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MVT (Mathematical Visualization Toolkit): 
The MVT (The Mathematical Visualization Toolkit) can be used online or downloaded, and can graph 2- and 3-D functions, functions in other coordinate systems (e.g. polar, spherical, cylindrical), and vector and gradient fields (including 3-D vector fields!). It also has a number of built in applications — like revolving curves around axes, fourier series approximations, and riemann sums. This is an incredible teaching and visualization tool.


  1. Operations on functions
  2. Trigonometric functions
  3. Slope of a line
  4. Derivative and tangent line
  5. Constant, linear, quadratic and cubic approximations
  6. Numerical integration
  7. Polar plotter
  8. A smooth function
  9. One-sided derivatives at a point of non-differentiability
  10. Secant lines for a function with two non-differentiable points (applet by Daniel J. Heath)
  11. Secant lines approaching a point of differentiability
  12. Secant lines approaching a point of non-differentiability (the one-sided derivatives don’t match)
  13. Secant lines approaching another point of non-differentiability (vertical tangent; the derivative is infinite)
  14. Chain rule applet (from a multimedia calculus course by Scott Sarra)
  15. First and second derivatives applet (by Scott Sarra)
  16. More first and second derivatives, with parameters you can tweak (applet from Maths Online)
  17. Derivatives of axsin xcos x (applets by Daniel J. Heath)
  18. Converging to the number e (applet from IES, Manipula Math); note that the simulation doesn’t let you go far enough to approach that close to e
  19. Zooming in on a tangent line (animation by Douglas N. Arnold)
  20. Linear approximation of sin x at 0 (applet from UBC Calculus Online)
  21. Finding a function’s extremum, applet from Maths Online
  22. Rolle’s theorem and the mean value theorem (applet from IES, Manipula Math)Riemann sums
  23. Definite integrals and the integral function 1
  24. Definite integrals and the integral function 2
  25. Numerical Integration Simulation (by Joseph L. Zachary)
  26. Volumes of Solid 1
  27. Volumes of Solid 2
  28. Solid of revolution 1
  29. Direction field applet (by Scott Sarra)
  30. More direction field applets (from UBC Calculus Online)
  31. Yet another direction field applet (from IES, Manipula Math)
  32. Parametric equation applet (by Scott Sarra)
  33. Another parametric equation applet (from IES, Manipula Math)
  34. Computing arc length (AVI) (animation by Przemyslaw Bogacki and Gordon Melrose)
  35. Approximating arc length (applet by Daniel J. Heath)
  36. Taylor approximations (applet by Daniel J. Heath)
  37. Power series grapher


  • Using the Graphing Tools
    1. Introduction to using the applets.
    2. Limitations of Graphing Software
  • Continuity and Limits
    1. An Informal, Graphical View of Continuity
    2. Intermediate Value Theorem
    3. Informal View of Limits
    4. One- and Two-Sided Limits and When Limits Fail to Exist
    5. Limits at Infinity
    6. Table View of Limits
    7. Formal Definition of Limits
    8. Definition of Continuity Using Limits
  • Introduction to the Derivative
    1. Average Velocity and Speed
    2. Instantaneous Velocity
    3. Derivative at a Point
    4. Derivative Function
    5. A Tabular View of the Derivative
    6. Second Derivative
    7. A Tabular View of the Second Derivative
    8. Differentiability
    9. Twice Differentiable
    10. Making a Piecewise Function Continuous and Differentiable
  • Differentiation Short Cuts
    1. Constant, Line, and Power Functions
    2. Exponential Functions
    3. Trigonometric Functions
    4. Constant Multiple
    5. Combinations: Sum, and Difference
    6. Combinations of Functions: Product and Quotient
    7. Composition of Functions (the Chain Rule)
    8. Transformations of Functions
    9. Inverses of Functions
    10. Hyperbolic Functions
    11. Linear Approximation
    12. Mean Value Theorem
  • Applications of Differentiation
    1. Curve Analysis: Basics
    2. Curve Analysis: Special Cases
    3. Curve Analysis: Global Extrema
    4. Optimization: Maximize Volume
    5. Extreme Value Theorem
    6. Related Rates
    7. L’Hopital’s Rule
    8. Parametric Derivatives
    9. Polar Derivatives
    10. Motion on a Line
    11. Motion in the Plane
  • Introduction to the Definite Integral
    1. Approximating Distance Traveled With a Table
    2. Approximating Distance Traveled With a Graph
    3. Riemann Sums and The Definite Integral
    4. Fundamental Theorem of Calculus
    5. Average Value
    6. Properties of Definite Integrals
  • Constructing Antiderivatives
    1. Antiderivatives from Slope and the Indefinite Integral
    2. Accumulation Functions
    3. Basic Antiderivatives
    4. Introduction to Differential Equations
    5. Second Fundamental Theorem of Calculus
    6. Functions Defined Using Integrals
    7. Equations of Motion
  • Integration Techniques
    1. Substitution
    2. Midpoint and Trapezoid Riemann Sums
    3. Improper Integrals
  • Applications of Integration
    1. Areas by Slicing
    2. Volumes of Revolution
    3. Volumes of Known Cross Section
    4. Arc Length
    5. Area of Polar Curve
  • Differential Equations
    1. Slope Fields
    2. Euler’s Method
    3. Separation of Variables
    4. Growth, Decay and the Logistic Equation
  • Sequences and Series
    1. Sequences
    2. Series
    3. Integral Test
    4. Comparison Test
    5. Limit Comparison Test
    6. Ratio Test
    7. Alternating Series and Absolute Convergence
    8. Power Series & Interval of Convergence
    9. Taylor Series & Polynomials
    10. Lagrange Remainder
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