# hewng Savea work Revert to Last HesponseMy Notes Ask Your leacFind an equation of the tangert line to the function y4x2 at the point P(1,4).EXAMPLE 1now only one point, P, on t, whereas we need two points toSOLUTION We will be able to find an equation of the tangent line t as soon as we know its slope m. The difficulty is that we kcompute the slope. But observe that we can compute an approximation to m by choosing a nearby point Qx, 42) on the graph (as in the figure) and computing the slope mag of theA secant line, from the Latin word secans, meaning cutting, is a line that cuts (intersects) a curve more than once.]We choose x# 1 so that Q P. Then,mpQFor instance, for the point Q(1.5,9) we haveVideo Example. ThisThe 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 meo is tosuggests that the slope of the tangent line t should be m21241.5 10.56001.01 8.040 99 7.960.001 8.004 .999 7.9968.4We say that the slope of the tangent line is the limit of the slopes of the secant lines, and we express this symbolically by writinglimmPO=mandlim 4x--4Assuming that this is indeed the slope of the tangent line, we use the point-slope form of the equation of a line (see Appendix B) to write the equation of the tangent line through (1, 4) as(x 1)The graphs below illustrate the limiting process that occurs in this example. As Q approaches Palong the graph, the corresponding secant lines rotate about P and approach the tangentline tQ appeoaches P lrom the righeTalk to Tutorhere to search8:26 PMM3/22/20193

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Step 1

To find an equation of the tangent line to the fnction y = 4x2 at the point P(1,4).

Compute an approximation to m by choosing a nearby point Q(x,4x2 ) on the graph .

Choose x≠1 so that Q≠P. Find the value of m for the point Q(1.5, 9).

Step 2

The table shows the values of m for several values of x close to 1. The closer P is to Q, the closer x is to 1.

The closer mPQ is to 8.

Step 3

The slope of the tangent line is the limit...

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