Precalculus A Fuctional Approach to Graphing and Problem Solving

Section 4.4: Exponential Functions
Essential Ideas
  • Squeeze Theorem
    Suppose b is a real number greater than 1. Then, for any real number x, there is a unique real number b x. Moreover, if h and k are any two rational numbers such that h < x < k , then
                                   bh < b x < b k

  • Extended Laws of Exponents
    Let a and b be real numbers and let P and Q be positive real numbers except that the form 00 and division by zero are excluded.
           First law (Additive):                  b pbq = b p+q
           Second law (Subtractive):     b p/bqbp-q
           Third law (Multiplicative):     (bq)p = b pq
           Fourth law (Distributive):     (ab)p = apbp
           Fifth law (Distributive):        (a/b)p = ap/bq

  • The function f is an exponential function if
                                        f (x) = b x
    where b is a positive constant other than 1 and x is any real number. The number x is called the exponent and b is called the base.

  • See the directory of curves for the graph of the exponential function.

  • Compound Interest: A = P(1 + r /n)nt
    Continuous Compounding: A = Pert
    where e is the natural base or Euler's number. That is, as n increases without bound, the number e is the irrational number that is the limiting value of the formula (1 + 1/n)n.

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© 2011 Karl J. Smith. All rights reserved.