A tangent line is a line which approximates a curve well near a particular point. This project investigates tangent lines graphically. The calculator part of the investigation only works if the calculator can graph the curve, which requires that the curve have some sort of formula.

First, use your calculator to graph the curve with equation y = sin(x) + 2cos(x).

Now zoom in on the point (0, 2) until the graph looks like a straight line.

It may take several applications of the ZoomIn utility before the graph begins to look like a straight line. Remember to keep the point (0,2) on the screen. Once the graph looks straight to you, use F3:Trace to create a cursor that stays on the graph. Move it to two different points and compute the slope of the graph.

Use the slope you just computed and the coordinates (0, 2) to find an equation for the line tangent to the curve at the point (0, 2).
 

Exercises:

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

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

3. 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.)

4. What happens with the curve which has the equation y = 2x² when x is less than 2 and the equation y = x⁴ - 10 when x is greater than or equal to 2?
        (You enter this two-part formula on the calculator as when(x<2,2x^2,x^4-10).)

5. What happens with the curve with which has the equation y = 2x² when x < 2 and the equation y = x⁴ - 8 when x => 2?

6. The calculator will automatically find tangent lines to graphs it's drawn.
        Use the F5:Math menu; Tangent is item A.
        You can either move the cursor to the point on the curve that you want, or you can type in its x-coordinate.
        Then ENTER will draw the tangent line and give you its equation.
        Use this utility to check your earlier answers.

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