Fundamentals of Calculus

Three Areas of Calculus:

1. Limits

   • Used to evaluate functions as they approach a certain value.
   • Example: To evaluate ( f(x) ) as ( x ) approaches 2, use limits if ( f(x) ) is undefined.
   • Useful for understanding behavior near a point.

2. Derivatives

   • Give the slope of the tangent line to a function at a point.
   • Represented as ( f'(x) ).
   • Useful for calculating rates of change.

3. Integration

   • Opposite of derivatives; finds the area under a curve.
   • Useful for calculating accumulation over time.
   • Derivatives and integrals are inverse processes.

Detailed Explanation of Concepts:

Limits

• Example Problem: Evaluate ( \frac{x^2 - 4}{x - 2} ) at ( x = 2
  • Direct substitution results in ( \frac{0}{0} ) (indeterminate).
  • As ( x ) approaches 2, function value approaches 4.
  • Factor using difference of squares: ( (x+2)(x-2) ).
  • Cancel ( (x-2) ) and use substitution.

Derivatives

• Power Rule: ( \frac{d}{dx}[x^n] = nx^{n-1} ).
• Slope of a tangent line:
  • Derivative provides slope at a specific ( x ).
  • Tangent line touches curve at one point, whereas secant line touches at two points.
• Example:
  • Find derivative of ( x^3 ): ( f'(x) = 3x^2 ).
  • Slope at ( x = 2 ) is 12.

Integration

• Anti-derivative: Process of finding original function from its derivative.
• Formula: Add 1 to power and divide.
• Example:
  • Integral of ( 4x^3 ) gives ( x^4 + C ).
• Uses:
  • Calculates total accumulation.
  • Example problem: Water tank volume change using definite integral.

Application and Examples:

Water Tank Example

• Given: Function ( a(t) = 0.01t^2 + 0.5t + 100 ).
• Calculations:
  • Evaluate at ( t = 0, 9, 10, 11, 20 ).
  • Find rate of change using derivative.
  • Use secant and tangent lines to approximate slope.

Integration Example

• Function: ( r(t) = 0.5t + 20 ).
• Calculate water accumulation from ( t = 20 ) to ( t = 100 ) using definite integral.
• Apply geometry to find area under curve: rectangle and triangle.

Key Takeaways:

• Limits provide value as ( x ) approaches a certain point.
• Derivatives measure instantaneous rate of change.
• Integrals measure total accumulation.
• Importance of understanding basic concepts in calculus for problem-solving.

Conclusion

• These concepts are fundamental in a calculus course.
• Practice by solving more problems on limits, derivatives, and integration to master these topics.
• Refer to additional resources for further study.