- Pearson Interactive Calculus Figures
- 1. Functions
- 2. Limits and Continuity
- 3. Derivatives
- 4. Applications of Derivatives
- 5. Integrals
- 6. Applications of Definite Integrals
- 7. Integrals and Transcendental Functions
- 8. Techniques of Integration
- 9. First Order Diff Equations
- 10. Infinite Sequences and Series
- 11. Parametric Eqns and Polar Coords
- 12. Vectors and the Geom of Space
- 13. Vector Valued Fns, Motion in Space
- 14. Partial Derivatives
- 15. Multiple Integrals
- 16. Integrals and Vector Fields
- Marc Renault Miscellaneous Figures
- Archived (original) applets

# Pearson Interactive Calculus Figures

- Author:
- Marc Renault

- Topic:
- Calculus

These figures were created under the direction of Pearson (pearson.com) to accompany Thomas' Calculus and forthcoming titles. 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

### 2. Limits and Continuity

- Secants and Tangents
- Dealing with points missing from the domain
- Functions with no limit at a point
- Precise definition of the limit
- Precise Definition of a Limit 2
- 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
- The derivative as a function
- Intuitive derivative
- Identify the derivative
- Graph the derivative function
- Derivatives of exponential functions
- When does a function NOT have a derivative at a point?
- Derivative rules: constant multiple and sum rules
- Derivatives of exponential functions
- 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

- First and second derivatives as slopes of tangent lines
- Identify the first and second derivatives
- First derivative theorem for local extreme values
- Rolle's Theorem
- Mean value theorem
- Same derivative means functions differ by a constant
- Graph f, given its first derivative
- Graph f, given its SECOND derivative
- First derivative test for local extrema
- Graphing the derivative of a function
- 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

### 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 I, 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 illustrator
- Integrating with respect to y

### 6. Applications of Definite Integrals

- 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)
- Circular cross sections: region between two parabolas
- Triangle and square cross sections: semicircle
- Right isosceles triangle cross sections with circular base
- Cylindrical shells around the y-axis
- Estimating arc length: polygonal paths
- The area of a surface of revolution
- Pappus's volume 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
- Slope field grapher
- Euler's method
- Euler's method exploration
- Euler's method grapher/solver
- 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

### 12. Vectors and the Geom of Space

- 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
- Three planes and three lines through a point
- The distance formula for points in space
- The equation of a sphere
- The standard unit vectors
- Force diagrams
- Quadric surfaces explorer with cross sections
- The six basic quadric surfaces with cross sections
- The angle between two planes
- The dot product and the angle between two vectors
- The dot product and perpendicular vectors
- The dot product and projections

### 13. Vector Valued Fns, Motion in Space

- Space curves
- Space curve grapher
- 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
- Exploring where trajectories crest
- The involute of a circle
- The unit tangent and principal unit normal vectors
- The osculating circle
- The TNB frame
- The tangent vector r'(t)

### 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
- 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

### 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