Fundamentals of Calculus

Overview

• Three main areas of calculus: Limits, Derivatives, Integration.
• Limits help evaluate functions as x approaches a certain value.
• Derivatives provide the slope of a function at some value.
• Integration is the opposite of derivatives and finds the area under a curve.

Limits

• Purpose: Evaluate a function as x approaches a specific value.
• Example:
  • Function: ( f(x) = \frac{x^2 - 4}{x - 2} )
  • Direct calculation at x = 2 gives indeterminate form (0/0).
  • Use limits: ( \lim_{{x \to 2}} \frac{x^2 - 4}{x - 2} = 4 )
  • Factor and simplify ( x^2 - 4 ) to ((x+2)(x-2)).
  • Conclusion: As ( x \to 2, f(x) \to 4 )._

Derivatives

• Definition: Function that gives the slope of the original function.
• Power Rule: Derivative of ( x^n = n \times x^{n-1} ).
  • Examples:
    • ( \frac{d}{dx}x^2 = 2x )
    • ( \frac{d}{dx}x^3 = 3x^2 )
    • Tangent line: Line that touches the curve at one point.
    • Secant line: Line touching the curve at two points.
• Finding the Slope:
  • Slope of tangent line at ( x = 2 ) for ( f(x) = x^3 ) is 12 (calculated using derivative ( f'(2) )).
  • Use limits and secant line approximation for slope.

Integration

• Definition: Process of finding the antiderivative or the integral.
• Integration Formula:
  • Add 1 to the exponent and divide by the new exponent.
  • Add a constant of integration.
• Relation to Derivatives:
  • Integration is the reverse process of differentiation.
• Example:
  • Integral of ( 4x^3 ) is ( x^4 + C ).
• Practical Application: Calculate accumulated quantity over time.
  • Example: Water tank problem calculated using definite integral for net water change.

Comparison: Derivatives vs. Integration

• Derivatives:
  • Calculate instantaneous rate of change or slope of tangent line.
  • Division of y by x values.
• Integration:
  • Calculate accumulated quantity or area under the curve.
  • Multiplication of y by x values.

Application Examples

• Rate of Change Application:
  • Function ( A(t) = 0.01t^2 + 0.5t + 100 ) for water in a tank.
  • Calculate rate of change at ( t = 10 ) using derivatives.
  • Derivative ( A'(t) = 0.02t + 0.5 ) shows rate of increase of water.
• Accumulation Application:
  • Function ( R(t) = 0.5t + 20 ) for water flow rate.
  • Calculate accumulated water from ( t = 20 ) to ( t = 100 ) using integration.
  • Evaluate definite integral from 20 to 100 for net change.

Summary

• Limits: Allow evaluation of function at approaching values.
• Derivatives: Provide slope and rate of change.
• Integration: Computes accumulated changes and area under curves.
• Understand the basic concepts and applications of limits, derivatives, and integration for calculus.