Fundamentals of Calculus Lecture Notes

Key Areas of Calculus

1. Limits

   • Evaluate what happens to a function as x approaches a certain value.
   • Useful for determining behavior of functions when direct evaluation fails.

2. Derivatives

   • Represent the slope of a function at a point, indicating the rate of change.
   • Denoted as f'(x), derived from f(x).
   • Calculated using the power rule: d/dx(x^n) = nx^(n-1).

3. Integration

   • Opposite of derivatives, used to find the area under a curve.
   • Represents accumulation over time.
   • Integral of f'(x) returns f(x).

Detailed Explanation of Concepts

Limits

• Example: Evaluating limit as x approaches 2 for ( f(x) = \frac{x^2 - 4}{x - 2} ):
  • Direct substitution gives indeterminate form 0/0.
  • Factor numerator using difference of squares: ((x + 2)(x - 2)).
  • Cancel common factors and evaluate limit using direct substitution: ( x + 2 |_{x=2} \Rightarrow 4 ).

Derivatives

• Tangent vs. Secant Lines
  • Tangent: Line touching curve at one point, slope given by derivative.
  • Secant: Line touching curve at two points, slope approximates derivative.
• Power Rule Applications:
  • (d/dx(x^2) = 2x)
  • (d/dx(x^3) = 3x^2)
• Example: Derivative (f(x) = x^3):
  • First derivative (f'(x) = 3x^2).
  • Tangent slope at (x=2) is (12).

Integration

• Equivalent to anti-differentiation.
• Example: Integral of (4x^3) returns (x^4 + C).
• Difference from Derivatives:
  • Derivatives: Determine rate by dividing y by x.
  • Integration: Determine area by multiplying y by x.

Applications

Example: Water in a Tank

• Function: ( a(t)=0.01t^2+0.5t+100 ) (gallons in a tank).
• Evaluate at several times to find water volume.
• Derivatives for Rate of Change:
  • Find ( a'(t) = 0.02t + 0.5 ).
  • At (t=10), rate of change is (0.7 \text{ gallons/minute}).
• Integration for Accumulation:
  • ( r(t) = 0.5t + 20 ): Find total accumulation from (t=20) to (t=100) using integration.
  • Integrate (r(t)) to find (4000 \text{ gallons accumulated}).

Summary

• Limits allow function evaluation as x approaches a certain value.
• Derivatives provide instantaneous rate of change and slope of tangent lines.
• Integration calculates total accumulation by finding area under curves.
• Understanding these concepts is foundational to calculus.