Fundamentals of Calculus

Overview

• The lecture covers three key areas of calculus:
  1. Limits
  2. Derivatives
  3. Integration

1. Limits

• Definition: Limits help evaluate a function as it approaches a particular value.
• Example: Evaluating the limit as x approaches 2 for the function f(x) = (x² - 4) / (x - 2)
  • Direct substitution leads to indeterminate form 0/0.
  • Calculate limits by approaching the value:
    • f(2.1) = 4.1
    • f(2.01) = 4.01
  • As x approaches 2, f(x) approaches 4.
  • Limit Expression:
    • Factor the function: f(x) = (x + 2)(x - 2)/(x - 2)
    • Limit as x approaches 2 of (x + 2) = 4.

2. Derivatives

• Definition: Derivatives represent the slope of the original function at a given value.
• Notation: The derivative of f(x) is f'(x).
• Power Rule:
  • For f(x) = xⁿ, f'(x) = n*xⁿ⁻¹.
• Example: Finding the derivative of x², x³, and x⁴:
  • f'(x²) = 2x, f'(x³) = 3x², f'(x⁴) = 4x³.
• Slope of Tangent Line:
  • A tangent line touches the curve at one point; a secant line touches at two points.
  • Example: For f(x) = x³, f'(2) = 3(2)² = 12 (slope at x = 2).
• Using Secant Lines to Approximate Tangents:
  • Use points close to x = 2, such as 1.9 and 2.1, to calculate the average slope.
• Limit Expression for Derivative:
  • f'(2) = lim (h -> 0) [f(2 + h) - f(2)] / h.

3. Integration

• Definition: Integration is the reverse process of differentiation, finding the area under the curve.
• Antiderivative:
  • If f'(x) = 4x³, then ∫4x³dx = x⁴ + C (adding constant of integration).
• Key Differences:
  • Derivatives give instantaneous rates of change; integration calculates overall accumulation.
  • Derivatives: divide y by x; Integration: multiply y by x.

Example Problem

• Given function A(t) = 0.01t² + 0.5t + 100, find water in tank at specific times:
  • A(0) = 100, A(9) = 105.31, A(10) = 106, A(11) = 106.71, A(20) = 114.
• Rate of Change: Find A'(10).
  • A'(t) = 0.02t + 0.5; A'(10) = 0.7 gallons/min.

Another Example

• Rate Function: R(t) = 0.5t + 20, calculate accumulation from t = 20 to t = 100:
  • Use definite integral: ∫R(t)dt from 20 to 100.
  • Calculate areas using rectangles and triangles to find total accumulation.

Conclusion

• Recap:
  • Limits help evaluate functions at certain values.
  • Derivatives give slopes of tangent lines and rates of change.
  • Integration determines accumulated values over time.
• Encouragement to practice with additional resources linked in the description.