Lecture Notes: Integration and Antiderivatives

Introduction

• Integration & Antiderivative Connection: The lecture begins by reconnecting integration to antiderivatives.
• Definitions:
  • Antiderivative: Previously introduced concept.
  • Integration: Involves finding the limit of a Riemann sum.

Fundamental Theorem of Calculus - Part 1

• Continuous Function Requirement: If function ( f ) is continuous on interval ([a, b]).
• Defining the Function: Capital ( F(x) ) as the definite integral from ( a ) to ( x ) of ( f(t) dt ).
• Theorem Statement:
  • ( F(x) ) is continuous and differentiable.
  • ( F'(x) = f(x) ) (Derivative of ( F ) yields original function ( f )).
• Differentiation and Integration as Inverses
  • Operations of differentiation and integration cancel each other out.
  • Example: Derivative of ( \int_{a}^{x} e^{t} \cos(t) dt ) results in ( e^{x} \cos(x) ).

Analyzing Function Behavior

• Example Function: ( F(x) = \int_{0}^{x} \cos(t) dt ).
• Finding Local Extrema:
  • Derivative ( F'(x) = \cos(x) ).
  • Critical Points: Where ( \cos(x) = 0 ) (( \pi/2, 3\pi/2 )).
  • Behavior around critical points determined by test points.

Intervals of Increase/Decrease and Concavity

• Example: ( F(x) = \int_{1}^{x} (t^2 - 7t + 10) dt ).
• Derivative Analysis:
  • Derivative ( F'(x) = x^2 - 7x + 10 ).
  • Critical Points: ( x = 2, 5 ).
  • Second Derivative: ( F''(x) = 2x - 7 ).
  • Concavity and increase/decrease behavior determined using test points.

Applying the Fundamental Theorem

• Tangent Line for Function:
  • Derivative ( F'(x) = 3x^2 - x ).
  • Example Tangent Line: ( y - 0 = 10(x - 2) ).
• General Case (Chain Rule Implication):
  • Upper limit as a function: ( F'(x) = f(u(x)) \times u'(x) ).

Second Part of Fundamental Theorem

• Evaluation of Definite Integrals: Develops a method to evaluate using antiderivatives.
• Example Calculation:
  • Define function: ( F(x) = \int_{2}^{x} (e^{t} + 2t) dt ).
  • Antiderivative leads to simplified evaluation.

Conclusion

• Differentiation and integration are inverse operations.
• Connection between antiderivatives and definite integrals.
• Anticipation of the second part of the fundamental theorem to simplify calculating definite integrals.