“Never Believe The Math Teacher”

# An Approach to Calculus

## 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.

• #### 14. Derivatives of exponential and logarithmic functions

• The system of natural logarithms.
• The general power rule.