Fundamentals of Calculus Lecture Notes

Overview

• Three main areas of calculus:
  1. Limits
  2. Derivatives
  3. Integration

1. Limits

• Purpose: Evaluate functions as they approach a certain value.
• Example: To evaluate ( f(x) ) as ( x ) approaches 2 for ( f(x) = \frac{x^2 - 4}{x - 2} ).
  • Direct substitution leads to indeterminate form ( \frac{0}{0} ).
  • Use factoring to simplify and evaluate limit:
    • Factor to ( (x + 2)(x - 2) ), cancel ( x - 2 ).
    • ( \lim_{{x \to 2}} (x + 2) = 4 ).
• Application: Predicts function behavior near specific points._

2. Derivatives

• Concept: Derivatives represent the slope of a function at a given point.
  • Example: ( f'(x) ) for ( f(x) = x^n ) is ( nx^{n-1} ).
  • Power Rule: Derivative of ( x^n ) is ( nx^{n-1} ).
• Tangent and Secant Lines:
  • Tangent Line: Touches curve at one point; derivative gives slope.
  • Secant Line: Touches curve at two points; used to approximate tangent slope.
• Example: For ( f(x) = x^3 ), ( f'(x) = 3x^2 ).
  • Calculate the slope at ( x=2 ): ( f'(2) = 12 ).
  • Use secant line for approximation: ( \frac{f(3) - f(1)}{3-1} = 13 ).

3. Integration

• Concept: Opposite of differentiation, calculates area under the curve.
  • Antiderivative: Process of finding the original function.
  • Example: Integral of ( 4x^3 ) returns ( x^4 + C ).
• Application: Determines accumulation over time.
  • Example Problem: Calculate water accumulation in a tank:
    • Given ( A(t) = 0.01t^2 + 0.5t + 100 ).
    • Evaluate at different times for the amount of water.
    • Derivative finds the rate of change at ( t=10 ).
    • Use integration for total accumulation from ( t=20 ) to ( t=100 ).

Summary

• Limits: Understand approach to value.
• Derivatives: Calculate rate of change, slope of tangent line.
• Integration: Find total accumulation, area under the curve.

Additional Resources

• Consider exploring more problems and specific topics in calculus through additional video playlists and practice problems linked in the description.