Tangent line

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For a curve

The tangent line to a curve at a point is the best local straight line appropximation to the curve at the point.

For the graph of a function

Suppose f is the graph of a function of one variable and x_0 is a point in the domain of f such that f is continuous at x_0. The tangent line through the point (x_0,f(x_0)) to the graph of f is defined as follows:

  • If the derivative f'(x_0) exists and is finite, it is given by the equation:

\! y - f(x_0) = f'(x_0)(x - x_0)

  • If both the one-sided limits for the derivative give the same sign of infinity (i.e., either both give +\infty or both give -\infty) then we say we have a vertical tangent and the equation is:

\! x = x_0

  • If neither of the above are true, there is no well defined tangent line through the point.

For a parametric description

See parametric differentiation for more.

Consider a curve described parametrically by x = f(t), y = g(t). At t = t_0, the tangent line (geometrically, to the point (f(t_0),g(t_0))), is defined as follows:

  • If the derivatives f'(t_0) and g'(t_0) exist as finite numbers and are not both zero:

\! f'(t_0)(y - g(t_0)) = g'(t_0)(x - f(t_0))

  • If both the derivatives are zero, then we try to see if \lim_{t \to t_0} g(t_0)/f(t_0) exists and is finite, and if so, take that as the derivative that we plug into the equation for the tangent line. If it is a single signed infinity, we get a vertical tangent.


Tangent line may intersect the curve at multiple points

The tangent line to a curve:

  • may intersect the curve at points other than the point of tangency, and
  • may or may not be tangent to the curve at these other points of intersection.

Thus, the definition of tangent line that we could use for the circle (namely, the line that intersects the curve at that point only) does not work in general.

Moreover, lines other than the tangent line may intersect the curve at a unique point. For instance, for the graph of a function, the vertical line through the point intersects the curve at that point alone, even though the vertical line is rarely the tangent line.

Tangent line and crossing the curve

If a curve has a well defined tangent line at a point, then it is usually the case (for nice functions at most points) that the tangent line does not cross the curve near the point, i.e., the part of the curve close to the point of tangency, lies completely to one side of the tangent line.

Moreover, if a well defined tangent line exists, then any line other than the tangent line must cut across the curve.

However, there are examples of situations where the tangent line does cut across the curve. The simplest examples are those involving a point of inflection where the curve changes its sense of concavity, such as the cube map at the origin. More complicated examples involve points where the second derivative of the function is oscillating sign very rapidly very close to the point, such as functions of the form \left \lbrace \begin{array}{rl}x^2 \sin(1/x), & x \ne 0 \\ 0, & x = 0 \\\end{array}\right..

Relation with Taylor series and Taylor polynomials

If f is differentiable at a point x_0 in its domain, then the equation of the tangent line is of the form y = P_1(f,x_0)(x), where P_1(f,x_0) is the first Taylor polynomial of f about x_0. In other words, it is the best approximation of f locally about x_0 by a linear function.