Subsection 2.1.1 Slopes of Continuous Functions, Tangent Lines and Secant Lines
Consider a curve \(C\) that is the graph of \(y = f(x)\) where \(f\) is a continuous function, and is intuitively “smooth”. Then, let \(P = (x_0, y_0)\) be a point on the graph so that \(y_0 = f(x_0)\text{,}\) and assume that \(P\) is not an endpoint of the graph.
Recall that the tangent line to a circle is a line which touches the circle at exactly one point. It has the property that it is perpendicular to a radius segment which extends out to the point where it intersects the tangent line.
Intuitively, the
tangent line to a curve at a point passes through that point, and has the same slope, steepness, or “direction” as the curve, at the point of tangency. For example, consider
this Desmos applet 1 . The word tangent comes from the Latin word for “touching”, intuitively because a tangent touches the curve at a single point, barely grazing it, instead of cutting across a curve in two places.
To determine the slope of the tangent line at \((x_0, f(x_0))\text{,}\) we will consider the slope of the line between \((x_0, f(x_0))\) and some nearby point, say \((x_0 + h, f(x_0 + h))\text{,}\) where \(h\) is a small, non-zero number. This line between two points on the graph of a function is called a secant line. Then, the slope of this line is,
\begin{align*}
m_{\text{sec}} \amp = \frac{\Delta y}{\Delta x}\\
\amp = \frac{f(x_0 + h) - f(x_0)}{(x_0 + h) - x_0}\\
\amp = \frac{f(x_0 + h) - f(x_0)}{h}
\end{align*}
In summary,
\begin{equation*}
\boxed{m_{\text{sec}} = \frac{f(x_0 + h) - f(x_0)}{h}}
\end{equation*}
The quantity \(\frac{f(x_0 + h) - f(x_0)}{h}\) is sometimes called a difference quotient, because the numerator represents the difference in \(y\)-values, the denominator represents a difference in \(x\)-values, so it represents a quotient of differences. As \(h\) approaches 0, this approximate slope will be closer and closer to the slope of the tangent line. However, notice that if \(h = 0\text{,}\) then the expression becomes,
\begin{equation*}
\frac{f(x_0 + 0) - f(x_0)}{0} = \frac{f(x_0) - f(x_0)}{0} = \frac{0}{0}
\end{equation*}
An expression of the form \(\frac{0}{0}\) is called an indeterminate form. However, we can still consider the behavior of this expression as \(h \to 0\text{.}\) In calculus, we define the slope of the graph as the limit of the slope of the secant line, as \(h \to 0\text{.}\)
Subsection 2.1.2 The Tangent Line, Slope of Functions
Definition 2.1.1.
Let \(f\) be continuous at \(x_0\text{.}\) Then, the slope of \(f\) at \(a\) is the number given by the limit,
\begin{equation*}
\boxed{m = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}}
\end{equation*}
provided the limit exists.
Then, the tangent line to \(f\) at \(a\) is the line with slope \(m\) and that passes through the point \((x_0, f(x_0))\text{.}\) We also say that the line is tangent to \(f\) at \(x_0\text{.}\)
Then, an equation of the tangent line is, using point-slope form,
\begin{equation*}
y - f(x_0) = m(x - x_0) \qquad \text{or, solving for $y$} \qquad y = m(x - x_0) + f(x_0)
\end{equation*}
Intuitively, \(\lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}\) provides a “formula” for the slope of a function at the particular point \(x_0\text{.}\)
Intuitively, if you zoom in on the graph, the graph will resemble a straight line. Then, the slope of this straight line is the slope of the tangent. Difference quotients were first written by Fermat in 1629, who applied it to polynomials. However, different quotients can be applied to any continuous function.
