Tangent lines and derivatives

Section 2.2 Tangent lines and derivatives

In the previous section we introduced the concept of derivative through kinematics. Here we see that the derivative also has a deep interpretation in geometry, as calculating the slope of the tangent line to the graph of a function. We study our first “tangent line problem”, which leads us to the question of how to evaluate (and define) limits.

Objectives

You should be able to:

Describe and illustrate the connection between the velocity and tangent problems.
Explain the difference between the slope of a secant line connecting two points on a curve and the slope of the tangent line to a curve at a point.
Describe the limit process that arises in the calculation of the slope of a tangent line.
Use correct notation for the limit process, to represent the slope of a tangent line as an appropriate limit of the slope of a secant line.
Calculate the equation of the tangent line for simple functions.

Subsection 2.2.1 Instructional videos

Subsection 2.2.2 Key concepts

Concept 2.2.1. Secant line.

Given two points \(P_1(c,f(c))\) and \(P_2(x_2,f(x_2))\) on the graph of a function \(f\text{,}\) the slope of the secant line through \(P_1\) and \(P_2\) is given by

\begin{equation*} m_{P_1 P_2} = \frac{f(x_2) - f(c)}{x_2 - c}. \end{equation*}

Equivalently, defining \(x_2 = c+h\text{,}\) it can be written as

\begin{equation*} m_{P_1 P_2} = \frac{f(c+h) - f(c)}{h}. \end{equation*}

This is known as the difference quotient of the function \(f\) at \(x=c\text{.}\)

Concept 2.2.2. Tangent line.

The slope of the tangent line to the curve \(y=f(x)\) at \(P_1(c,f(c))\) is obtained by taking the limit \(x_2 \to c\) or, equivalently, \(h \to 0\text{:}\)

\begin{align*} m =\amp \lim_{x_2 \to c} \frac{f(x_2) - f(c)}{x_2 - c}\\ =\amp \lim_{h \to 0} \frac{f(c+h)- f(c)}{h}. \end{align*}

Thus, \(m\) is precisely equal to the derivative of \(f\) at \(x=c\text{;}\) that is, \(m = f'(c)\text{.}\)

MATH 144: Calculus for the Physical Sciences I