Tangent Lines to Curves

Suppose a secant line $ s $ intersects a curve at two distinct points $ P $ and $ Q $. As the point $ Q $ moves closer and closer to $ P $ along the curve, the secant line approaches a limiting position. This limiting line is called the tangent line $ t $ to the curve at point $ P $.
example

A tangent line is the line that locally follows the same direction as the curve at a given point.

In other words, if you zoom in near the point of tangency, the curve and the tangent line look almost identical over a very small region.

A tangent line does not always touch the curve at only one point. For many curves, the tangent line may intersect the curve again at other points.

example of a tangent line

However, some curves have special geometric properties. For example, in circles and ellipses, the tangent line has exactly one point of contact with the curve.

Example. In a circle, the tangent line at a point is perpendicular to the radius $ r $ drawn to the point of tangency $ P $. The line touches the circle at exactly one point.
example of a tangent line to a circle

Finding the Equation of a Tangent Line with Derivatives

If the curve is the graph of a function $ y=f(x) $, the tangent line at a point can be found using derivatives.

The derivative $ f'(x_0) $ represents the slope of the tangent line at the point whose x-coordinate is $ x_0 $.

Once the slope is known, the tangent line can be written using the point-slope form:

$ y-f(x_0)=f'(x_0)(x-x_0) $

or equivalently

$ y=f'(x_0)(x-x_0)+f(x_0) $

Example

Consider the parabola

$ f(x)=x^2 $

We want to find the tangent line at the point where $ x_0=1 $.

First, compute the corresponding point on the curve:

$ f(1)=1^2=1 $

So the point of tangency is $ P(1,1) $

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Now write the general equation of a line passing through that point:

\[ (y-y_0)=m \cdot (x-x_0) \]

Substituting the coordinates of $ P $:

\[ (y-1)=m \cdot (x-1) \]

Next, compute the derivative of the function:

$ f'(x)=2x $

Evaluate the derivative at $ x_0=1 $:

$ f'(1)=2 $

Therefore, the slope of the tangent line is

$ m=2 $

Substitute this value into the equation of the line:

$ y-1=2(x-1) $

Simplify the equation:

$ y-1=2x-2 $

$ y=2x-1 $

Therefore, the tangent line to the parabola $ y=x^2 $ at the point $ P(1,1) $ is

$ y=2x-1 $

The tangent line describes the local behavior of the function near the point of tangency.

example

And so on.