1. Increasing and decreasing functions and extrema
Some terminology
The derivative test for monotonicity and extrema
The proof of Fermat's theorem


2. Understanding the mean value theorem
Rolle's theorem and the mean value theorem


3. Determining concavity and extrema
The second derivative and extrema
The proof of the second derivative test for local extrema


4. Drawing the graph of a function


5. Using maxima and minima in real applications
Optimization
Applications to economics


6. The importance of inverse functions
Inverse trigonometric functions


7. Using the derivative of an inverse function
The general expression
The derivatives of inverse trigonometric functions
The proof of theorem 1 (optional)


8. Applying the natural exponential function and the natural logarithm
The natural logarithm


9. Exponential functions with arbitrary bases
Logarithmic functions with arbitrary bases
Arbitrary powers of x


10. Orders of magnitude in exponential functions
Logarithmic growth
The natural exponential function as a limit of polynomials


11. Using exponential functions in growth and decay rates
The solution of the differential equation y = ky
Compound interest


12. Introduction to hyperbolic and inverse
Hyperbolic functions
Inverse hyperbolic functions


13. Using L'Hopital's rule for indeterminate forms
The indeterminate form 0/0
The indeterminate form [infinity] / [infinity]
The indeterminate form 0 / [infinity]
The indeterminate forms 1[infinity], [infinity]0 and 00
The indeterminate form [infinity] - [infinity]


Index.