A very short post today. Audrey Weeks, author of Calculus in Motion, sent me this link to a BBC article by Melissa Hogenboom discussing the 10 most beautiful equations. Each equation can be clicked for more details and some of the links even have videos discussing the equation. Click here for You decide: What is the most beautiful equation? After reading about them, you can vote for your favorite and see the results so far.
A very typical calculus problem is given the equation of a function, to find information about it (extreme values, concavity, increasing, decreasing, etc., etc.). This is usually done by computing and analyzing the first derivative and the second derivative. All the textbooks show how to do this with copious examples and exercises. I have nothing to add to that. One of the “tools” of this approach is to draw a number line and mark the information about the function and the derivative on it.
A very typical AP Calculus exam problem is given the graph of the derivative of a function, but not the equation of either the derivative or the function, to find all the same information about the function. For some reason, student find this difficult even though the two-dimensional graph of the derivative gives all the same information as the number line graph and, in fact, a lot more.
Looking at the graph of the derivative in the x,y-plane it is easy to very determine the important information. Here is a summary relating the features of the graph of the derivative with the graph of the function.
| Feature | the function |
 > 0 |
is increasing |
| < 0 |
is decreasing |
| changes – to + |
has a local minimum |
| changes + to – |
has a local maximum |
| increasing |
is concave up |
| decreasing |
is concave down |
| extreme value |
has a point of inflection |
Here’s a typical graph of a derivative with the first derivative features marked.
Here is the same graph with the second derivative features marked.
The AP Calculus Exams also ask students to “Justify Your Answer.” The table above, with the columns switched does that. The justifications must be related to the given derivative, so a typical justification might read, “The function has a relative maximum at x = -2 because its derivative changes from positive to negative at x = -2.”
| Conclusion | Justification |
| y is increasing | > 0 |
| y is decreasing | < 0 |
| y has a local minimum | changes – to + |
| y has a local maximum | changes + to – |
| y is concave up | increasing |
| y is concave down | decreasing |
| y has a point of inflection | extreme values |
For notes on vertical asymptotes see
For notes on horizontal asymptotes see Other Asymptotes
Is God a Mathematician? by Mario Livio begins
When you work in cosmology … one of the facts of life becomes the weekly letter, e-mail, or fax from someone who wants to describe to you his own theory of the universe (yes, they are invariably men). The biggest mistake you can make is to politely answer that you would like to learn more. This immediately results in an endless barrage of messages. So how can you prevent the assault? The particular tactic I found to be quite useful (short of the impolite act of not answering at all) is to point out the true fact that as long as his theory is not precisely formulated in the language of mathematics, it is impossible to assess its relevance. This response stops most amateur cosmologists in their tracks. … Mathematics is the solid scaffolding that holds together any theory of the universe.
Is God a Mathematician? discusses the question of whether mathematics was invented or discovered. Dr. Livio’s other popular books include The Accelerating Universe (cosmology), The Golden Ratio: The Story of Phi, the World’s most Astounding Number, and The Equation that Couldn’t be Solved: How Mathematical Genius Discovered the Language of Symmetry. All are excellent reads for teachers and students.
Teaching Calculus 