Proof of Fundamental Theorem of Calculus (ii)
Instructions
Use this dynamic activity to explore the demonstration of the proof of Fundamental Theorem of Calculus (ii). Move the point on x-axis to adjust the value of point c. Adjust so that |x-c|<, |f(x)-f(c)|<. Change to show [F(x)-F(c)] and [x-c] value. Approximately find the point c such that the F'(c) shown in the activity equals to the value of f(c). Observe the locus of [F(x)-F(c)]/[x-c] as well as the geometric interpretation of F'(c).
Reflection Questions
1. In order to show that F'(c)=f(c), rewrite F'(c) = lim (xc) [(F(x)-F(c))/(x-c)] and try to show the limit equals to f(c). Which is given an >0, we must provide a >0 such that if |x-c|<, |(1/(x-c))(Integral(xc)f(t)dt)|<. 2. Consider the continuity of f, observe the two - and - bands. 3. Use Abbott, S. (2015) pp.235-236 as reference for the formal proof.