Exploration 1: A Locally Linear Function and Its Tangent Line

Use this interactive figure to explore the concept of linearization. As you "zoom in" on a graph, the graph begins to look like a straight line. In the left pane, drag point A, B, C, D, and E, to change the graph of the function. Next, drag point P to change the position of the tangent line. Finally, drag the center point M of the magnifying square and observe the magnified view of the function in the pane on the right. Use the slider at the bottom of the left pane to increase the magnification by a factor of 1, 2, 4, 8, ...and so on (by a factor of 2n). When you zoom in on the tangent line, the function and the line will become almost indistinguishable.