Fundamentals of Calculus

Introduction

• Three Main Areas of Calculus:
  • Limits
  • Derivatives
  • Integration

1. Limits

• Purpose: Helps evaluate a function as a variable approaches a value.
  • Useful when a function is undefined at a certain point.
• Example:
  • Function: ( f(x) = \frac{x^2 - 4}{x - 2} )
  • ( f(2) ) results in ( \frac{0}{0} ) (indeterminate)
  • Approach: Use limits to find value as ( x \rightarrow 2 ).
  • Result: ( \lim_{{x \to 2}} \frac{x^2 - 4}{x - 2} = 4 ).
  • Process: Factor to ((x + 2)(x - 2)), cancel terms, substitute ( x = 2 )._

2. Derivatives

• Purpose: Provides the slope of the tangent line to a function at a given point.
  • Useful for calculating rates of change.
• Basic Rule: Power rule for differentiation.
  • ( \frac{d}{dx} x^n = nx^{n-1} )
• Example:
  • ( f(x) = x^3 ), ( f'(x) = 3x^2 )
  • Find slope at ( x = 2 ): ( f'(2) = 12 ).
• Tangent vs. Secant Lines:
  • Tangent Line: Touches a curve at one point.
  • Secant Line: Cuts across a curve at two points.
• Connecting Limits and Derivatives:
  • Use limits to find derivative (e.g., slope of tangent).
  • Formula: ( \lim_{{x \to a}} \frac{f(x) - f(a)}{x - a} )_

3. Integration

• Purpose: Opposite of differentiation; finds accumulated value over time or area under a curve.
• Antiderivative Formula:
  • ( \int x^n dx = \frac{x^{n+1}}{n+1} + C )
• Example:
  • ( \int 4x^3 dx = x^4 + C )
• Comparing Derivatives and Integration:
  • Derivatives: Rate of change, slope calculation.
  • Integration: Accumulation, area calculation.

Application Examples

Instantaneous Rate of Change

• Function: ( A(t) = 0.01t^2 + 0.5t + 100 )
• Calculate changes at different times using derivative.
• Verify using secant lines for approximation.

Accumulation Over Time

• Function: ( R(t) = 0.5t + 20 )
• Calculate total accumulation from ( t = 20 ) to ( t = 100 ) using definite integral.
• Graphically represents as area under curve.

Conclusion

• Summary:
  • Limits evaluate functions as ( x \rightarrow a ).
  • Derivatives calculate instantaneous rates of change.
  • Integration determines accumulation over time by finding area under curves.
  • Understand basic ideas behind each concept for calculus foundation.
• Further Learning: Practice problems and additional resources available for deeper understanding.