Find an equation of the tangent line to the function y = 4x4 at the point P(1, 4). Solution We will be able to find an equation of the tangent line 9 as soon as we know its siope m. The difficulty is that we know only one point, P, on 5', 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, 4x4) on the parabola (as in the figure below) and computing the slope mm 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.] C 1 G) We choose x 1 so that Q i P. Then, 4x4 4 m = 7. PQ X 1 For instance, for the point Q(1.5, 20.25) we have the following. The tables below show the values 01'me 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 mm is to :l . This suggests that the slope of the tangent line 3 should be m = :l . x mPQ x mPQ 2 60 O 4 1.5 32.5 .5 7.5 1.1 18.564 .9 13.756 1.01 16.242 .99 15.762 1.001 16.024 .999 15.976 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. . . 4x4 4 Qllinp(mPQ) = m and lll'T'l (7) = [:l. Xel X1 Assuming that this is indeed the slope of the tangent line, we use the pointslope form of the equation of a line (see Appendix B) to write the equation of the tangent line through (1, 4) as the following. y(l:])=(:])(x1) 0r y=(:])x(:l)