Fundamentals of Calculus

Overview

• Three key areas of calculus: Limits, Derivatives, Integration

1. Limits

• Definition: Limits help evaluate a function as it approaches a certain value.
• Example: Evaluating f(2) for f(x) = (x² - 4) / (x - 2).
  • f(2) = 0/0 (indeterminate).
  • Use limits to find behavior as x approaches 2.
  • Example Values:
    • f(2.1) = 4.1
    • f(2.01) = 4.01
• Limit Calculation:
  • Factor f(x): x² - 4 = (x + 2)(x - 2)
  • Cancel (x - 2):
    • limit as x approaches 2 of (x + 2) = 4
• Conclusion: Limit exists even though f(2) does not.

2. Derivatives

• Definition: Derivatives represent the slope of the original function at a certain value.
  • Notation: f'(x) for the derivative of f(x).
• Power Rule:
  • If f(x) = xⁿ, then f'(x) = n*xⁿ⁻¹.
  • Examples:
    • f(x) = x² → f'(x) = 2x
    • f(x) = x³ → f'(x) = 3x²
    • f(x) = x⁴ → f'(x) = 4x³
• Tangent Line:
  • A line that touches the curve at one point.
  • Slope of the tangent line = f'(x) at that point.
• Secant Line:
  • Touches the curve at two points.
  • Slope: (y2 - y1) / (x2 - x1)
• Example Calculation:
  • For f(x) = x³, find the slope at x = 2:
    • f'(2) = 12
  • Slope indicates that the curve increases by 12 for each unit increase in x.
• Using Limits to Find Derivative:
  • Limit as x approaches a value of (f(x) - f(a)) / (x - a)
  • Related to the slope of the tangent line.

3. Integration

• Definition: The process of finding the area under the curve or the antiderivative of a function.
• Relationship to Derivatives: Integration is the opposite of differentiation.
  • Example: If f'(x) = 4x³, then ∫4x³ dx = x⁴ + C.
• Key Formula:
  • ∫f'(x) dx = f(x) + C
  • Add constant C since the derivative of a constant is 0.
• Comparison of Derivatives and Integrals:
  • Derivatives ➔ Slope of the tangent line (instantaneous rate of change).
  • Integrals ➔ Area under the curve (accumulation over time).

Practical Applications

• Example Problem:
  • Function: a(t) = 0.01t² + 0.5t + 100 (gallons of water in tank over time t).
  • Calculate values at specific times (t=0, 9, 10, 11, 20).
  • Determine rate of change at t=10 using derivatives.
• Integration Example:
  • Rate function: r(t) = 0.5t + 20.
  • Integrate over interval [20, 100] to find total accumulation of water in the tank over time.

Summary

• Limits: Evaluate behavior of functions as they approach certain values.
• Derivatives: Calculate instantaneous rates of change and slopes of tangent lines.
• Integration: Determine total accumulation or area under curves.

Additional Resources

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• Links in description for more practice problems.