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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...
- 1. Discontinuities
- 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
- 1. dy/dx
- 3. y'= slope of tangent/@ (a, f(a)) to y=f(x)
- 4. f'(x) = roc @ x= a
- 5. df(x)/x
- 6. Dxf(x)
Notation for second derivitive
- 1. f''(x)
- 2. y''
- 3. d2y/dx2
Notation for Composite Functions
- 1. f(g(x))
- 2. fog
- 3. fog(x)