When different parts of the domain have different functions.
Infinite Limit
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
Continuous Function
When x=a passes through the graph with no interruptions.
1. f(a) exists
2. limf(x) as x approaches a = f(a)
Tangent
A line that most resembles a graph at point P.
Secant
A line that crosses two points on the graph.
The Derivitive Function
The slope of a tangent
Differentiability
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...
1. Discontinuities
2. Vertical tangent
2. Sudden change in slope
Implicit Differentiation
Differentiating both sides with respect to x when y is embedded in the equation.
Composition
The process of combining two functions to create a new one.
Decomposition
The process of identifying two functions such that they create the given function through composition.
Euler's Number
The special number that makes its derivitive when x is 0 = one.
e=2.71828182
Natural logarithm
The logarithm with base e. Written ln(x). Its function is also the inverse of y=e^x
Logarithmic Differentiation
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.
Absolute Min
if f(c) _< f(x) fo any x value in the domain of f(x), then f(c) is called absolute minimum.
Absolute Max
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
Fermat's Thereum
if f(x) has a local max or min at c, then either f'(x)=0 or f'(x)=undefined.
Local max/min
When f(c)_> or _< f(x) in the neighbourhood of a value.
Concave up
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.
Concave Down
The graph y=f(x) is called Concave Down on the interval (a,b) if the graph lies below its tangents on the interval.