Definitions of Integral and Function
Function
A function is a relationship or rule that assigns to each element from one set (the domain) exactly one element from another set (the codomain or range).
Key Properties:
Input-Output Relationship: For every input \(x\) in the domain, there is exactly one output \(f(x)\).
Notation: Written as \(f: X \to Y\), where \(X\) is the domain and \(Y\) is the codomain.
Examples:
Linear function: \(f(x) = 2x + 3\)
Quadratic function: \(f(x) = x^2\)
Trigonometric function: \(f(x) = \sin(x)\)
Integral
An integral represents the accumulation of quantities (e.g., areas under curves) and is the inverse operation of derivatives.
Types of Integrals:
Indefinite Integral:
Represents a family of functions (includes \(+C\)).
Notation: \(\int f(x) \, dx = F(x) + C\), where \(F(x)\) is the antiderivative of \(f(x)\).
Example: \(\int 2x \, dx = x^2 + C\).
Definite Integral:
Computes net accumulation over \([a, b]\).
Notation: \(\int_a^b f(x) \, dx\).
Example: \(\int_0^2 2x \, dx = 4\) (area under \(2x\) from 0 to 2).
Applications:
Area/volume calculations.
Work, displacement, and solving differential equations.
Relationship (Fundamental Theorem of Calculus):
\(\frac{d}{dx} \left( \int f(x) \, dx \right) = f(x)\).
Difference Definition of the Derivative (Using df)
The derivative of a function \(f\) at a point \(x_0\) can be expressed using the difference quotient with differential notation:
Where:
\(df\) = infinitesimal change in \(f\)
\(dx\) = infinitesimal change in \(x\)
\(\Delta x\) = finite change in \(x\)
Integral Form (Fundamental Theorem of Calculus)
For an integral \(F(x) = \int f(x) \, dx\), the derivative is:
Notes
Subscripts (like \(x_0\)) are rendered correctly.
Greek letters (\(\Delta\), \(\delta\)) are supported.
Limits and fractions are properly formatted.