5.1 Tangent Lines

The tangent line is a terrific tool for better understanding the graph of a function \( z \mathrel{\overset{f}{\leftarrow}} x \). We may center a tangent line at any point on the graph. The tangent line matches both the height and the slope of the graph at the center point. Here's a graph with four tangents drawn in as dotted lines.

Near its center, the tangent is the line that most closely resembles the graph.

Definition.

Suppose \( z \mathrel{\overset{f}{\leftarrow}} x \) is a function. The tangent to \( f \) is the linear function \( z_T \depends x_T \) given by

\[ \begin{align} z_T - z &= \operatorname{\overset{\mathit{x}}{\slope}} z \mult (x_T - x) \end{align} \]

where

\[ \begin{align} &x : \Number, & &z : \Number, & &\operatorname{\overset{\mathit{x}}{\slope}} z : \Number \end{align} \]

are the constants found at the center.

Let's make sense of this definition by working through an example.

Example.

Consider the cubing function,

\[ \begin{align} z &\depends x \\ z &= x^3, \end{align} \]

and its slope

\[ \begin{align} \operatorname{\overset{\mathit{x}}{\slope}} z &= 3 x^2. \end{align} \]

Let's find three tangent lines by centering at \( x = 1 \), at \( x = -1 \), and at \( x = 0 \).

Centering at \( x = 1 \).

By centering at \( x = 1 \), we find constants

\[ \begin{align} &x = 1, & &z = 1, & &\operatorname{\overset{\mathit{x}}{\slope}} z = 3. \end{align} \]

The equation for the tangent line reads

\[ \begin{align} z_T - 1 &= 3 \mult (x_T - 1). \end{align} \]

Centering at \( x = -1 \).

Here the constants for the tangent line are

\[ \begin{align} &x = -1, & &z = -1, & &\operatorname{\overset{\mathit{x}}{\slope}} z = 3. \end{align} \]

The equation for the tangent line reads

\[ \begin{align} z_T + 1 &= 3 \mult (x_T + 1). \end{align} \]

Centering at \( x = 0 \).

At \( x = 0 \) we find

\[ \begin{align} &x = 0, & &z = 0, & &\operatorname{\overset{\mathit{x}}{\slope}} z = 0. \end{align} \]

The tangent line can be written as

\[ \begin{align} z_T - 0 &= 0 \mult (x_T - 0) \end{align} \]

or more simply

\[ \begin{align} z_T &= 0. \end{align} \]

This is an equation for the \( x \)-axis: all points with height zero.

The tangent line gives us a new geometric way to understand level points.

Important.

A level point is any point on a graph that has a horizontal tangent line.

We see, once again, that the cubing function has a level point at \( x = 0 \). If we zoomed in around the origin, the cubing function would look more and more like a horizontal line.