alphabetize
    Toggle On
    Toggle Off
  • Front First
    Toggle On
    Toggle Off
  • Both Sides
    Toggle On
    Toggle Off
  • Read
    Toggle On
    Toggle Off
Reading...
Front

Card Range To Study

through

image

Play button

image

Play button

image

Progress

1/39

Click to flip

Use LEFT and RIGHT arrow keys to navigate between flashcards;

Use UP and DOWN arrow keys to flip the card;

H to show hint;

A reads text to speech;

39 Cards in this Set

  • Front
  • Back
Fractal
A dimension that is non-integer
"Smooth"
C^1
i.e. Both the function and its first derivative are differentiable and continuous.
C^k denotes all derivatives up until the kth derivative are differentiable/continuous.
Benoit Mandelbrot / Mandelbrot Set
Created the Mandelbrot Set (includes all the Julia sets). Discovered fractals. Worked at IBM.
f(z)=z^2+c is the Mandelbrot set.
Helge von Koch
Discovered the snowflake curve. Triangle in which each side is elongated to form a new triangle, and so on. Mathematically infinite perimeter, conceptually finite area.
Gaston Julia
Discovered the Julia sets (parts of the Mandelbrot set). Each worked on the idea of recursion: that the output of the function would be reused as the input, an infinite number of times.
Georg Cantor
Discovered the cantor set. Starts with a line segment, the middle third of which is removed. The middle third of the two new segments are removed and so on. No matter how far in you look, it is always the same pattern.
Formula for energy requirement based on mass
km^(3/4) where m = mass of any living organism. (Significant because it supports the principle that bodies are produced with fractal coding, allowing larger animals to need proportionally less energy)
Metaknowledge
An awareness of what you know and what you don't know.
Product Rule
if y=uv, and u&v are differentiable.
y'=u'v+uv'
Quotient Rule
if y=u/v, and u&v are differentiable/v does not equal 0.
y'=(u'v-uv')/v^2
What is dy of 3x^2 + 7?
6xdx
This is an example of remembering to include dx if the question is asking for dy and not dy/dx
d/dx e^u
e^u du/dx
Name three functions that are their own derivative (or better yet one all encompassing function).
e^x, -e^x, and 0
(Bonus points for Se^x where S is a parameter)
sin'u
cosu du/dx
cos'u
-sinu du/dx
tan'u
sec^2 u du/dx
cot'u
-csc^2 u du/dx
sec'u
secutanu du/dx
csc'u
-cscucotu du/dx
If y = f(x) what is f^-1(x)? (f inverse)
x = f(y)
(vice versa also true)
Parameter
An adjustable constant
Weierstrass Function
Continuous everywhere, but differentiable nowhere.
Slope Field
"Directory" of how family curves are changing.
Lattice Points
Ordered pairs that are determined by integers.
Diffeq.
Differential Equation: any equation that includes 1 or more derivatives.
d/dx sin^-1 u
1/√(1-u^2) du/dx
d/dx cos^-1 u
-1/√(1-u^2) du/dx
d/dx tan^-1 u
1/(1+u^2) du/dx
What is antiderivative notation?
∫f(x)dx = antiderivative of f(x)
(Ex. ∫7/(x^2+1) = 7tan^-1 x + C)
Parametric Chain Rule
dy/dx = dy/dt ÷ dx/dt
Derivative of the Inverse of a Function
If y=f^-1 x
d/dx(f^-1 x)=1/(f'(f^-1(x)))
Implicit Differentiation
Differentiate both sides and set the equal to each other. For dy/dx, treat y as a function and write its derivative as y'. Isolate and solve for y'.
What is the relationship between continuity and differentiability?
Continuity is necessary but not sufficient for differentiability.
Write j'(p) in all other forms.
j'(p) = dj/dp = d/dp(j) = lim of ∆j/∆p as ∆p approaches 0 = [j(p)]' = Dp j
Artifacts
Anything left behind as a result of some other process.
What are the two types of diffeq's?
ODEs (ordinary diffeq's) which are used in calculus and PDEs (partial diffeq's) which are used in HLAVC.
What is the notation to find rel. min/max?
y' set = 0
(set is written above =)
What is the linearization of f(x) at x=c?
y≈f(c)+f'(c)(x-c) or, equivalently, y≈f(c)+f'(c)dx
Define the differentials dx and dy.
dx=∆x
dy=f'(x)dx