One of the fundamental themes of calculus is to find the slope of the tangent line to a curve at a point. To see how this can be done, consider the point (2, 4) on the graph of the quadratic function f(x) = x2, as shown in the figure.

(a) Find the slope m1 of the line joining (2, 4) and (3, 9).
(b) Find the slope m2 of the line joining (2, 4) and (1, 1).
(c) Find the slope m3 of the line joining (2, 4) and (2.1, 4.41).
(d) Find the slope mh of the line joining (2, 4) and (2 + h, f (2 + h)) in terms of the nonzero number h.
(e) Evaluate the slope formula from part (d) for h = ˆ’1, 1, and 0.1. Compare these values with those in parts (a)-(c).
(f) What can you conclude the slope mtan of the tangent line at (2, 4) to be? Explain.

Transcribed Image Text:

(2, 4)/ 4 3- 2- 1- 3 2 1 1 2 3