Fundamentals of Calculus Lecture Notes

Overview

This lecture covers three main areas of calculus: limits, derivatives, and integration.

Limits

• Purpose: Evaluate the behavior of a function as it approaches a certain value.
  • Example: Evaluate (f(x)) as (x) approaches 2 when (f(2)) is undefined or indeterminate.
• Concept: If (f(x) = \frac{x^2 - 4}{x - 2}), direct substitution at (x = 2) results in 0/0, an indeterminate form.
• Finding Limits:
  • Factor and simplify the function.
  • Use direct substitution after simplification.
  • Example: ( \lim_{{x \to 2}} \frac{x^2 - 4}{x - 2} = \lim_{{x \to 2}} \frac{(x + 2)(x - 2)}{x - 2} = \lim_{{x \to 2}} (x + 2) = 4 )

Derivatives

• Purpose: Determine the slope of a function at a specific point.
• Definition: Derivative of (f(x)) is (f'(x)).
• Power Rule: (\frac{d}{dx} [x^n] = nx^{n-1})
  • Examples:
    • (\frac{d}{dx} [x^2] = 2x)
    • (\frac{d}{dx} [x^3] = 3x^2)
• Tangent vs. Secant Lines:
  • Tangent line: Touches the curve at one point.
  • Secant line: Touches the curve at two points.
  • Slope of tangent line: ( m = f'(x) )
• Example: Given (f(x) = x^3), find (f'(2))
  • (f'(x) = 3x^2)
  • (f'(2) = 3(2^2) = 12)

Integration

• Purpose: Measure the area under a curve; find the accumulation over time.
• Relationship to Derivatives: Integration is the reverse process of differentiation.
• Antiderivative: (\int 4x^3 , dx = x^4 + C)
  • General formula: (\int x^n , dx = \frac{x^{n+1}}{n+1} + C)
• Definite vs. Indefinite Integrals:
  • Definite Integral: Provides a numerical value representing the area under the curve between two points (e.g., ( \int_{20}^{100} (0.5t + 20) , dt )).
  • Indefinite Integral: Provides a function (e.g., ( \int (0.5t + 20) , dt = 0.25t^2 + 20t + C )).

Example Problems

1. Instantaneous Rate of Change:
   • Function: (a(t) = 0.01t^2 + 0.5t + 100)
   • Derivative: (a'(t) = 0.02t + 0.5)
   • At (t = 10): (a'(10) = 0.7) gallons per minute.
2. Accumulation of Water:
   • Rate function: (R(t) = 0.5t + 20)
   • Find total volume from (t = 20) to (t = 100):
   • (\int_{20}^{100} (0.5t + 20) , dt = 4000 ) gallons.

Summary

• Limits: Help evaluate functions as (x) approaches a value.
• Derivatives: Provide instantaneous rates of change and slopes of tangent lines.
• Integration: Finds area under the curve and measures accumulation.
• Key Concepts:
  • Limits: Evaluated using factoring and simplification.
  • Derivatives: Found using rules like the power rule.
  • Integration: Opposite of differentiation, used to find total accumulation or area under the curve.