On Tuesday, 12/5, we began class by presenting the HW10 Challenge Problems in their entirety.
We then completed proofs both parts of the Fundamental Theorem of Calculus (Theorem 7.5.1).
Proof of part i) involved using properties of partitions and lower and upper sums.
Proof of part ii) was accomplished by proving two claims:
- is continuous on
- if is continuous at .
Finally, we went over problem 1 from the Examples and Applications section of the lecture notes. This problem examines a situation that at first glance, violates the Fundamental Theorem of Calculus. However, upon further analysis Fundamental Theorem of Calculus does not actually apply to the given setting.