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Exponenetial and Logarithmic Functions
Chapter Four
§4.1 Exponential Function (Review)
Exponential function: 指数函数 if b is a positive number other than 1 (b>0, b≠1), there is a unique function called the exponential function with base b that is defined by
f(x)=bx for every real number x
yx bb
§4.1 The natural Exponential Base
Definition: The natural exponential function is
Where
n 10 100 1000 10,000 100,000
2.59374 2.70481 2.71692 2.711815 2.718271(1 )nn
§4.1 Continuous Compounding of Interest
If P is the initial investment (the principal) and r is the interest rate (expressed as a decimal), the balance B after the interest is added will be
B=P+Pr=P(1+r) dollars
to be continued
§4.1 Continuous Compounding of Interest
§4.1 Present value
§4.1 Exponential Growth and Decay
§4.2 Logarithmic Function (对数函数 )
§4.2 Graphs of Logarithmic Function
§4.2 The Natural Logarithm
§4.2 Doubling Time
§4.2 Half Time
§4.3 Differentiation of Logarithmic and Exponential Function
to be continued
to be continued
§4.3 Differentiation of Exponential Function
Differentiate both sides of the equation
to be continued
§4.3 Exponential Growth and Decay
§4.3 Logarithmic Differentiation
Taking the derivatives of some complicated functions can be simplified by using logarithms. This is called logarithmic differentiation.
to be continued
§4.3 Logarithmic Differentiation
The relative rate of change of a quantity Q(x) can be computed by finding the derivative of lnQ.
'( )(ln )( )
d Q xQdx Q x
§4.4 Additional Exponential Models
Curve Sketching:
to be continued
2 min
-------- ++++++( x0
to be continued
to be continued
0
max
++++++ --------x
-------- ++++++
-1 Inf
1Inf
Sign of ++++++x
§4.4 Optimal Holding time
§4.4 Optimal Holding time
to be continued
§4.4 Learning Curve
Learning Curve
§4.4 Learning Curve