Calculus Basics

Calculus

Introduction

• Calculus is a branch of mathematics that deals with the study of continuous change.
• It consists of two main branches: Differential Calculus and Integral Calculus.

Differential Calculus

• Limits: The concept of a limit is central to calculus. It involves determining the behavior of a function as the input (or x-value) approaches a specific value.
• Derivatives: The derivative of a function represents the rate of change of the function with respect to its input.
  • 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)
    • Quotient Rule: If f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2
• Applications of Derivatives:
  • Finding the Maximum and Minimum Values of a function
  • Determining the Rate of Change of a function
  • Optimization Problems

Integral Calculus

• Definite Integrals: The definite integral of a function represents the area between the curve of the function and the x-axis over a specific interval.
• Fundamental Theorem of Calculus: The definite integral of a function can be evaluated using the antiderivative of the function.
  • Basic Integration Rules:
    • Power Rule: ∫x^n dx = (x^(n+1))/(n+1) + C
    • Constant Multiple Rule: ∫k f(x) dx = k ∫f(x) dx
• Applications of Integrals:
  • Finding the Area between curves
  • Volume of Solids using disks and washers
  • Work and Energy problems

Multivariable Calculus

• Partial Derivatives: The partial derivative of a function of multiple variables represents the rate of change of the function with respect to one of its variables.
• Double and Triple Integrals: The definite integral of a function of multiple variables represents the volume under a surface or the area of a region.
• Vector Calculus: The study of vectors and their applications in calculus, including gradient, divergence, and curl.

Calculus

Introduction

• Calculus is the study of continuous change, divided into two main branches: Differential Calculus and Integral Calculus.

Differential Calculus

• Limits are crucial in calculus, involving the behavior of a function as the input approaches a specific value.
• Derivatives represent the rate of change of a function with respect to its input.
• 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)
  • Quotient Rule: If f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2
• Applications of Derivatives:
  • Finding the Maximum and Minimum Values of a function
  • Determining the Rate of Change of a function
  • Optimization Problems

Integral Calculus

• Definite Integrals represent the area between the curve of the function and the x-axis over a specific interval.
• Fundamental Theorem of Calculus: The definite integral of a function can be evaluated using the antiderivative of the function.
• Basic Integration Rules:
  • Power Rule: ∫x^n dx = (x^(n+1))/(n+1) + C
  • Constant Multiple Rule: ∫k f(x) dx = k ∫f(x) dx
• Applications of Integrals:
  • Finding the Area between curves
  • Volume of Solids using disks and washers
  • Work and Energy problems

Multivariable Calculus

• Partial Derivatives represent the rate of change of a function of multiple variables with respect to one of its variables.
• Double and Triple Integrals represent the volume under a surface or the area of a region.
• Vector Calculus involves the study of vectors and their applications in calculus, including gradient, divergence, and curl.