Tangent lines

01Match a point and a slope#

A tangent line is the affine function that matches a differentiable curve's value and slope at one chosen input. It gives a simple local model of the curve near that input.

A line with slope $m$ through $(a,b)$ changes its output by $m(x-a)$ when the input moves from $a$ to $x$. Its equation is $y=b+m(x-a)$.

For a tangent to $f$ at $a$, use $b=f(a)$ and $m=f'(a)$:

02Keep the base point visible#

Take $f(x)=x^2$ at $a=2$. The curve passes through $(2,4)$ and has derivative $f'(2)=4$. Its tangent is $T(x)=4+4(x-2)=4x-4$.

Substituting $x=2$ checks the point: $T(2)=4$. The line's slope is $4$, checking the derivative. A line $4x$ has the right slope but misses the point.

03Local agreement#

A tangent need not stay on one side of a curve or intersect it only once. For $f(x)=x^3$, the tangent at zero is $y=0$, and the curve crosses that line at zero. Tangency is defined by local value and derivative agreement.

Constructing the tangent requires a finite derivative at the base point. The line is an exact object; using it to estimate the curve away from that point is a separate approximation step.