The Tangent Problem

The word tangent itself is derived from the Latin word tangens, which means “touching.” In mathematics and in geometry, a tangent line is a line drawn that touches a curve at an angle perfectly parallel to that particular section of the curve. (The curve can be anything to a drawn circle on a page, or the line on a graph.)

However, the tangent line is no ordinary line drawn, but in its conception, a very unique state. It is a line drawn that precisely touches a curve at one, single, minute point — and no other. If zoom in on our curve and our tangent line until each are drawn down to a single string of atomic-sized pixels, and our tangent line is precisely parallel to the single pixel that it is touching.

If we have a point on a curve, then the tangent line is the specific line drawn that touches the curve precisely at that point. Precision.

Say the point is (2,1) on some curve. The tangent at P(2,1) touches the point P(2,1) precisely. At P(2.00000, 1.00000). Not at (2.5, 1.3). Not even at (2.00030, 1.00007) if those points even exist on the corresponding curve. But you get my point. These numbers are specific.

The significance of this is that the tangent line is the slope of the graph at that specific point. When real-world data is plotted on the x- and y- axes of a graph, then the slope has meaning at a given point. It might tell you the rate of population growth at a precise moment in time, if you have population on the y and time in years on the x, for example.

Euclid said that a tangent is a line that intersects a circle once and only once. This statement remains true today for a circle, though not for more complicated curves. We can draw any number of lines through the graphs of sine or cosine functions, none of which are tangents.

We can find the equation for tangent line, t, once we know slope m. If we know one point on a curve, we need a second to compute slope. Points P and Q let us say. We can approximate the slope by taking a second x-value we will call $x_2$ and computing its y-coordinate. Then we can compute the slope of the secant line.

The secant line is a line that intersects a curve more than once. From secans, Latin for “cutting”, the secant uses two points one the curve that it cuts through. If we can bring them closer and closer together, our secant line will approximate the slope of the original point, if we bring them significantly closer together.

Notes

• Example parabola $y = x^2$ with P(1,1). Second example flash unit on camera.

References

1. Early Transcendentals. Eight Edition. James Stewart.

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