Calculus Overview: Differential and Integral

Branch of mathematics focused on change and motion.
Foundations of calculus laid by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century.
Divided mainly into two branches: Differential Calculus and Integral Calculus.

Concerned with the concept of the derivative.
Derivative Definition:
- Measures the rate of change of a function.
- Defined as the limit of the average rate of change as the interval approaches zero.
Notation:
- f'(x), dy/dx, Df.
Rules:
- Power Rule: d/dx(x^n) = nx^(n-1).
- Product Rule: d/dx(u*v) = 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 slopes of tangents to curves.
- Analyzing motion (velocity and acceleration).
- Optimization problems (maxima and minima).

Focuses on the concept of the integral.
Integral Definition:
- Represents the accumulation of quantities, often area under curves.
Notation:
- ∫ f(x) dx.
Types of Integrals:
- Definite Integral: ∫[a, b] f(x) dx - computes the area between f(x) and the x-axis from a to b.
- Indefinite Integral: ∫ f(x) dx - represents a family of antiderivatives.
Fundamental Theorem of Calculus:
- Connects differentiation and integration.
- If F is an antiderivative of f, then ∫[a, b] f(x) dx = F(b) - F(a).
Techniques of Integration:
- Substitution Method.
- Integration by Parts.
- Partial Fraction Decomposition.
Applications:
- Calculating areas, volumes, and averages.
- Solving problems in physics and engineering.

Core concept that underpins calculus.
Formal Definition:
- The value that a function approaches as the input approaches a certain value.
Notation:
- Lim as x approaches c of f(x) = L.
Properties of Limits:
- Limit of a sum, difference, product, and quotient.
- Squeeze Theorem for finding limits.
One-Sided Limits:
- Left-hand limit and right-hand limit.

A function is continuous if:
- The limit as x approaches c equals f(c).
- No breaks, jumps, or asymptotes present in the graph.
Importance in calculus:
- Continuous functions are easier to analyze and integrate.

Important in the study of series and sequences.
A series is convergent if the sum approaches a finite limit.
A series is divergent if it does not approach a finite limit (e.g., harmonic series).

Extension of calculus to functions of multiple variables.
Key concepts include:
- Partial derivatives.
- Multiple integrals (double and triple integrals).
- Gradient, divergence, and curl in vector calculus.

Concerned with the concept of the derivative which measures the rate of change of a function
Derivative is defined as the limit of the average rate of change of a function as the interval approaches zero
Derivative notation: f'(x), dy/dx, Df
Common Derivative Rules:
- Power Rule: d/dx(x^n) = nx^(n-1)
- Product Rule: d/dx(u*v) = 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 include:
- Finding slopes of tangents to curves
- Analyzing motion (velocity and acceleration)
- Optimization problems (maxima and minima)

Focuses on the concept of the integral
Integral represents the accumulation of quantities, often the area under curves
Notation: ∫ f(x) dx
Types of Integrals:
- Definite Integral: ∫[a, b] f(x) dx - computes the area between f(x) and the x-axis from a to b
- Indefinite Integral: ∫ f(x) dx - represents a family of antiderivatives
Fundamental Theorem of Calculus:
- Connects differentiation and integration
- If F is an antiderivative of f, then ∫[a, b] f(x) dx = F(b) - F(a)
Techniques of Integration include:
- Substitution Method
- Integration by Parts
- Partial Fraction Decomposition
Applications:
- Calculating areas, volumes, and averages
- Solving problems in physics and engineering

Core concept underpinning Calculus
Defined as the value that a function approaches as the input approaches a certain value
Notation: Lim as x approaches c of f(x) = L
Limit Properties:
- Limit of a sum, difference, product, and quotient
- Squeeze Theorem for finding limits
One-Sided Limits:
- Left-hand limit and right-hand limit

A function is continuous if:
- The limit as x approaches c equals f(c)
- No breaks, jumps, or asymptotes present in the graph
Importance in calculus: Continuous functions are easier to analyze and integrate

Important in the study of series and sequences
A series is convergent if the sum approaches a finite limit
A series is divergent if it does not approach a finite limit (e.g., harmonic series)

Extension of Calculus to functions of multiple variables
Key concepts include:
- Partial derivatives
- Multiple integrals (double and triple integrals)
- Gradient, divergence, and curl in vector calculus