Table of Contents

1. 1. Functions
   1. Graph Transformations
   2. Practice Evaluating Trig Functions
   3. Graphs of trigonometric functions
   4. Graphing calculator
   5. Domain and range
   6. Exponential functions
   7. The definition of e
   8. Exponential growth and decay
   9. Graphs of inverse functions
   10. Exponential and logarithmic functions
   11. Trigonometric functions and their inverses
   12. Graphs of logarithmic functions
   13. Half-life function
   14. Transformations of Sine and Cosine
2. 2. Limits and Continuity
   1. Secants and Tangents
   2. Dealing with points missing from the domain
   3. Functions with no limit at a point
   4. Precise Definition of a Limit
   5. One-sided limits
   6. The Intermediate Value Theorem
   7. Zooming in on zeros
   8. Limit as x approaches infinity
   9. Precise definition of the limit as x approaches infinity
   10. Horizontal asymptotes
   11. Oblique asymptotes
   12. Precise definition of infinite limit
3. 3. Derivatives
   1. Derivative as the slope of the tangent line
   2. The slope of a curve at a point
   3. Intuitive derivative
   4. Identify the derivative - game
   5. Graph the derivative function
   6. When does a function NOT have a derivative at a point?
   7. Derivative rules: constant multiple and sum rules
   8. Derivatives of exponential functions
   9. The second derivative as the slope of tangent lines
   10. Motion of a point along a line
   11. Simple harmonic motion
   12. Simple harmonic motion
   13. Implicit differentiation
   14. Tangent lines and normal lines
   15. Linearization
   16. Average and Instantaneous Rate of Change
4. 4. Applications of Derivatives
   1. Identify extrema on a graph
   2. First derivative theorem for local extreme values
   3. Rolle's Theorem
   4. Mean value theorem
   5. First derivative test for local extrema
   6. Graph f, given its first derivative
   7. First and second derivatives as slopes of tangent lines
   8. Identify the first and second derivatives
   9. Graph f, given its SECOND derivative
   10. Concavity, points of inflection, and tangent lines
   11. Graphs of the first and second derivatives
   12. Maximizing the volume of an open-top box
   13. Minimizing the surface area of a cylinder
   14. Maximizing the area of a rectangle inscribed in a semicircle
   15. Maximizing the area of a rectangle inscribed in an isosceles right triangle
   16. Inscribing a right circular cone inside a sphere
   17. Maximizing triangular area, given two sides and an included angle
   18. Maximizing the volume of a box to meet USPS shipping standards
   19. Maximizing the volume of a trough
   20. Paper folding
   21. Constructing cones
   22. Maximizing the area of a rectangle inscribed in a 3-4-5 triangle
   23. Shortest beam
   24. Minimizing the distance from a point to a graph
   25. Maximizing the area of an isosceles triangle inside a parabolic arc
   26. Maximizing the volume of a right circular cone inscribed in another right circular cone
   27. Newton's method
   28. Antiderivatives
   29. Same derivative means functions differ by a constant
5. 5. Integrals
   1. Approximating area with finite sums
   2. Average value of a nonnegative continuous function
   3. The Riemann sum
   4. Find the Riemann sum
   5. The definite integral
   6. Properties of the definite integral
   7. Definite integral: the sum property
   8. FTC - The integral as a function
   9. Geometric interpretation of the definite integral
   10. The average value of a function
   11. Special definite integrals of even and odd functions
   12. Integrating even and odd functions
   13. The area bounded by graphs of two functions
   14. Area between two curves as an integral
   15. Integrating with respect to y
6. 6. Applications of Definite Integrals
   1. Triangle and square cross sections: semicircle
   2. Right isosceles triangle cross sections with circular base
   3. Circular cross sections: region between two parabolas
   4. The Cavalieri Principle
   5. The Disk method (about x-axis)
   6. Solids of revolution: the disk method (x-axis)
   7. Solids of revolution: the disk method (y-axis)
   8. Solids of revolution: the washer method (x-axis)
   9. Solids of revolution: the washer method (y-axis)
   10. Cylindrical shells around the y-axis
   11. Cylindrical shells proof exploration
   12. Arc Length
   13. The area of a surface of revolution
   14. Linear discrete system, moment and center of mass
   15. Linear continuous system, moment and center of mass
   16. Planar discrete system, moments and center of mass
   17. Centroid of a planar region and planar curve
   18. Pappus's volume theorem
   19. Pappus's surface area theorem
   20. Visualizing surfaces perpendicular to a curve
   21. Circular spiral surface
   22. Solids of revolution around a line
7. 7. Integrals and Transcendental Functions
   1. Definition of natural logarithm
   2. The effect of k on exp(kx), exponential growth and decay
   3. Graphs of hyperbolic functions
   4. Comparing the growth of functions
8. 8. Techniques of Integration
   1. Numerical integration: the Trapezoidal Rule and Simpson's Rule
   2. Improper integrals of Type I
   3. Improper integrals of Type II
   4. The sine-integral function
   5. Discrete and continous probability density functions
   6. Normal probability density function
9. 9. First Order Diff Equations
   1. Slope fields: viewing solution curves
   2. The slope field of a first-order differential equation
   3. Euler's method
   4. Euler's method exploration
   5. Euler's Method Grapher/Solver
   6. RL Circuit
   7. Orthogonal trajectories to a circle
   8. Newton's law of cooling
   9. Graphical solutions of autonomous equations
   10. A competitive-hunter model: trout and bass
   11. A predator-prey model: whales and krill
   12. Motion with Resistance Proportional to Velocity
10. 10. Infinite Sequences and Series
    1. Visualizing sequences three ways
    2. Properties of sequences
    3. The sandwich theorem for sequences
    4. The continuous function theorem for sequences
    5. Geometric series
    6. Geometric series and figures
    7. p-series and the integral test
    8. The comparison test
    9. The limit comparison test
    10. Absolute convergence, ratio and root tests
    11. Alternating series
    12. Power series and convergence
    13. Taylor polynomials
11. 11. Parametric Eqns and Polar Coords
    1. Parametric graph exploration
    2. Parametric equations grapher
    3. The cycloid and trochoid
    4. Epitrochoids and hypotrochoids
    5. Tangents to a parametric curve
    6. The centroid of a parametric curve
    7. Surface of revolution of a parametric curve
    8. Polar coordinates: plotting points
    9. Graph polar functions
    10. Polar functions: graphs and derivatives
    11. Area in a polar curve
    12. Area in a polar curve - approximate with sectors
    13. Area between two polar curves
    14. Length of a polar curve
    15. Conic sections as intersection of cone and plane
    16. The parabola
    17. The ellipse
    18. Equations for ellipses
    19. The hyperbola
    20. Equations for hyperbolas and their asymptotes
    21. Conics: eccentricity and directrices
    22. Conics: focus at origin, drag the directrix
    23. Polar equations for conics
    24. Polar equations for lines and circles
12. 12. Vectors and the Geom of Space
    1. Planes and Lines in 3 Dimensions
    2. The distance formula for points in space
    3. The equation of a sphere
    4. Component form and magnitude of a vector
    5. Vector algebra in the plane
    6. The standard unit vectors
    7. Force diagrams
    8. The dot product - 2D
    9. The dot product and projections
    10. The dot product and perpendicular vectors
    11. Triple scalar product (volume of a parallelepiped)
    12. Vector equation of a line
    13. Intersection of two planes
    14. Cylinders
    15. Quadric surfaces
    16. Quadric surfaces II
    17. Equation of a plane and the dot product
    18. Quadric surfaces explorer
    19. The six basic quadric surfaces
    20. The line of intersection of two planes
    21. The cross product
    22. The triple scalar product, or box product
    23. Quadric surfaces explorer with cross sections
    24. The six basic quadric surfaces with cross sections
    25. The angle between two planes
13. 13. Vector Valued Fns, Motion in Space
    1. Space Curves
    2. Planar curves: v(t), a(t), T, and N
    3. The helix
    4. The derivative of a vector function
    5. Exploring r(t), v(t), and a(t)
    6. Exploring r(t), v(t), and a(t) in projectile motion
    7. Projectile Motion
    8. Exploring where trajectories crest
    9. The involute of a circle
    10. The unit tangent and principal unit normal vectors
    11. The osculating circle
    12. The tangent vector r'(t)
    13. The TNB Frame
    14. Velocity and acceleration in polar coordinates
14. 14. Partial Derivatives
    1. Surface and a level curve
    2. More level curves
    3. The limit of a function of two variables along a path
    4. The wave equation
    5. Partial derivatives as slopes of tangent lines
    6. The directional derivative
    7. The gradient vector with a surface
    8. The gradient vector and level curves
    9. Gradient vector to a surface; tangent plane
    10. Tangent line to intersecting surfaces
    11. Second derivative test for extreme values
    12. Taylor polynomials in two variables
    13. Constrained variables illustration
    14. Lagrange multipliers
15. 15. Multiple Integrals
    1. Double integrals as volumes
    2. MOD x-First iterated integral over a rectangle
    3. MOD y-First iterated integral over a rectangle
    4. x-First iterated integral over a rectangle
    5. y-First iterated integral over a rectangle
    6. Double integral over a general region
    7. Double integrals in polar form
    8. Solids described by cylindrical coordinates
    9. More solids by cylindrical coordinates
    10. Solids described by spherical coordinates
16. 16. Integrals and Vector Fields
    1. Line integrals
    2. Line integrals with piecewise components
    3. Vector fields in the plane
    4. Vector fields in space
    5. Vector fields and line integrals - plane curves
    6. Vector fields and line integrals - space curves
    7. Vector fields and line integrals - piecewise defined curve
    8. Circulation and flux
    9. Parametrizing a surface
    10. Tangent plane to a parametrized surface
    11. Area of a parametrized surface
    12. Surface integral of a scalar function
    13. Surface integral of a vector field (flux)
    14. Center of mass of a parametrized surface
    15. Surface integral of a vector field (flux)
17. Marc Renault Miscellaneous Figures
    1. 1 Trig Triangles Demo
    2. Trig Values and the Unit Circle
    3. Average and Instantaneous Velocity
    4. 3.1 Definition of the derivative at a point
    5. First Riemann Sums
    6. Riemann Sum Calculator
    7. Bug?
18. Archived (original) applets
    1. Rolle's theorem illustrated
    2. Mean value theorem illustrated
    3. Combining functions; shifting and scaling graphs
    4. Secant lines compared to tangent lines
    5. Maximizing the volume of an open-top box
    6. Maximizing the area of a rectangle inscribed in an isosceles right triangle
    7. Minimizing the distance from a point to a graph
    8. The average value of a nonnegative continuous function
    9. Area between two curves illustrator
    10. When does a function NOT have a derivative?
    11. Integrating with respect to y
    12. Numerical integration
    13. Newton's method
    14. Domain & range of functions
    15. Triangle and square cross sections: semicircle
    16. Estimating arc length: polygonal paths
    17. Precise definition of the limit
    18. The derivative as a function
    19. Derivatives of exponential functions
    20. Graphing the derivative of a function
    21. Slope field grapher
    22. Parametric equations grapher
    23. The cycloid
    24. dy/dx for parametric curves
    25. Area between two polar curves
    26. Arc length in polar coordinates
    27. Euler's method grapher/solver
    28. Polar equations of conics
    29. Three planes and three lines through a point
    30. The dot product and the angle between two vectors
    31. The TNB frame
    32. Space curves
    33. Space curve grapher
    34. Equations for Ellipses and Hyperbolas
    35. Cylinders