The GIF Activity that we worked on this week in AP Calc helped introduce us to derivatives by having us explore slopes of tangent and secant lines. The activity that we worked on before creating our GIFs required that we find the slope of four secant lines through different points on a power function. Each line went through the point A and through either point B, C, or D and represented the average rate of change of the secant line through those points. We discovered that as the secant line went through the points closer to point A, the line evened out and more closely represented the tangent line and the instantaneous rate of change at point A, or the derivative. This activity helped introduce us to the idea illustrated in the first GIF, in which the secant lines are shown as a moving point approaches the set point. During the activity we were also given the equation y=m(x-x1) + y1, a variation of the point-slope formula that would help us to create the equations for our graphs and GIFs.
At first, my group struggled with the basics of creating the graphs and GIFs because we had never had the chance to work with Desmos or Snag It before. Through trial and error and a lot of practice, we were able to construct the graphs with Desmos and record animated GIFs. One thing that we struggled with when creating the first graph was figuring out how to create a sliding point. We had graphed the function and the point (2,2), and through more exploration and scratch work on paper, discovered that the sliding point and secant line function must contain variables. This realization helped us to better understand the concept of moving functions and how they must include variables to account for all of the points that the graph travels through.
When we began working on our second graph, we knew that we no longer needed a set point and a moving point, but two moving points on the function. Through trial and error and by getting our thoughts out on paper, we were able to create a secant line that moves and intersects both moving points with the equation: g(x)= f(a) - f(b) (x-a) + f(a)
a-b
Through this activity, we learned that an analysis of secant lines helps us to find the slope of a tangent line at a point. By finding the slope of many different secant lines that go through the same base point on the function, we can find the slope of the tangent line at that point, or the derivative.
At first, my group struggled with the basics of creating the graphs and GIFs because we had never had the chance to work with Desmos or Snag It before. Through trial and error and a lot of practice, we were able to construct the graphs with Desmos and record animated GIFs. One thing that we struggled with when creating the first graph was figuring out how to create a sliding point. We had graphed the function and the point (2,2), and through more exploration and scratch work on paper, discovered that the sliding point and secant line function must contain variables. This realization helped us to better understand the concept of moving functions and how they must include variables to account for all of the points that the graph travels through.
When we began working on our second graph, we knew that we no longer needed a set point and a moving point, but two moving points on the function. Through trial and error and by getting our thoughts out on paper, we were able to create a secant line that moves and intersects both moving points with the equation: g(x)= f(a) - f(b) (x-a) + f(a)
a-b
Through this activity, we learned that an analysis of secant lines helps us to find the slope of a tangent line at a point. By finding the slope of many different secant lines that go through the same base point on the function, we can find the slope of the tangent line at that point, or the derivative.
I also found this super helpful webpage about secant and tangent lines and the limit definition of a derivative: