Unit 7
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Lesson 1: Exponential Functions and Base e
Definition of exponential function (指数)
f(x)=a⋅b^x, where b>0, b≠1 and a∈R
Transformations:
f(x)=a⋅b^((x−h)+k
Base e, meaning of e
Modeling with Exponentials.
A(t)=P(1+r/n)^nt
Lesson 2: Graphing Logarithmic Functions Base e
Definition of logarithmic functions (对数)
Logarithmic form
Exponential form
Common log (base 10) vs. natural log (base e)
Transformations
Lesson 3: Property of Logs and Solving Log Equations
Laws of Logarithms
Product rule
Quotient rule
Power rule
Inverse rule
Change of base rule
Solving exponential and logarithmic equations
Equate exponents or equate logs with same base
Rewrite as an exponent or rewrite as a log
Factor or solving quadratic equations
Take log of both sides or combine exponents
Lesson 4: Investigating derivatives using first principles
The slope (斜率) of a curve at a point
Average rate of change
Instantaneous rate of change
Slope function, or derivative (导数) function
Method of first principle
Power conjecture
Lesson 5: Determining the equation of a tangent line (切线) using the derivative
Finding equation of the tangent line
Given f(x) and x = x0, first find f'(x0) as slope, and (x, f(x)) as a point on the line
How to find the equation of a line, from a point and a slope
Lesson 6: Modeling with Exponential, logarithmic and logistic function
Exponential growth
Exponential decay
Gaussian model
Natural logarithm model
Common logarithm model
Logistic growth