1.1 - Introducing Calculus: Can Change Occur at an Instant?

The Origins of Calculus

Two mathematicians are credited with developing calculus independently during the 17th century.

Isaac Newton (England, born 1643) - Developed calculus methods while studying physics and motion

Gottfried Leibniz (Germany, born 1646) - Developed similar concepts around the same time - Much of our modern notation comes from Leibniz

The controversy over who “invented” calculus first still goes on today. Most historians think both developed these ideas at the same time, since math had reached a point where calculus was the natural next step.

The Central Question of Calculus

Can change occur at an instant?

Can we measure how fast something is changing at an exact moment in time, not just over an interval? This question drives everything we study in calculus.

Average Rate of Change

The average rate of change measures how a quantity changes over an interval.

\text{Average Rate of Change} = \frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}

This is just the slope between two points on a curve.

Example: Running a race

A runner’s distance from the starting line is tracked over time. At

t = 0

minutes, the runner is at

D(0) = 0

meters. At

t = 10

minutes, the runner is at

D(10) = 1500

meters.

Average speed for the entire run:

\frac{D(10) - D(0)}{10 - 0} = \frac{1500 - 0}{10} = 150 \text{ meters per minute}

The word “per” always signals a rate of change (miles per hour, dollars per item, and so on).

Secant Lines

When we connect two points on a curve, we get a secant line. The slope of that secant line equals the average rate of change between those two points.

1.1 - Introducing Calculus: Can Change Occur at an Instant?

Practice setup

For a function

h(t)

representing height over time:

Average rate of change from

t = 1

t = 4

\frac{h(4) - h(1)}{4 - 1}

Average rate of change from

t = 5

t = 8

\frac{h(8) - h(5)}{8 - 5}

Notice we don’t need to actually compute the answer. Setting up the fraction correctly is what shows you understand the concept.

Instantaneous Rate of Change

Average rate of change tells us what happens over an interval. Instantaneous rate of change tells us what’s happening at a single moment. This is the idea that separates calculus from algebra.

Approximating it

To estimate the instantaneous rate of change at

t = 3

, we can take the average rate of change over a very small interval near

t = 3

\frac{f(3.001) - f(3)}{3.001 - 3}

The smaller the interval, the better the approximation. When the two points are incredibly close together (like 2.9999 and 3), the secant line becomes nearly indistinguishable from the tangent line.

Why not shrink the interval all the way to zero?

If we try to use the same point twice (

b = a

), the average rate of change turns into

\frac{f(a) - f(a)}{a - a} = \frac{0}{0}

which is undefined. The average rate of change divides by the change in the input, so it only works when that change isn’t zero. One point by itself can’t give us a slope.

Limits fix this. Instead of plugging in

b = a

, we look at the value the average rates of change approach as

b

gets closer and closer to

a

\text{Instantaneous rate of change at } x = a = \lim_{b \to a} \frac{f(b) - f(a)}{b - a}

That’s how calculus answers the central question: it uses limits to describe change at an instant. Most of Unit 1 is about how limits work, and in Unit 2 this exact limit becomes the derivative.

Tangent Lines

A tangent line touches a curve at exactly one point (locally). The slope of the tangent line at a point equals the instantaneous rate of change at that point.

Example: estimating from a graph

Given a distance vs. time graph, estimate the instantaneous rate of change at

t = 5

minutes:

Draw the tangent line at the point where $t = 5$

Find the slope of that tangent line using rise over run

Include the right units

Suppose the tangent line rises 400 meters when we move 2 minutes to the right:

\text{Instantaneous rate of change} \approx \frac{400 \text{ meters}}{2 \text{ minutes}} = 200 \text{ meters per minute}

That approximation represents the speed at exactly

t = 5

minutes.

Function Notation in Context

In real world problems, functions represent meaningful quantities.

Example: deer population

Let

P(t)

represent the deer population in a state forest, where

t

is years since 1900.

$P(75)$ is the deer population in the year 1975.

$P(60) - P(40)$ is the change in deer population from 1940 to 1960 (could be positive or negative).

$\dfrac{P(60) - P(40)}{60 - 40}$ is the average rate of change of the deer population per year from 1940 to 1960.

\frac{P(60) - P(40)}{20} = \text{average change in deer per year over that 20-year period}

$\dfrac{P(51) - P(50)}{51 - 50}$ is an estimate of the instantaneous rate of change in 1950. Since the interval is only one year, it gives a decent approximation of how quickly the population was changing that year.

$\dfrac{P(50.0001) - P(50)}{0.0001}$ is an even better estimate of the instantaneous rate of change in 1950. The tiny interval makes this nearly equal to the true instantaneous rate.

Average vs. Instantaneous

Average Rate of Change	Instantaneous Rate of Change
Over an interval	At a specific point
Slope of a secant line	Slope of a tangent line
$\frac{f(b) - f(a)}{b - a}$ where $b \neq a$	Limit as the interval shrinks to zero
Easy to calculate	Requires calculus (coming soon)

Key Takeaways

Average rate of change: $\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}$ measures change over an interval

The slope of a secant line equals the average rate of change between two points

Instantaneous rate of change measures how fast something is changing at an exact moment

The slope of a tangent line equals the instantaneous rate of change at a point

Using a very small interval gives a good approximation of the instantaneous rate of change

An interval of zero gives $\frac{0}{0}$ (undefined), so calculus uses a limit of average rates of change to get the instantaneous rate of change

The word “per” signals a rate of change (velocity, cost per item, population per year)

Newton and Leibniz both developed calculus independently around 1670

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