A tangent line is a line which approximates a curve well near a particular point. This project investigates tangent lines numerically. The investigation only works if we can find points on the curve arbitrarily close to the point we're interested in. In practice, that requires that the curve have some sort of formula.

First, use your calculator to graph the curve with equation y = sin(x) + 2cos(x). Clear the Y= screen first; F1,8 will do it.

Next, we want a formula for the slope between the point (0, 2) on this graph and some nearby point.
The calculator has named this function y1, so we want (y1(x) - y1(0))/(x - 0).
So that we can easily look at other points later, enter this formula for y2 as (y1(x)-y1(a))/(x-a)|a=0.
(The vertical line is pronounced with or such that.)

Now go to the TABLE screen.
It should show three columns labeled x, y1, and y2.
The x-values displayed are determined by the TblSet menu.

The entry under y2 next to x = 0 should be undef. (Why?)

The nearby entries in the y2 column give slopes of lines through (a, y1(a)) and nearby points on the graph.

Adjust the entries in the TblSet menu until you can get a good estimate of the slope of the tangent line to the graph at the point (0, 2).
 

Exercises:

1. Repeat the process at a couple of different points on the curve.

2. Repeat the process for the curve with equation y = 3.127sin(1.3x) + 2.485cos(1.8x) at some convenient points on the graph.

3. Repeat the process for the curve with equation y = x^x at the point (2, 4).

4. Repeat the process for the curve with equation y = sin(1/x) at the point (0,0).
    (This formula doesn't make sense when x=0, but the calculator usually doesn't mind.)

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