The activity at the beginning of hour helped my understanding of the generalization represented in the first gif by giving me a good idea how this concept of secant/tangent line works. It helped to see how to get a tangent line of 1 point, and it was very simple process. All you had to to was find the slope between the points and continue doing that with points closer to the point where you want to find the tangent line. I did struggle a lot during the activity, though most of it was due to technology (I direct you to the video at the end of my post), I did struggle with recreating the second gif. I thought the point that you put into the y=m(x1-x)+y1 was (a,b), not (a, f(a)). But once I figured that out, it was smooth sailing from there. The changes made from the first graph to the second graph were that you put the point (a, f(a)) for the x and y in the slope and you had to add the point (b, f(b)) to the graph. The set up from the first two graphs helped with making my own graph by having the points prepared to move with the function with the slope formula all prepared. All I had to do was make a cool looking graph. The analysis of secant lines help up determine the tangent line of a function by finding out that as we get closer to the point, we can approximate the tangent line more accurately. Finding the secant lines allow us to approximate a better slope for the tangent line and gets us closer to the answer. The links for each of the gifs are below:
https://docs.google.com/file/d/0B9vhBEIDe5D3WTNPa1U1c0pacGVKRjI1MkRvWlo4QnY3dFNj/edit AKA 1st Gif
https://docs.google.com/file/d/0B9vhBEIDe5D3WTNPa1U1c0pacGVKRjI1MkRvWlo4QnY3dFNj/edit AKA 1st Gif
https://docs.google.com/file/d/0B9vhBEIDe5D3MFNtd0syTERkTnZYRlE2WEtZQVB4T1pWaG44/edit AKA Bonus Gif
https://docs.google.com/file/d/0B9vhBEIDe5D3MjA1NW9ZRXRYVmVJY2lHTHlBLWhXbE9qdDNj/edit AKA My Gif