Find an equation of the tangent line to the function y 3x² at the point P(1, 3). Solution We will be able to find an equation of the tangent line as soon as we know its slope m. The difficulty is that we know only one point, P, on , whereas we need two points to compute the slope. But observe that we can compute an approximation to m by choosing a nearby point Q(x, 3x²) on the parabola (as in the figure below) and computing the slope meo of the secant line PQ. (A secant line, from the Latin word secans, meaning cutting, is a line that cuts (intersects) a curve more than once.] V We choose x = 1 so that Q P. Then, -3x²-3- 3*2 mpQ For instance, for the point Q(1.5, 6.75) we have the following. ✔) -3 mpQ 6.75 1.5 lim Q-P X 2 1.5 ✔ 1.1 The tables below show the values of mpo for several values of x close to 1. The closer Q is to P, the closer x is to 1 and, it appears from the tables, the closer mp is to 6 mpQ 9 7.5 6.3 3.75 X (mp) - m and lim .S 1.01 6.030 .99 1.001 6.003 mpQ 3 0 .S .9 5.700 4.5 5.970 .999 5.997 -7.5 We say that the slope of the tangent line is the limit of the slopes of the secant lines, and we express this symbolically by writing the following. 3x² -6 @ ✔ ✓ This suggests that the slope of the tangent line should be m-0 x.