Calculus Basics

Branches of Calculus

• Differential Calculus: deals with the study of rates of change and slopes of curves
• Integral Calculus: deals with the study of accumulation of quantities

Key Concepts

• Limits: the behavior of a function as the input (or x-value) approaches a specific value
• Derivatives: measure of how a function changes as its input changes
• Integrals: measure of the accumulation of a function over a given interval

Derivatives

• 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 Derivatives:
  • Finding Maximum and Minimum Values
  • Determining the Rate at which a Quantity Changes

Integrals

• Types of Integrals:
  • Definite Integrals: have a specific upper and lower bound
  • Indefinite Integrals: do not have specific bounds
• Rules of Integration:
  • Substitution Method
  • Integration by Parts
  • Integration by Partial Fractions
• Applications of Integrals:
  • Finding the Area Between Curves
  • Finding the Volume of Solids

Theorems and Properties

• Fundamental Theorem of Calculus: relates the derivative of an antiderivative to the original function
• Mean Value Theorem: states that a function must have at least one critical point in a given interval
• Intermediate Value Theorem: states that a continuous function must take on all values between its minimum and maximum values

Branches of Calculus

• Calculus has two main branches: Differential Calculus and Integral Calculus
• Differential Calculus deals with rates of change and slopes of curves
• Integral Calculus deals with accumulation of quantities

Key Concepts

• Limits involve the behavior of a function as the input approaches a specific value
• Derivatives measure how a function changes as its input changes
• Integrals measure the accumulation of a function over a given interval

Derivatives

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 Derivatives

• Derivatives are used to find maximum and minimum values
• Derivatives are used to determine the rate at which a quantity changes

Integrals

Types of Integrals

• Definite Integrals have a specific upper and lower bound
• Indefinite Integrals do not have specific bounds

Rules of Integration

• Substitution Method is used to integrate functions
• Integration by Parts is used to integrate functions
• Integration by Partial Fractions is used to integrate functions

Applications of Integrals

• Integrals are used to find the area between curves
• Integrals are used to find the volume of solids

Theorems and Properties

• The Fundamental Theorem of Calculus relates the derivative of an antiderivative to the original function
• The Mean Value Theorem states that a function must have at least one critical point in a given interval
• The Intermediate Value Theorem states that a continuous function must take on all values between its minimum and maximum values