Fundamentals of Calculus

Overview

• The lecture covers three fundamental areas of calculus:
  • Limits
  • Derivatives
  • Integration

1. Limits

• Definition: Limits help evaluate a function as a variable approaches a specific value, even if the function is undefined at that point.
• Example: Evaluating ( f(2) ) for ( f(x) = \frac{x^2 - 4}{x - 2} )
  • Direct substitution gives ( \frac{0}{0} ) (indeterminate).
  • Limit evaluation shows that as ( x ) approaches 2, ( f(x) ) approaches 4.
• Finding Limits:
  • Factor the function when possible.
  • Example using difference of squares:
    • ( x^2 - 4 = (x + 2)(x - 2) )
  • Cancelling out factors leads to direct substitution to find the limit.

2. Derivatives

• Definition: Derivatives represent the slope of a function at a specific point, indicating the rate of change.
• Notation: ( f' ) (read as "f prime") for the derivative of ( f(x) ).
• Power Rule for Derivatives:
  • Derivative of ( x^n ) is ( n x^{n-1} ).
  • Examples:
    • ( f(x) = x^2 ) => ( f'(x) = 2x )
    • ( f(x) = x^3 ) => ( f'(x) = 3x^2 )
• Tangent vs. Secant Lines:
  • Tangent Line: touches the curve at one point (slope = derivative).
  • Secant Line: touches the curve at two points (average rate of change).
• Example of Evaluating Derivative:
  • For ( f(x) = x^3 ), find the slope of the tangent at ( x = 2 ):
    • ( f'(2) = 3(2^2) = 12 )
  • The slope indicates that for every unit increase in ( x ), ( y ) increases by 12.

3. Integration

• Definition: Integration is the opposite of differentiation, finding the area under a curve (anti-derivative).
• Notation: The integral of ( f'(x) ) gives back ( f(x) ) plus a constant of integration (C).
• Finding Integrals:
  • To integrate ( 4x^3 ), use the formula: add 1 to the exponent, divide by the new exponent, and add C.
  • Example: Integral of ( 4x^3 = x^4 + C )
• Comparison of Derivatives and Integration:
  • Derivatives calculate the instantaneous rate of change (slope).
  • Integrations calculate accumulation (area under the curve).
  • Division vs. Multiplication: Differentiation involves dividing (rise/run) while integration involves multiplying (base * height).

Applications of Calculus

• Example Problem: Finding the amount of water in a tank over time uses both derivative and integral concepts.
  • Function: ( A(t) = 0.01t^2 + 0.5t + 100 )
  • Evaluate amount of water at various times.
  • Derivative gives the rate of change of water volume.
• Definite Integrals: Used to find total accumulation over an interval.
  • Integrating ( r(t) = 0.5t + 20 ) from ( t = 20 ) to ( t = 100 ).
  • Calculate area under the curve using geometric shapes.

Conclusion

• Key Points to Remember:
  • Limits help evaluate functions as they approach certain values.
  • Derivatives provide slopes and rates of change at points.
  • Integration calculates total accumulation over intervals.
• Further Resources: Links in the description for additional practice problems and calculus topics.