Proof of Darboux's Theorem
- Author:
- Nick Wasserman
Instructions
Use this dynamic activity to explore the proof of Darboux's Theorem. You can change the value of the interval [a, b] and the slide the value M between f'(a) and f'(b). Observe the generated function g(x)=f(x)-Mx. Find such point c that f'(c)=M by slide c to match M on the y-axis.
Reflection Questions
1. What is the relation between f'(x) and g'(x)? 2. If f'(a)>M>f'(b), what does it mean to g'(a) and g'(b)? 3. When f'(c)=M, what is special about g(c) and g'(c)? 4. How to write the proof from the demonstration of this activity?