The Lessons

1. Continuous versus discrete

The definition of a “continuous” quantity.


2. Limits
 A sequence of rational numbers.
 The definition of the limit of a variable.
 The limit of a function.
 Theorems on limits. Limits of polynomials.

3. Continuous functions
 The definition of “a function is continuous at a value of x.”

4. Infinity (∞)
 The definition of “a variable becomes infinite.”
 Limits of rational functions.

5.The Derivative
 The slope of a tangent line to a curve.
 The difference quotient and the definition of the derivative.
 Notations for the derivative.
 The equation of a tangent to a curve.

6. Rules for derivatives

 The derivative of a constant. The derivative of y = x.
 The product rule.
 The power rule.
 The derivative of the square root function.

7. The chain rule
 The derivative of “a function of a function.”

8. More rules for derivatives
 The quotient rule.
 Implicit differentiation.
 The derivative of inverse functions

9. Instantaneous velocity & Related rates
 The second derivative

10. Maximum and minimum values
 The turning points of a graph. Critical values.

11. Applications of maximum and minimum values

12. Derivatives of trigonometric functions

13.Derivatives of inverse trigonometric functions

14. Derivatives of exponential and logarithmic functions
 The system of natural logarithms.
 The general power rule.

15.Evaluating e
Appendix 1. Are the real numbers really numbers?
Appendix 2. Is a line really composed of points?