The process of getting closer to a value.
When different parts of the domain have different functions.
When f(x) approaches infinity was x gets closer to a value. When the limit as x approaches a = infinity there is a VA at x=a and the limit does not exist.
Limit at infinity
As x approaches infinity f(x) approaches a value=HA or infinity=DNE
When x=a passes through the graph with no interruptions.
1. f(a) exists
2. limf(x) as x approaches a = f(a)
A line that most resembles a graph at point P.
A line that crosses two points on the graph.
The Derivitive Function
The slope of a tangent
The ability to find the derivitive.
f(x) is differentiable if..
1. f'(a) exists
2. f(x) is continuous at x=a
f(x) is not differentiable at...
2. Vertical tangent
2. Sudden change in slope
Differentiating both sides with respect to x when y is embedded in the equation.
The process of combining two functions to create a new one.
The process of identifying two functions such that they create the given function through composition.
The special number that makes its derivitive when x is 0 = one.
The logarithm with base e. Written ln(x). Its function is also the inverse of y=e^x
When there is a function that is like x^x that is non polynomial, non exponential, apply ln to each side and move the exponent to the front then do Implicit Differentiation.
if f(c) _< f(x) fo any x value in the domain of f(x), then f(c) is called absolute minimum.
if f(c) _> f(x) for any x value in the domain of f(x), than f(c) is called absolute maximum. It is usually within a closed interval
if f(x) has a local max or min at c, then either f'(x)=0 or f'(x)=undefined.
When f(c)_> or _< f(x) in the neighbourhood of a value.
The graph of y=f(x) is called Concave Up on the interval (a,b) if the graph lies above its tangents on the interval.
The graph y=f(x) is called Concave Down on the interval (a,b) if the graph lies below its tangents on the interval.
Point of Inflection
When the function changes concavity at x=a.
Notation for Derivitive
3. y'= slope of tangent/@ (a, f(a)) to y=f(x)
4. f'(x) = roc @ x= a
Notation for second derivitive
Notation for Composite Functions
All about Calculus :)