Section 4.2 Worksheet:

Assigned problems: Exercises pp. 238-239, #4, 6, 12, 15, 23, 33 (due Friday, 2/28).

1. What are the three hypotheses of Rolle's theorem?

2. Express Rolle's Theorem in your own words.

3. In what sense is the Mean Value Theorem (MVT) a generalization of Rolle's Theorem?

4. The proof of the MVT is based on Rolle's Theorem (typical of proofs to use simple results to prove more complex ones). Look over the proof of the MVT, and explain how we turn the conditions necessary for the application of the MVT into the conditions necessary for the application of Rolle's Theorem.

5. Who was Joseph-Louis Lagrange?

Notes:

1. Calculus is not about formulas; it is about understanding nature. If a phenomenon in nature is connected, then a model for the phenonenon might be a continuous function; if the phenomenon is smooth, then a differentiable function. Formulas are very useful, of course, but only in the service of knowledge.
2. If you have a complex mathematical object (e.g. function), but are only interested in it in a small region of its domain, then you may want to approximate that object by some sort of simpler linear object (e.g. a tangent line). In this class, you should both understand this idea, and then be able to apply derivatives (and other mathematical knowledge) to find the simpler linear object.
3. Formulas without understanding are useless. If that's all you get out of calculus, then you're of less value (from a mathematical perspective!) than a good calculator.