Rates of Change (The 2nd Pillar of Calculus)

Before we start talking about instantaneous rate of change, let's talk about average rate of change. A simple example is average velocity. If you drive 180 miles in 3 hours, then your average speed is 60 mph. We get this by dividing the distance traveled by the time:

$v_{avg} = \frac{\Delta s}{\Delta t},$ where

$\Delta s$ is the distance traveled and

$\Delta t$ is the time elapsed. We use the Greek letter

$\Delta$ to mean "change in". If we start at position

$s(t_0)$ at time

$t_0$ and end up at position

$s(t_1)$ at time

$t_1$ , then

$\begin{eqnarray*} \Delta s & = & s(t_1) - s(t_0), \cr \Delta t & = & t_1 - t_0. \end{eqnarray*}$ If you plot position against time on a graph,

$v_{avg}$ is the slope of a secant line.

To get the instantaneous veloctity at a particular time $t=a$ , we average over shorter and shorter time intervals. That is, we compute the average velocity between time $a$ and $a + \Delta t$ , and then take a limit as $\Delta t \to 0$ . On a graph, this is taking the slope of secant lines between points that are getting closer and closer. In the limit, we get the slope of a tangent line.

Whether we think in terms of velocity or slope, we get a limit:

$\lim_{\Delta t \to 0} \frac{s(a+\Delta t)-s(a)}{\Delta t}.$ This can also be written as:

$\lim_{x \to a} \frac{s(x)-s(a)}{x-a},$ where

$x=a+\Delta t$ . This quantity (if the limit exists) is called the derivative of

$s(t)$ at time

$t=a$ .

In the animations below, $a=3$ , the $x$ coordinate of point $P$ ; and $a+\Delta t$ is the $x$ -coordinate of point $Q$ . The red tangent line is the limit of the blue secant lines. Note that $\Delta t$ can be positive (first animation) or negative (second animation). For the derivative to exist, the limits as $\Delta t \to 0^+$ and $\Delta t \to 0^-$ must give the same answer.

(Animations from Barry McQuarrie)

The Six Pillars of Calculus

Trigonometry Review

Exponential Functions

Logarithms and Inverse functions

Intro to Limits

Limit Laws and Computations

Continuity and the Intermediate Value Theorem

Limits at Infinity

Rates of Change

The Derivative Function

Basic Differentiation Rules

Product and Quotient Rules

Derivatives of Trig Functions

The Chain Rule

Implicit Differentiation

Derivatives of Logs

Derivatives in Science

Related Rates

Linear Approximation and Differentials

Differentiation Review

Absolute and Local Extrema

The Mean Value and other Theorems

ff vs. f′f'

Indeterminate Forms and L'Hospital's Rule

Optimization

Newton's Method

Anti-derivatives

The Area under a curve

Definite Integrals

The Fundamental Theorem of Calculus

Average and instantaneous velocity

$f$ vs. $f'$