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 limit of a variable. The limit of a function. Theorems on limits. Limits of polynomials.
The definition of "a function continuous at a value of x."
The definition of "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.
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. Velocity and rates of change
Instantaneous velocity.
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
Exponential growth. The system of natural logarithms. The general power rule.
15. Evaluating e
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