Mathematics

Calculus I

Precalculus

  • Exploring Precalculus
  • Precalculus Textbook
  • Tangent Lines and Rates of Change
  • Top of Page

    • Limits

  • Calculating Limits
  • Computing Limits
  • Continuity
  • Continuity
  • Continuity of Functions of One Variable
  • Definition of the Limit
  • Determining Limits Using L'Hopital's Rule
  • Indeterminate Forms and L'Hopital's Rule
  • Infinite Limits
  • Limits
  • Limit Definition of a Definite Integral
  • Limits of Exponential Functions
  • Limit of a Function
  • Limits of a Function as x Approaches a Constant
  • Limits of a Function as x Approaches Plus or Minus Infinity
  • Limits at Infinity, Part I
  • Limits at Infinity, Part II
  • Limits Involving Infinity
  • Limit Properties
  • One-Sided Limits
  • Precises Limits of a Function as x Approaches a Constant
  • Proof of Trig Limits
  • Proof of Various Limit Properties
  • Squeeze Principle
  • Types of Infinity
  • Top of Page

    • Derivatives

  • Basic Differentiation Formulas
  • Chain Rule
  • Chain Rule
  • Chain Rule
  • Definition of Derivative
  • Derivatives of Exponential and Logarithmic Functions
  • Derivatives of Exponential and Logarithmic Functions Proofs
  • Derivative as a Function
  • Derivatives of Hyperbolic Functions
  • Derivatives of Logarithmic and Exponential Functions
  • Derivatives of Inverse Trig Functions
  • Derivatives of Inverse Trig Functions Proofs
  • Derivatives and Rates of Change
  • Derivatives Rules Proofs
  • Derivatives of Trig Functions
  • Derivatives of Trig Functions Proofs
  • Derivatives Using the Limit Definition
  • Differentiation Formulas
  • Differentiation of Inverse Trigonometric Functions
  • Differentiation of Trigonometric Functions
  • Differentiation Using the Chain Rule
  • Differentiation Using the Product Rule
  • Differentiation Using the Quotient Rule
  • Graphing Functions Using First and Second Derivatives
  • Higher Order Derivatives
  • Implicit Differentiation
  • Implicit Differentiation
  • Interpretation of Derivative
  • Logarithmic Differentiation
  • Logarithmic Differentiation
  • Maximum and Minimum Values
  • Maximum and Minimum Values Using Derivatives
  • Product and Quotient Rules
  • Product and Quotient Rules
  • Quotient Rule
  • Proof of Various Derivative Properties
  • Related Rates
  • Top of Page

    • Derivative Applications

  • Applications of Maximum and Minimum Values
  • Business Applications
  • Critical Points
  • Curve Sketching
  • Derivatives and the Shapes of Graphs
  • Finding Absolute Extrema
  • Implicit Differentiation Problems
  • Instantaneous Velocity and Related Rates
  • L'Hospital's Rule and Indeterminant Forms
  • Linear Approximations
  • Logistic Functions
  • Logistic Functions Examples
  • Maximum and Minimum Problems
  • Minimum and Maximum Values
  • Mean Value Theorem
  • Mean Value Theorem
  • Newton's Method
  • Optimization Problems
  • Optimization Problems, Part I
  • Optimization Problems, Part II
  • Proofs of Derivative Applications
  • Rates of Change
  • Shape of a Graph, Part I
  • Shape of a Graph, Part II
  • Top of Page

    • First Order Differential Equations

  • Bernoulli Differential Equations
  • Definitions
  • Differentials
  • Differential Equations Notes
  • Direction Fields
  • Equilibrium Solutions
  • Euler's Method
  • Exact Equations
  • Intervals of Validity
  • Linear First Order Differential Equations
  • Modeling with First Order Differential Equations
  • Separable Equations
  • Substitutions
  • Top of Page

    • Second Order Differential Equations

  • Basic Concepts
  • Complex Roots
  • Euler Equations
  • Fundamental Sets of Solutions
  • Non-Homogeneous
  • Variation of Parameters
  • Undetermined Coefficients
  • Wronskian
  • Real Roots
  • Repeated Roots
  • Reduction of Order
  • Top of Page

    • Higher Order Differential Equations

  • Homogeneous Linear Higher Order Differential Equations
  • Undetermined Coefficients
  • Variation of Parameters
  • Top of Page

    • Differential Equations Applications

  • Linear Approximations and Differentials
  • Mechanical Vibrations
  • Top of Page

    • Differential Equation Systems

  • Complex Eigenvalues
  • Differential Equations
  • Eigenvalues and Eigenvectors
  • Modeling with DEs
  • Non-homogeneous Systems
  • Phase Plane
  • Real Eigenvalues
  • Repeated Eigenvalues
  • Solutions to Systems
  • Systems of Differential Equations
  • Top of Page

    • Partial Differential Equations

  • Heat Equation
  • Heat Equation with Non-zero Temperature Boundaries
  • Laplace's Equation
  • Separation of Variables
  • Solving the Heat Equation
  • Terminology for Separation of Variables
  • Vibrating String
  • Wave Equation
  • Top of Page

    • Laplace Transforms

  • Convolution Integral
  • Dirac Delta Function
  • Inverse Laplace Transforms
  • IVPs with Step Functions
  • Laplace Transforms
  • Laplace Transform Definition
  • Non-constant Coefficient IVPs
  • Solving IVPs
  • Step Functions
  • Table of Laplace Transforms
  • Top of Page

    • Fourier Series and Boundary Value Problems

  • Boundary Value Problems
  • Convergence of Fourier Series
  • Fourier Series
  • Fourier Cosine Serices
  • Fourier Sine Series
  • Periodic Functions and Orthogonal Functions
  • Top of Page

    • Resources

  • Calculus I Review and Practice Problems
  • Calculus II Review and Practice Problems
  • Calculus III Review and Practice Problems
  • Common Calculus Errors
  • eCalculus Resources for the Calculus Student
  • Fundamental Theorem of Calculus
  • MIT Open Courseware Calculus Textbook
  • Top of Page