Fundamentals of Calculus

Overview of Calculus Areas

1. Limits

   • Helps evaluate a function near a certain point.
   • Allows understanding of what happens as x approaches a certain value, even if the function is undefined at that point.

2. Derivatives

   • Provide the slope of the original function at a specific point.
   • The derivative of f(x) is represented as f'(x).
   • Useful for calculating rates of change.

3. Integration

   • The opposite of differentiation.
   • Finds the area under the curve, useful for calculating accumulation over time.
   • Derivative and integral relationship:
     • If f'(x) = slope, then ∫f'(x)dx = f(x) + C (the antiderivative).

Detailed Breakdown

Limits

• Example: Evaluate f(x) = (x² - 4) / (x - 2) at x = 2.
  • Direct substitution gives 0/0 (indeterminate).
  • Use limits:
    • As x approaches 2, f(x) approaches 4.
    • Limit notation: ( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4 ).
  • Factorization aids in evaluating the limit.

Derivatives

• Power Rule: If f(x) = x^n, then f'(x) = n*x^(n-1).
  • Example:
    • For f(x) = x², f'(x) = 2x.
    • For f(x) = x³, f'(x) = 3x².
• Tangent vs. Secant Lines:
  • Tangent lines touch a curve at one point; secant lines touch at two points.
  • Slope of tangent line = derivative at that point.
  • Example:
    • For f(x) = x³, slope at x = 2 is f'(2) = 12.
• Estimating Tangent Slope:
  • Use secant line slopes approaching the tangent point.
  • Example: Estimate slope near x = 2 using points like (1.9, 2.1).

Integration

• Anti-Differentiation: The process of finding the integral.
  • Example:
    • To find the integral of 4x³, apply anti-differentiation rules:
      • ∫4x³dx = (4/4)x⁴ + C = x⁴ + C.
• Definite vs. Indefinite Integrals:
  • Definite integrals have limits and yield a number; indefinite integrals do not.
  • Example problem:
    • Calculate water amount over time using integral of rate function.

Application Example

• Water amount represented by: a(t) = 0.01t² + 0.5t + 100.
  • Calculate values at specific times (t=0, 9, 10, 11, 20).
  • Derivative a'(t) helps find the instantaneous rate of change, especially at t=10.
• Use definite integral to calculate accumulation from t=20 to t=100.

Summary

• Limits help evaluate function behavior.
• Derivatives indicate instantaneous rates of change (slope of tangent line).
• Integration determines accumulation or area under curves (anti-differentiation).

Additional Resources

• Links for practice problems on limits, derivatives, and integration will be available in the video description.