Fundamentals of Calculus

Overview

• Calculus consists of three main areas:
  1. Limits
  2. Derivatives
  3. Integration

1. Limits

• Purpose: Evaluate functions as they approach a certain value.
• Example: When a function is undefined at x = 2, limits help predict behavior as x approaches 2.
• Process:
  • For example, with f(x) = (x² - 4)/(x - 2), direct evaluation at x=2 gives 0/0 (indeterminate).
  • As x approaches 2, f(x) approaches 4.
  • Factor expression to simplify and solve using limits.
• Key Concept: Limits allow the prediction of function behavior near undefined points.

2. Derivatives

• Purpose: Determine the slope of a function at a given point.
• Key Formula: Power rule - Derivative of xⁿ is n * xⁿ⁻¹.
• Example Problems:
  • Derivative of x² is 2x.
  • Derivative of x³ is 3x².
• Tangent vs. Secant Lines:
  • Tangent Line: Touches curve at one point, slope = derivative at that point.
  • Secant Line: Passes through two points on a curve.
• Example: Derivative of x³ is 3x², slope at x=2 is 12.
• Limits and Derivatives Connection: Use limits to evaluate the slope of tangent lines (using secant line approximations).*

3. Integration

• Purpose: Calculate the area under a curve (anti-differentiation).
• Key Formula: For xⁿ, integral is (xⁿ⁺¹)/(n+1) + C.
• Derivatives vs. Anti-Derivatives:
  • Derivatives: Indicate instantaneous rate of change.
  • Integration: Indicates total accumulation over time.
• Example: Determine water accumulation over time using definite integrals.
• Definite vs. Indefinite Integrals:
  • Definite Integrals: Include limits, result in a number.
  • Indefinite Integrals: Result in a function with a constant.

Application Examples

• Water Tank Problem:
  • Function: A(t) = 0.01t² + 0.5t + 100.
  • Tasks:
    • Calculate water amount at different times.
    • Determine rate of change at t=10 using derivatives.
    • Verify using secant line approximations.
• Integration Example:
  • Function: r(t) = 0.5t + 20.
  • Calculate water accumulation from t=20 to t=100 using definite integral.
  • Graph and understand area under the curve interpretation.

Summary

• Limits: Evaluate function behavior near specific points.
• Derivatives: Calculate instant rates of change and understand slope.
• Integration: Evaluate total accumulation and area under curves.

Conclusion

• Mastery of limits, derivatives, and integration forms the basis of calculus.
• Further practice recommended through additional problems and exercises.

Resources

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