GIFs! Or as I like to call them, "gifes", because I'm not gonna enter that debate.
The activity we did at the beginning of the hour was to determine the slope of the tangent line at a certain point on the graph. The activity helped to point out that, while we can't immediately find the slope of the point, we can find the slope of lines with other points that get closer and closer to that point. (lots of points were involved in this process) The activity served to further our understanding of what is presented in the first GIF: the idea that, as points get closer in distance to the desired point, the slope of the lines of those points gets closer to the slope of the tangent line of the desired point. (side note: forgive me for the many prepositions; I could not figure out a better way to word that.)
There were two main causes of struggle during this quest for derivative knowledge. One was the fact that I was using a Mac and I apparently can't type fast. This caused a depressing number of issues during this activity. The second issue (and the more math related one) was figuring out how to create a function for the graph. Although we were taught how to use Desmos, we had no instructions on how to translate what we knew the graph had to do into a graphable function. Thankfully (unlike the issue of my inability to use a Mac), we were able to solve this quickly. We realized that we could take the generic graph model, y=m(x-x1)+y1, and modify it to suit our needs. The formula we ended up graphing was g(x)=(f(a)-2)/(a-2)*(x-2)+2, which had a set point of (2,2) and a movable point of (a.f(a)).
There were very few changes we had to make from the first GIF to the second; all we had to do was to make the function of the line more generic. Instead of having only one movable point and a set point, the new function we created had two moving points and no set points. The equation for the line turned out to be g(x)=(f(a)-f(b))/(a-b)*(x-b)+b, where the two points were (a,f(a)) and (b,f(b)).
The setup for the first two GIFs worked really well for the function we created; in fact, there was very little to change between the second GIF and the third GIF. All we did was change the equation of f(x). The g(x) equation was essentially kept the same.
The purpose of this entire lesson was to show how looking at secant lines (lines that cross the circle at two points) can help us determine the slope and equation of a tangent line (a line that crosses the circle at one point). By looking at secant lines whose points get extremely close to the desired point, we can determine the slope of the line tangent to the desired point.
The activity we did at the beginning of the hour was to determine the slope of the tangent line at a certain point on the graph. The activity helped to point out that, while we can't immediately find the slope of the point, we can find the slope of lines with other points that get closer and closer to that point. (lots of points were involved in this process) The activity served to further our understanding of what is presented in the first GIF: the idea that, as points get closer in distance to the desired point, the slope of the lines of those points gets closer to the slope of the tangent line of the desired point. (side note: forgive me for the many prepositions; I could not figure out a better way to word that.)
There were two main causes of struggle during this quest for derivative knowledge. One was the fact that I was using a Mac and I apparently can't type fast. This caused a depressing number of issues during this activity. The second issue (and the more math related one) was figuring out how to create a function for the graph. Although we were taught how to use Desmos, we had no instructions on how to translate what we knew the graph had to do into a graphable function. Thankfully (unlike the issue of my inability to use a Mac), we were able to solve this quickly. We realized that we could take the generic graph model, y=m(x-x1)+y1, and modify it to suit our needs. The formula we ended up graphing was g(x)=(f(a)-2)/(a-2)*(x-2)+2, which had a set point of (2,2) and a movable point of (a.f(a)).
There were very few changes we had to make from the first GIF to the second; all we had to do was to make the function of the line more generic. Instead of having only one movable point and a set point, the new function we created had two moving points and no set points. The equation for the line turned out to be g(x)=(f(a)-f(b))/(a-b)*(x-b)+b, where the two points were (a,f(a)) and (b,f(b)).
The setup for the first two GIFs worked really well for the function we created; in fact, there was very little to change between the second GIF and the third GIF. All we did was change the equation of f(x). The g(x) equation was essentially kept the same.
The purpose of this entire lesson was to show how looking at secant lines (lines that cross the circle at two points) can help us determine the slope and equation of a tangent line (a line that crosses the circle at one point). By looking at secant lines whose points get extremely close to the desired point, we can determine the slope of the line tangent to the desired point.