Chapter 2 | Limits
127
Example 2.1 illustrates how to find slopes of secant lines. These slopes estimate the slope of the tangent line or, equivalently, the rate of change of the function at the point at which the slopes are calculated. Example 2.1 Finding Slopes of Secant Lines Estimate the slope of the tangent line (rate of change) to f ( x ) = x 2 at x =1 by finding slopes of secant lines through (1, 1) and each of the following points on the graph of f ( x ) = x 2 . a. (2, 4)
⎛ ⎝ 3
⎞ ⎠
9 4
b.
2 ,
Solution Use the formula for the slope of a secant line from the definition. a. m sec = 4−1 2−1 =3
9 4 −1 3 2 −1
b. m sec = = 5 2 =2.5 The point in part b. is closer to the point (1, 1), so the slope of 2.5 is closer to the slope of the tangent line. A good estimate for the slope of the tangent would be in the range of 2 to 2.5 ( Figure 2.7 ).
Figure 2.7 The secant lines to f ( x ) = x 2 at (1, 1) through (a) (2, 4) and (b) ⎛ ⎝ 3 2 , 9 4 ⎞ ⎠ provide successively closer approximations to the tangent line to f ( x ) = x 2 at (1, 1).
Made with FlippingBook - professional solution for displaying marketing and sales documents online