Fundamentals of Calculus Lecture Notes

Overview

Purpose: Evaluate a function as x approaches a certain value.
Example Problem: Evaluate ( f(x) = \frac{x^2 - 4}{x - 2} ) at ( x = 2 ).
- Direct substitution gives ( \frac{0}{0} ), an indeterminate form.
- Use limits to find what the function approaches:
  - As x approaches 2, function approaches 4.
- Factor and simplify:
  - ( \frac{(x+2)(x-2)}{x-2} \rightarrow x+2 )
  - Substitute x = 2 to get limit = 4.
Key Point: Limits help understand behavior of functions at points where evaluation isn't straightforward.

Purpose: Calculate the rate of change or slope of the tangent line at a point.
Basic Rule: Power Rule
- Derivative of ( x^n ) is ( nx^{n-1} ).
- Examples:
  - ( x^2 \rightarrow 2x )
  - ( x^3 \rightarrow 3x^2 )
Understanding Tangent Lines:
- Tangent line touches a curve at one point; slope of tangent equals derivative at that point.
- Secant line: intersects at two points; used to approximate tangent slope.
Example: Find slope of tangent for ( f(x) = x^3 ) at ( x = 2 ).
- Derivative: ( f'(x) = 3x^2 )
- Calculate: ( f'(2) = 12 )
Limit and Derivative Connection:
- Use limits to find derivative: ( \lim_{x\to a} \frac{f(x) - f(a)}{x-a} )

Purpose: Opposite of differentiation, calculate accumulation (e.g., area under curve).
Example:
- Derivative of ( x^4 ) is ( 4x^3 ), thus integral of ( 4x^3 ) is ( x^4 + C ).
Integral Formula:
- ( \int x^n dx = \frac{x^{n+1}}{n+1} + C )
Comparison:
- Derivatives give instantaneous rate (slope), integration gives total accumulation (area).

Water Tank Problem:
- Function: ( a(t) = 0.01t^2 + 0.5t + 100 )
- Objective: Calculate water amount at various times.
- Rates: Calculate change rate using derivatives.
- Accumulation: Use integration to find water accumulation between ( t = 20 ) to ( t = 100 )
Use definite integrals for specific intervals, indefinite for generic antiderivatives.

Practice problems and video playlists for further learning and application of calculus concepts.