Derivatives

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					Derivatives
Overriding principal is the concept of change (speed, weather, populations,..)

   Average speed
           An object is dropped from a 500 ft building. From physics we know that the
           distance above the ground is d (t )  500  16 t 2 . Sketch a graph of distance
           versus time. Find time until object hits the ground. Compute average speed
           on the interval [a, b] using
                                                d (b)  d (a)
                                        v ave 
                                                    ba
           Geometrically the average speed is the slope of the secant line.

   I.      Instantaneous velocity
           Greeks realized that at any given instant even a moving object is frozen.
           Newton and his contemporaries gave up on the notion of speed at an instant
           and replaced it with average speeds over shorter and shorter intervals.
           Compute instantaneous speed at the time t  a using
                                                     d (t )  d (a)
                                        vinst  lim                 .
                                                t a      ta
           Geometrically the instantaneous speed is the slope of the tangent line.

   II.     The derivative at a point
           Can apply the same process to any function. The instantaneous rate of change
           of y  f (x) at the point x  a is defined by
                                              f ( x)  f ( a )
                             f ' (a)  lim                     or letting h  x  a
                                       x a        xa
                                                         f ( a  h)  f ( a )
                                        f ' (a)  lim
                                                    h 0          h

   III. The derivative as a function
           We can apply the process of finding the derivative to many different x values
           and in fact for every x value there is a corresponding value for the derivative.
           For any function y  f (x) we define the derivative function
                                                f ( x  h)  f ( x )
                                f ' ( x)  lim
                                           h 0          h
     There are several equivalent notations for the derivative function including
                                         dy df
                       f ' ( x), y' ( x), , , Dx f ( x), Dx y
                                         dx dx

IV. Interpretations of the derivative
     As a slope and as a rate of change.
                                                 dC
     Ex. Suppose C is cost and t is time. Then      is the rate of change of cost.
                                                 dt

V.   Graphical derivatives
     Function           Derivative
     Increasing         Positive
     Decreasing         Negative
     Constant           Zero
     Concave up         Increasing
     Concave down       Decreasing
     Inflection point   Max/Min
     Smooth max/min     Zero
     HA (x=a)           HA (x=a)
     VA (y=a)           VA (y=0)
     Sharp corner       undefined

VI. Algebraic shortcuts for computing derivatives
     Table of known derivatives
      Power rule
         d n
            x  nx n 1
         dx
      Trigonometric functions
         d
            sin x  cos x
         dx
         d
            cos x   sin x
         dx
         d
            tan x  sec2 x
         dx
         d
            cot x   csc2 x
         dx
         d
            sec x  sec x tan x
         dx
         d
            csc x   csc x cot x
         dx
    Combinations
     Sums, Differences, Products, Quotients
       d
           f ( x)  s ( x)  d f ( x)  d s ( x)
       dx                          dx            dx
       d
           f ( x)  s ( x)  d f ( x)  d s ( x)
       dx                          dx            dx
       d
           f ( x)  s ( x)  f ( x)  d s ( x)  s ( x)  d f ( x) d
       dx                               dx                  dx
                                    d                  d
                          d ( x )  n( x )  n( x )  d ( x )
       d  n( x )                  dx                dx
           d ( x)  
       dx                              d ( x)  2


     Chain rule
        d
            f ( g ( x))  f ' ( g ( x))  g ' ( x)
       dx
     Implicit differentiation

VII. Focus on theory
     Limits
     Intuitive definition
     lim f ( x)  L means that as x gets close to a , f (x) gets close to L .
     x a
      Existence of limits (limits can fail to exist if there are small jumps,
      unbounded behavior or excessive oscillations)
      Rigorous definition
     lim f ( x)  L means that for every   0 there exists a   0 such that
     x a

      f ( x)  L   whenever x  a   .

     Continuity
     Intuitively a function is continuous if you can sketch its graph without lifting
     your pencil. From a rigorous perspective a function is continuous at x  a if
     (i) f (a) is defined
     (ii) lim f ( x) exists
            xa

     (iii) lim f ( x)  f (a)
            x a



     Differentiability
     A function is differentiability if it possesses the property of local linearity.
     Differentiability  continuity
     Continuity  differentiability
                                                                         f (b)  f (a)
     Differentiable functions have the Mean value property f ' (c) 
                                                                             ba
           for some value of c  (a, b) . This means there is a point where the slope of
           the tangent line is equal to the slope of the secant line or that there is point
           where the instantaneous rate of change is equal to the average rate of change.

   VIII. Homework assignment

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