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An Approach

to

CALCULUS

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The Lessons

11.  Continuous versus discrete

The definition of a "continuous" quantity.

12.  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.

13.  Continuous functions

The definition of "a function is continuous at a value of x."

14.  Infinity ()

The definition of "a variable becomes infinite."
Limits of rational functions.

15.  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.

16.  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.

17.  The chain rule

The derivative of a function of a function.

18.  More rules for derivatives

The quotient rule.
Implicit differentiation.
The derivative of inverse functions.

19.  Instantaneous velocity and Related rates

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?

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