MATH 140 Study Guide - Final Guide: Riemann Sum, List Of Night Buses In London, Power Rule

My Not EXAMPLE 1 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 t as soon as we know its slope m. The difficulty observe that we can compute an approximation to m by choosing a nearby point Qix, ax) on the graph (as in the figure) and computing the slope mpo of the secant line PO. [A secant line, from the Latin word secans, meaning cutting, is a line that cuts (intersects) a curve more than once.] is that we know only one point, P, on t, whereas we need two points to compute the slope. But we choose x# 1 so that Q* P. Then, 4x44 x-1 PQ For instance, for the point Q(1.5, 20.25) we have .5 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 mpo is to tangent line t should be m = This suggests that the slope of the 2 60 04 1.5 32.5 57.5 1.1 18.5649 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