Slope of the Tangent Line (darker slopes for visibility)
Points A and C are moveable. The tangent to the graph of f(x) = x^2 at the point C is in blue. The slope of this tangent line is in blue and designated as mTan. The secant line connecting points A and C is in green. The slope of this secant line is in green and designated as mSec. 1. Select a location for point C. 2. Move point A towards point C.
What happens to the distance between the x-coordinates as A approaches C? What happens to the slope of the secant line as A approaches C? As A approaches C, how does the slope of the secant line compare to the slope of the tangent line? Interpretation: As A moves closer to C, the slope of the secant line connecting A and C approaches the slope of the tangent line at C. Thus, we can define the slope of the tangent line as the limit of the slope of the secant line as A approaches C (or as delta x approaches 0).