This table is regularly updated. Last modified: 9/27/00.
| Topic | Title | Description | References | Comments |
|---|---|---|---|---|
| Derivative as slope | Estimating the slope of a curve | Estimate the slope of a curve by zooming in with a graphing calculator and finding directly. Compare with value given by the derivative. | This is a standard "reform calculus" exercise. Ask AD for his version. | - |
| Derivative | Derivative in other sciences | It's nice to give examples of how the derivative turns up in areas at this point. | See eg [Stewart, p. 158]. | The students can be asked to each choose an example and present it to the class (MR). |
| Implicit differentiation | Cartography | lecture | - | - |
| Trigonometry (triangles) | Eratosthenes' measurement of the earth | - | This can be found in many sources, eg [Hahn, p. 9]. Ask AD for homework problems. | The students liked this. |
| Trigonometry | The weather in Fairbanks | A sine curve fits the daily temperature in Fairbanks. | Mathematics Teacher 70 (1977) p. 534-537 | This needs some homework problems. |
| Optimization | The bee eye as an example of optimum vision | An ommatitium is just the right diameter to see accurately without problems coming from diffraction. | [Alexander, p. 30-35],[Feynman I Lect 36, p. 6-9] It's interesting that these two references give different derivations of the result. | The differaction effect can be demonstrated by shining a laser through a pinhole. |
| Max-min | Rainbows | Explaining the rainbow, its position in the sky, its colors, the secondary bow, etc, is a good (and subtle) application of calculus. | A detailed description can be found in [MAA vol. 3, p. 42-55]. | The tricky part is that a critical point of a function leads to a concentration of light rays. |
| Exponential growth | Moldy bread | Grow mold on a piece of bread and measure daily the amount covered by mold. | [Blanchard p. 137] or ask AD for writeup for calculus students. | pretty amazing! |
| Exponential growth | E. coli | Section 11.6 in [Hahn] contains much information and some tables of data to which exponential growth curves can be fit. | - | - |
| Exponential decay | Carbon-14 dating | Carbon-14 has been used to date many things, for example the Shroud of Turin (see website XXX), the Lascaux cave paintings, and Stonehenge. | - | - |
| Exponential decay | Glottochronology | Words tend to leave a language similarly to exponential decay. This provides a way to date languages. | UMAP Module 334 (1982); R. Lees, "The basic of glottochronology" Language 29 (1953) 113-127; Conversational Calculus v. I, p. 208. | - |
| Newton's method | Fractals and chaos with Newton's method | The basins of attraction over the complex numbers are often fractals. And trying to find the real roots of x^2 + 1 leads to chaotic behavior. | There are many references for the fractals part, eg UMAP Module 716. For the chaos part, see [Strang, p. 144f] | This shows how a basic calculus topic can lead to interesting mathematics. |
| History | Calculus in the past at your school | Have students find out when calculus was first taught at your school. What book did they use, and what topics did they study? | This can be investigated in the school archives. | fun! |
| History | Ancient calculus texts | At Mt Holyoke we have an original copy of the 1748 text by Maria Agnesi, and Smith has a copy of the first English translation. | Some materials from Agnesi are available on our web site | - |