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Chapter 4 Review Important Terms_ Symbols_ and Concepts

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					        Chapter 11 Review
Important Terms, Symbols, Concepts

 11.1. The Constant e and Continuous
  Compound Interest
    The number e is defined as either one of the limits
                                               n
                       1
                                                   e  lim 1  s 
                                                                      1
            e  lim 1                                                  s
                n    n                             s0


        If the number of compounding periods in one year is
         increased without limit, we obtain the compound interest
         formula A = Pert, where P = principal, r = annual interest
         rate compounded continuously, t = time in years, and A =
         amount at time t.
Barnett/Ziegler/Byleen Business Calculus 11e                                  1
                       Chapter 11 Review

 11.2. Derivatives of Exponential and
  Logarithmic Functions
    For b > 0, b  1 d x       d x
                         e e x
                                   b  b x ln b
                      dx        dx

                                        d         1
                                           ln x      d
                                                         logb x 
                                                                   1 1
                                                                      ( )
                                        dx        x   dx          ln b x
        The change of base formulas allow conversion from base e
         to any base b > 0, b  1: bx = ex ln b, logb x = ln x/ln b.


Barnett/Ziegler/Byleen Business Calculus 11e                                2
                       Chapter 11 Review

 11.3. Derivatives of Products and Quotients
    Product Rule: If f (x) = F(x)  S(x), then
                                           dS dF
                            f ' ( x)  F        S
                                           dx dx
        Quotient Rule: If f (x) = T (x) / B(x), then
                                 B ( x)  T ' ( x)  T ( x)  B ' ( x)
                      f ' ( x) 
                                              [ B ( x)] 2
 11.4. Chain Rule
    If m(x) = f [g(x)], then m’(x) = f ’[g(x)] g’(x)

Barnett/Ziegler/Byleen Business Calculus 11e                             3
                       Chapter 11 Review

 11.4. Chain Rule (continued)
    A special case of the chain rule is the general power rule:

                       d
                           f x  n  n  f x  n 1  f ' ( x)
                       dx
        Other special cases of the chain rule are the following
         general derivative rules:
       d                   1                         d f ( x)
          ln [ f ( x)]          f ' ( x)              e      e f ( x ) f ' ( x)
       dx                f ( x)                      dx


Barnett/Ziegler/Byleen Business Calculus 11e                                         4
                       Chapter 11 Review

 11.5. Implicit Differentiation
    If y = y(x) is a function defined by an equation of the form
     F(x, y) = 0, we can use implicit differentiation to find y’
     in terms of x, y.
 11.6. Related Rates
    If x and y represent quantities that are changing with
     respect to time and are related by an equation of the form
     F(x, y) = 0, then implicit differentiation produces an
     equation that relates x, y, dy/dt and dx/dt. Problems of this
     type are called related rates problems.

Barnett/Ziegler/Byleen Business Calculus 11e                     5
                       Chapter 11 Review

  11.7. Elasticity of Demand
     The relative rate of change, or the logarithmic derivative,
      of a function f (x) is f ’(x) / f (x), and the percentage rate
      of change is 100  (f ’(x) / f (x).
     If price and demand are related by x = f (p), then the
      elasticity of demand is given by
                          p  f ' ( p)    relative rate of change of demand
           E ( p)                    
                            f ( p)          relative rate of change of price
          Demand is inelastic if 0 < E(p) < 1. (Demand is not
           sensitive to changes in price). Demand is elastic if
           E(p) > 1. (Demand is sensitive to changes in price).
Barnett/Ziegler/Byleen Business Calculus 11e                                   6

				
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