Introduction to Calculus: Differential Calculus

Introduction to Calculus

• Calculus is a branch of mathematics focused on change, studying how quantities change over time or space.
• It has two main branches: differential calculus and integral calculus.
• Differential calculus deals with rates of change, like finding the slope of a curve at a point.
• Integral calculus deals with accumulation of quantities, like finding the area under a curve.
• These branches are deeply interconnected and frequently used together to solve problems.

Differential Calculus

• The Derivative: The derivative of a function at a point represents the instantaneous rate of change of the function at that point.
• Geometric Interpretation: The derivative is the slope of the tangent line to the graph of the function at a given point.
• Notation: The derivative of a function f(x) is often denoted as f'(x) or dy/dx.
• Rules for Finding Derivatives: Specific rules exist for various functions:
  • Power rule (for polynomials)
  • Product rule (for products of functions)
  • Quotient rule (for quotients of functions)
  • Chain rule (for composite functions)
• Applications of Derivatives: Used to find:
  • Maximum and minimum values of functions (critical points).
  • Velocity and acceleration in physics.
  • Optimization problems (e.g., maximizing profit, minimizing cost).
  • Rates of change in diverse fields, including economics and biology.

Integral Calculus

• The Integral: The integral of a function represents the area under the curve of the function over a given interval.
• Geometric Interpretation: The definite integral of a function f(x) over an interval [a, b] gives the signed area between the graph of f(x) and the x-axis from x = a to x = b.
• Notation: The integral of a function f(x) from a to b is written as ∫_a^b f(x) dx.
• Fundamental Theorem of Calculus: Connects differential and integral calculus:
  • The integral of a function's derivative returns the original function, plus a constant.
  • The derivative of a function's integral returns the original function.
• Applications of Integrals: Used to find:
  • Areas and volumes of shapes.
  • Work and energy.
  • Probability and statistics.
  • Moments and centroids.
  • Arc length.

Key Concepts and Techniques

• Limits: The concept of a limit is fundamental to calculus, defining how a function behaves as its input approaches a particular value.
• Continuity: A continuous function has no breaks or jumps in its graph.
• Average Rate of Change: The average rate of change of a function over an interval is given by the slope of the secant line between the endpoints of the interval.
• Approximation: Calculus provides methods for approximating areas, volumes, and other quantities.

Real-World Applications

• Calculus is used in numerous fields, including:
  • Engineering (e.g., designing bridges, calculating forces).
  • Physics (e.g., motion, gravity).
  • Economics (e.g., optimization, cost analysis).
  • Computer science (e.g., image processing, machine learning).
  • Biology (e.g., population growth, chemical reactions).
• This highlights the broad applicability of calculus across disciplines.