Calculus Introduction and Differential Calculus

Calculus

Introduction

• Branch of mathematics that deals with the study of continuous change
• Developed by Sir Isaac Newton and German mathematician Gottfried Wilhelm Leibniz in the 17th century
• Consists of two main branches: Differential Calculus and Integral Calculus

Differential Calculus

Limits

• Concept of a limit: the value that a function approaches as the input gets arbitrarily close to a certain point
• Notation: lim x→a f(x) = L
• Properties of limits:
  • Linearity: lim x→a [af(x) + bg(x)] = alim x→a f(x) + blim x→a g(x)
  • Homogeneity: lim x→a [f(x)g(x)] = [lim x→a f(x)][lim x→a g(x)]

Derivatives

• Definition: the rate of change of a function with respect to its input
• Notation: f'(a) or (d/dx)f(x) at x=a
• Rules of differentiation:
  • Power rule: if f(x) = x^n, then f'(x) = nx^(n-1)
  • Product rule: if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x)
  • Chain rule: if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)

Applications of Differential Calculus

• Finding the maximum and minimum values of a function
• Determining the rate at which a quantity changes over time
• Analyzing optimization problems

Integral Calculus

Definite Integrals

• Definition: the area between a curve and the x-axis over a specific interval
• Notation: ∫[a, b] f(x) dx
• Properties of definite integrals:
  • Linearity: ∫[a, b] [af(x) + bg(x)] dx = a∫[a, b] f(x) dx + b∫[a, b] g(x) dx
  • Additivity: ∫[a, c] f(x) dx = ∫[a, b] f(x) dx + ∫[b, c] f(x) dx

Fundamental Theorem of Calculus

• States that differentiation and integration are inverse processes
• ∫[a, b] f(x) dx = F(b) - F(a), where F(x) is the antiderivative of f(x)

Applications of Integral Calculus

• Finding the area under curves and surfaces
• Solving problems involving accumulation of quantities
• Modeling real-world phenomena, such as physics and engineering problems

Calculus

Introduction

• Deals with the study of continuous change in mathematics
• Developed by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century
• Comprises two main branches: Differential Calculus and Integral Calculus

Differential Calculus

Limits

• Concept of a limit: the value a function approaches as the input gets arbitrarily close to a certain point
• Notation: lim x→a f(x) = L
• Properties: linearity, homogeneity, and more

Derivatives

• Definition: the rate of change of a function with respect to its input
• Notation: f'(a) or (d/dx)f(x) at x=a
• Rules of differentiation: power rule, product rule, chain rule, and more
• Power rule: if f(x) = x^n, then f'(x) = nx^(n-1)
• Product rule: if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x)
• Chain rule: if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)

Applications of Differential Calculus

• Finding maximum and minimum values of a function
• Determining the rate at which a quantity changes over time
• Analyzing optimization problems

Integral Calculus

Definite Integrals

• Definition: the area between a curve and the x-axis over a specific interval
• Notation: ∫[a, b] f(x) dx
• Properties: linearity, additivity, and more
• Linearity: ∫[a, b] [af(x) + bg(x)] dx = a∫[a, b] f(x) dx + b∫[a, b] g(x) dx
• Additivity: ∫[a, c] f(x) dx = ∫[a, b] f(x) dx + ∫[b, c] f(x) dx

Fundamental Theorem of Calculus

• States that differentiation and integration are inverse processes
• ∫[a, b] f(x) dx = F(b) - F(a), where F(x) is the antiderivative of f(x)

Applications of Integral Calculus

• Finding the area under curves and surfaces
• Solving problems involving accumulation of quantities
• Modeling real-world phenomena, such as physics and engineering problems