Outline
Pearson Interactive Calculus Figures
These figures were created under the direction of Pearson (pearson.com) to accompany Thomas' Calculus and forthcoming titles.
In particular, these figures are prominently featured in Pearson's Interactive Calculus https://www.pearson.com/en-us/subject-catalog/p/interactive-calculus-early-transcendentals-single-variable.
Figures were authored by Tim Brzezinski, Kevin Hopkins, Steve Phelps, and Marc Renault (in 2016) and by Marc Renault (since 2019). These figures are free and open for all to use and enjoy. Questions? Corrections? Fan letters? Contact Marc Renault msrenault@ship.edu.
Table of Contents
1. Functions
- Graph Transformations
- Practice Evaluating Trig Functions
- Graphs of trigonometric functions
- Graphing calculator
- Domain and range
- Exponential functions
- The definition of e
- Exponential growth and decay
- Graphs of inverse functions
- Exponential and logarithmic functions
- Trigonometric functions and their inverses
- Graphs of logarithmic functions
- Half-life function
- Transformations of Sine and Cosine
2. Limits and Continuity
- Secants and Tangents
- Dealing with points missing from the domain
- Functions with no limit at a point
- Precise Definition of a Limit
- One-sided limits
- The Intermediate Value Theorem
- Zooming in on zeros
- Limit as x approaches infinity
- Precise definition of the limit as x approaches infinity
- Horizontal asymptotes
- Oblique asymptotes
- Precise definition of infinite limit
3. Derivatives
- Derivative as the slope of the tangent line
- The slope of a curve at a point
- Intuitive derivative
- Identify the derivative - game
- Graph the derivative function
- When does a function NOT have a derivative at a point?
- Derivative rules: constant multiple and sum rules
- Derivatives of exponential functions
- The second derivative as the slope of tangent lines
- Motion of a point along a line
- Simple harmonic motion
- Simple harmonic motion
- Implicit differentiation
- Tangent lines and normal lines
- Linearization
- Average and Instantaneous Rate of Change
4. Applications of Derivatives
- Identify extrema on a graph
- First derivative theorem for local extreme values
- Rolle's Theorem
- Mean value theorem
- First derivative test for local extrema
- Graph f, given its first derivative
- First and second derivatives as slopes of tangent lines
- Identify the first and second derivatives
- Graph f, given its SECOND derivative
- Concavity, points of inflection, and tangent lines
- Graphs of the first and second derivatives
- Maximizing the volume of an open-top box
- Minimizing the surface area of a cylinder
- Maximizing the area of a rectangle inscribed in a semicircle
- Maximizing the area of a rectangle inscribed in an isosceles right triangle
- Inscribing a right circular cone inside a sphere
- Maximizing triangular area, given two sides and an included angle
- Maximizing the volume of a box to meet USPS shipping standards
- Maximizing the volume of a trough
- Paper folding
- Constructing cones
- Maximizing the area of a rectangle inscribed in a 3-4-5 triangle
- Shortest beam
- Minimizing the distance from a point to a graph
- Maximizing the area of an isosceles triangle inside a parabolic arc
- Maximizing the volume of a right circular cone inscribed in another right circular cone
- Newton's method
- Antiderivatives
- Same derivative means functions differ by a constant
5. Integrals
- Approximating area with finite sums
- Average value of a nonnegative continuous function
- The Riemann sum
- Find the Riemann sum
- The definite integral
- Properties of the definite integral
- Definite integral: the sum property
- FTC - The integral as a function
- Geometric interpretation of the definite integral
- The average value of a function
- Special definite integrals of even and odd functions
- Integrating even and odd functions
- The area bounded by graphs of two functions
- Area between two curves as an integral
- Integrating with respect to y
6. Applications of Definite Integrals
- Triangle and square cross sections: semicircle
- Right isosceles triangle cross sections with circular base
- Circular cross sections: region between two parabolas
- The Cavalieri Principle
- The Disk method (about x-axis)
- Solids of revolution: the disk method (x-axis)
- Solids of revolution: the disk method (y-axis)
- Solids of revolution: the washer method (x-axis)
- Solids of revolution: the washer method (y-axis)
- Cylindrical shells around the y-axis
- Cylindrical shells proof exploration
- Arc Length
- The area of a surface of revolution
- Linear discrete system, moment and center of mass
- Linear continuous system, moment and center of mass
- Planar discrete system, moments and center of mass
- Centroid of a planar region and planar curve
- Pappus's volume theorem
- Pappus's surface area theorem
- Visualizing surfaces perpendicular to a curve
- Circular spiral surface
- Solids of revolution around a line
7. Integrals and Transcendental Functions
8. Techniques of Integration
9. First Order Diff Equations
- Slope fields: viewing solution curves
- The slope field of a first-order differential equation
- Euler's method
- Euler's method exploration
- Euler's Method Grapher/Solver
- RL Circuit
- Orthogonal trajectories to a circle
- Newton's law of cooling
- Graphical solutions of autonomous equations
- A competitive-hunter model: trout and bass
- A predator-prey model: whales and krill
- Motion with Resistance Proportional to Velocity
10. Infinite Sequences and Series
- Visualizing sequences three ways
- Properties of sequences
- The sandwich theorem for sequences
- The continuous function theorem for sequences
- Geometric series
- Geometric series and figures
- p-series and the integral test
- The comparison test
- The limit comparison test
- Absolute convergence, ratio and root tests
- Alternating series
- Power series and convergence
- Taylor polynomials
11. Parametric Eqns and Polar Coords
- Parametric graph exploration
- Parametric equations grapher
- The cycloid and trochoid
- Epitrochoids and hypotrochoids
- Tangents to a parametric curve
- The centroid of a parametric curve
- Surface of revolution of a parametric curve
- Polar coordinates: plotting points
- Graph polar functions
- Polar functions: graphs and derivatives
- Area in a polar curve
- Area in a polar curve - approximate with sectors
- Area between two polar curves
- Length of a polar curve
- Conic sections as intersection of cone and plane
- The parabola
- The ellipse
- Equations for Ellipses and Hyperbolas
- The hyperbola
- Conics: Eccentricity and Directrices
- Conics: Focus at origin, drag the directrix
- Polar equations for conics
- Polar equations for lines and circles
12. Vectors and the Geom of Space
- Planes and Lines in 3 Dimensions
- The distance formula for points in space
- The equation of a sphere
- Component form and magnitude of a vector
- Vector algebra in the plane
- The standard unit vectors
- Force diagrams
- The dot product - 2D
- The dot product and projections
- The dot product and perpendicular vectors
- Triple scalar product
- Vector equation of a line
- Intersection of two planes
- Quadric surfaces
- Quadric surfaces II
- Equation of a plane and the dot product
- Cylinders
- Quadric surfaces explorer
- The six basic quadric surfaces
- The line of intersection of two planes
- The cross product
- The triple scalar product, or box product
- Quadric surfaces explorer with cross sections
- The six basic quadric surfaces with cross sections
- The angle between two planes
13. Vector Valued Fns, Motion in Space
- Space Curves
- Planar curves: v(t), a(t), T, and N
- The helix
- The derivative of a vector function
- Exploring r(t), v(t), and a(t)
- Exploring r(t), v(t), and a(t) in projectile motion
- Projectile Motion
- Exploring where trajectories crest
- The involute of a circle
- The unit tangent and principal unit normal vectors
- The osculating circle
- The tangent vector r'(t)
- The TNB Frame
- Velocity and acceleration in polar coordinates
14. Partial Derivatives
- Surface and a level curve
- More level curves
- The limit of a function of two variables along a path
- Partial derivatives as slopes of tangent lines
- The directional derivative
- The gradient vector with a surface
- The gradient vector and level curves
- Gradient vector to a surface; tangent plane
- Tangent line to intersecting surfaces
- Second derivative test for extreme values
- Taylor polynomials in two variables
- Constrained variables illustration
- Lagrange multipliers
15. Multiple Integrals
- Double integrals as volumes
- MOD x-First iterated integral over a rectangle
- MOD y-First iterated integral over a rectangle
- x-First iterated integral over a rectangle
- y-First iterated integral over a rectangle
- Double integral over a general region
- Double integrals in polar form
- Solids described by cylindrical coordinates
- More solids by cylindrical coordinates
- Solids described by spherical coordinates
16. Integrals and Vector Fields
- Line integrals
- Line integrals in the plane
- Vector fields in the plane
- Vector fields in space
- Vector fields and line integrals - plane curves
- Vector fields and line integrals - space curves
- Vector fields and line integrals - piecewise defined curve
- Circulation and flux
- Parametrizing a surface
- Tangent plane to a parametrized surface
- Area of a parametrized surface
- Surface integral of a scalar function
- Surface integral of a vector field (flux)
Marc Renault Miscellaneous Figures
Archived (original) applets
- Rolle's theorem illustrated
- Mean value theorem illustrated
- Combining functions; shifting and scaling graphs
- Secant lines compared to tangent lines
- Maximizing the volume of an open-top box
- Maximizing the area of a rectangle inscribed in an isosceles right triangle
- Minimizing the distance from a point to a graph
- The average value of a nonnegative continuous function
- Area between two curves illustrator
- When does a function NOT have a derivative?
- Integrating with respect to y
- Numerical integration
- Newton's method
- Domain & range of functions
- Triangle and square cross sections: semicircle
- Estimating arc length: polygonal paths
- Precise definition of the limit
- The derivative as a function
- Derivatives of exponential functions
- Graphing the derivative of a function
- Slope field grapher
- Parametric equations grapher
- The cycloid
- dy/dx for parametric curves
- Area between two polar curves
- Arc length in polar coordinates
- Euler's method grapher/solver
- Polar equations of conics
- Three planes and three lines through a point
- The dot product and the angle between two vectors
- The TNB frame
- Space curves
- Space curve grapher