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