Calculus Definitions

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Calculus Definitions
2013-12-23 12:47:47
Limits Derivitives Optimization Curve sketching

All about Calculus :)
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  1. Limit
    The process of getting closer to a value.
  2. Split Function
    When different parts of the domain have different functions.
  3. 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.
  4. Limit at infinity
    As x approaches infinity f(x) approaches a value=HA or infinity=DNE
  5. 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)
  6. Tangent
    A line that most resembles a graph at point P.
  7. Secant
    A line that crosses two points on the graph.
  8. The Derivitive Function
    The slope of a tangent
  9. 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
  10. Implicit Differentiation
    Differentiating both sides with respect to x when y is embedded in the equation.
  11. Composition
    The process of combining two functions to create a new one.
  12. Decomposition
    The process of identifying two functions such that they create the given function through composition.
  13. Euler's Number
    The special number that makes its derivitive when x is 0 = one. 

  14. Natural logarithm
    The logarithm with base e. Written ln(x). Its function is also the inverse of y=e^x
  15. 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.
  16. Absolute Min
    if f(c) _< f(x) fo any x value in the domain of f(x), then f(c) is called absolute minimum.
  17. 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
  18. Fermat's Thereum
    if f(x) has a local max or min at c, then either f'(x)=0 or f'(x)=undefined.
  19. Local max/min
    When f(c)_> or _< f(x) in the neighbourhood of a value.
  20. 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.
  21. 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.
  22. Point of Inflection
    When the function changes concavity at x=a.
  23. Notation for Derivitive
    • 1. dy/dx
    • 2.y'
    • 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)
  24. Notation for second derivitive
    • 1. f''(x)
    • 2. y''
    • 3. d2y/dx2
  25. Notation for Composite Functions
    • 1. f(g(x))
    • 2. fog
    • 3. fog(x)