Below you will find a collection of dynamic visualizations of calculus ideas.
To browse visualizations, simply click the buttons next to the categories and titles.
In every case you will run the visualization software by clicking on a button.
In some cases you may need to click a button many times to see a long-term pattern emerge
(for example, the slope of a secant line approaches the slope of tangent line).
To perform repeated iterations quickly, simply click on the button once.
You may then use the enter key to perform more iterations.
You may also hold down the enter key to see many many iterations in rapid succession.
Please note that for the time being, these visualizations work best when using a laptop or tablet.
It is my sincere hope that these visualizations help you to acquire a deeper, more visceral understanding of the ideas covered in a
calculus course.
Single Variable Calculus
Limits
Limit of a Function
Continuity at A Point
Discontinuity at A Point (Case 1)
$\varepsilon$-$\delta$ Definition (Limit Exists)
$\varepsilon$-$\delta$ Definition (Limit Does Not Exist)
Derivatives
The Derivative as a Limit
Average vs. Instantaneous Rate of Change
The Derivative as a Function
The Derivative of $f(x)=\sin x$
The Derivative of $f(x)=\cos x$
Newton's Method
Integrals
The Limit of a Riemann Sum
Arc Length as a Limit
Sequences and Series
Definition of Convergent Sequence
Graph of the $n$th Partial Sum of $\displaystyle \sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^n$
Graph of the $n$th Partial Sum of $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n}$
Graph of the $n$th Taylor Polynomial for $\displaystyle f(x)=\sin x$
Multivariable and Vector Calculus
Vector-Valued Functions and Curves in Space
Derivatives of Vector-Valued Functions 1
Derivatives of Vector-Valued Functions 2
The Principal Unit Normal Vector in the Plane
The Principal Unit Normal Vector in the Plane (Example)
The Circle of Curvature
The Circle of Curvature (Example)
The Circle of Curvature (Example)
Position, Velocity, and Acceleration Vectors
Position, Velocity, and Acceleration Vectors (Example)