Fundamentals of Calculus

Three Areas of Calculus

1. Limits

   • Evaluate a function as it approaches a certain value.
   • Useful for functions that are undefined at a particular point.

2. Derivatives

   • Determine the slope of a function at a particular value.
   • Represented as f'(x).
   • Useful for calculating rates of change.

3. Integration

   • Opposite process of differentiation.
   • Determine the area under a curve.
   • Calculated as the integral of f'(x), which is f(x).

Limits

• Example: f(x) = (x² - 4) / (x - 2)
  • Evaluating at x = 2 gives 0/0, indeterminate.
  • Use limits to see behavior as x approaches 2.
  • Factor and simplify to find the limit = 4.

Derivatives

• Definition: Provides slope of tangent line to the curve.
• Power Rule: Derivative of xⁿ is n * xⁿ⁻¹.
  • Example: Derivative of x² is 2x.
• Tangent vs. Secant Line
  • Tangent: touches at one point, slope = derivative.
  • Secant: touches at two points, slope calculated traditionally.
• Example: f(x) = x³
  • f'(x) = 3x²
  • Slope at x = 2 is 12.*

Calculating Derivatives via Limits

• Use limits to find slope: limit as x approaches a value.
• Factor and simplify expressions to use limits.
• Differentiation tells rate of change.

Integration

• Process of finding the anti-derivative.
  • Example: Integral of 4x³ gives x⁴ (plus constant).
• Used to find accumulation over time.
• Definite vs. Indefinite Integrals
  • Definite: Has limits, gives numerical result.
  • Indefinite: No limits, gives function.

Example Problem

• Function for water in tank: A(t) = 0.01t² + 0.5t + 100
• Calculate amount at t = 0, 9, 10, 11, 20.
  • Use integration for accumulation over time.

Rates of Change

• Derivative tells how fast water changes in tank.
• Integrals accumulate water over time.

Graphical Representation

• Integration finds area under curve.
• Divide region into geometric shapes (rectangle, triangle) to calculate area.

Summary

• Limits: Evaluate function behavior as it approaches a value.
• Derivatives: Calculate instantaneous rate of change.
• Integration: Determine accumulation over time by finding area under the curve.

Conclusion

• These concepts are fundamental in calculus.
• Recommended to check additional resources for practice and deeper understanding.