Calculus Overview and Differential Concepts

Branch of mathematics dealing with rates of change (differential calculus) and accumulation of quantities (integral calculus).
Fundamental in understanding changes in physical systems, economics, biology, and many other fields.

Concept: Focuses on the concept of the derivative, which represents the rate of change of a function.
Key Principles:
- Derivative: The limit of the average rate of change of a function as the interval approaches zero.
- Notation: Commonly denoted as f'(x) or dy/dx.
- Rules:
  - Power Rule: d/dx (x^n) = n*x^(n-1)
  - Product Rule: d/dx (uv) = u'v + uv'
  - Quotient Rule: d/dx (u/v) = (u'v - uv')/v^2
  - Chain Rule: d/dx (f(g(x))) = f'(g(x)) * g'(x)
Applications:
- Finding tangents to curves.
- Solving optimization problems (maxima and minima).

Concept: Focuses on the integral, which represents the accumulation of quantities and the area under curves.
Key Principles:
- Definite Integral: Represents the net area under a curve between two points a and b, denoted as ∫[a,b] f(x) dx.
- Indefinite Integral: Represents a family of functions and includes a constant of integration, denoted as ∫ f(x) dx = F(x) + C.
Fundamental Theorem of Calculus:
- Connects differentiation and integration.
- Part 1: If F is an antiderivative of f on [a, b], then ∫[a,b] f(x) dx = F(b) - F(a).
- Part 2: States that the derivative of the integral of a function is the function itself: d/dx ∫[a,x] f(t) dt = f(x).
Techniques of Integration:
- Substitution Method.
- Integration by Parts.
- Partial Fractions.
- Numerical Integration (Trapezoidal Rule, Simpson's Rule).

Fundamental concept for both derivatives and integrals.
Definition: The value that a function approaches as the input approaches some value.
Properties:
- Limit of a sum: lim (f(x) + g(x)) = lim f(x) + lim g(x).
- Limit of a product: lim (f(x) * g(x)) = lim f(x) * lim g(x).
- Limits involving infinity: Can determine horizontal and vertical asymptotes of functions.

Convergence: A sequence or series converges if it approaches a specific value as more terms are added.
Common Series:
- Geometric Series: ∑ ar^n converges if |r| < 1.
- Taylor Series: Represents functions as infinite sums of terms calculated from the values of their derivatives at a single point.

A mathematical branch concerned with rates of change and accumulation of quantities.
Essential for analyzing changes in various fields such as physics, economics, and biology.

Derivative Concept: Measures the rate of change of a function.
Key Principles:
- Derivative: Defined as the limit of the average rate of change as the interval narrows to zero.
- Common Notation: f'(x) or dy/dx.
- Rules:
  - Power Rule: d/dx (x^n) = n*x^(n-1).
  - Product Rule: d/dx (uv) = u'v + uv'.
  - Quotient Rule: d/dx (u/v) = (u'v - uv')/v^2.
  - Chain Rule: d/dx (f(g(x))) = f'(g(x)) * g'(x).
Applications:
- Used to compute tangents to curves and solve optimization problems to find maxima and minima.

Integral Concept: Focuses on the accumulation of quantities and calculating areas under curves.
Key Principles:
- Definite Integral: Represents net area under a curve from point a to b, written as ∫[a,b] f(x) dx.
- Indefinite Integral: Represents a family of functions, includes a constant of integration, formatted as ∫ f(x) dx = F(x) + C.
- Fundamental Theorem of Calculus:
  - Part 1: Relates integration to antiderivatives: ∫[a,b] f(x) dx = F(b) - F(a).
  - Part 2: Derivative of the integral of a function equals the function itself: d/dx ∫[a,x] f(t) dt = f(x).
Techniques of Integration:
- Methods include substitution, integration by parts, partial fractions, and numerical techniques like the Trapezoidal and Simpson's rules.

A core concept essential for understanding derivatives and integrals.
Definition: Values a function approaches as the input nears a certain value.
Properties:
- Limit of a sum: lim (f(x) + g(x)) = lim f(x) + lim g(x).
- Limit of a product: lim (f(x) * g(x)) = lim f(x) * lim g(x).
- Useful for determining horizontal and vertical asymptotes in functions.

Convergence: A series or sequence converges when it approaches a specific value as additional terms are added.
Common Series:
- Geometric Series: ∑ ar^n converges if the absolute value of r is less than 1.
- Taylor Series: Represents functions as infinite sums derived from their derivatives at a point.

Physics: Critical in analyzing motion, forces, and energy calculations.
Engineering: Utilized in the design and analysis of structures and system optimization.
Biology: Models population dynamics and growth rates effectively.
Economics: Provides tools for marginal analysis, cost functions, and profit maximization strategies.