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Introducing Calculus
Can Change Occur at an Instant?

Explore how we measure instantaneous rates of change by shrinking intervals to zero: the foundational idea behind limits and derivatives.

AP Calculus BC

What Does Change Mean?

In everyday life, we talk about change all the time: the speed of a car, the growth of a population, the temperature rising through the day. All of these describe how one quantity varies as another quantity (usually time) progresses.

Average vs Instantaneous Rate

Average Rate
Computed over an interval \([a, b]\): \(\displaystyle \frac{f(b) - f(a)}{b - a}\). This is the slope of the secant line.
Instantaneous Rate
Computed at a single point \(x = a\): the slope of the tangent line. Requires a limit as the interval shrinks to zero.

From Secant to Tangent

x y a b Secant b' Tangent b → a

The Limit Definition

We formalize "shrinking the interval to zero" with a limit. The instantaneous rate of change of \(f\) at \(x = a\) is:

If this limit exists, we say \(f\) is differentiable at \(a\), and the value of the limit is the slope of the tangent line at that point.

Visualizing the Difference Quotient

x y f(a+h) - f(a) h (a, f(a)) (a+h, f(a+h)) slope = difference quotient

Average vs Instantaneous: Side by Side

Average Rate Instantaneous Rate
Domain Over an interval \([a, b]\) At a single point \(x = a\)
Formula \(\dfrac{f(b)-f(a)}{b-a}\) \(\displaystyle\lim_{h\to 0}\dfrac{f(a+h)-f(a)}{h}\)
Geometric Meaning Slope of secant line Slope of tangent line
Requires Limit? No Yes

Why Does This Matter?

The concept of instantaneous rate of change is the foundation of all of calculus. It shows up everywhere in science and engineering:

Velocity
The instantaneous rate of change of position with respect to time.
Growth Rate
How fast a population, investment, or quantity is changing right now.
Slope of a Curve
The tangent line slope tells us the direction and steepness at any point.

How to Find \(f'(a)\) Using the Limit

1
Write the difference quotient: \(\dfrac{f(a+h)-f(a)}{h}\).
2
Substitute \(a + h\) into \(f\) and expand. Then subtract \(f(a)\).
3
Simplify the numerator. Factor out \(h\) so you can cancel the \(h\) in the denominator.
4
Take the limit as \(h \to 0\). The result is \(f'(a)\).

Example 1: \(f(x) = x^2\) at \(x = 3\)

1
Difference quotient: \(\dfrac{(3+h)^2 - 3^2}{h}\)
2
Expand: \(\dfrac{9 + 6h + h^2 - 9}{h} = \dfrac{6h + h^2}{h}\)
3
Simplify: \(\dfrac{h(6 + h)}{h} = 6 + h\)
4
Limit: \(\displaystyle\lim_{h \to 0}(6 + h) = \) \(6\)

Graph: Tangent to \(y = x^2\) at \(x = 3\)

x y 0 1 2 3 4 5 5 10 15 20 25 slope = 6 (3, 9)

Example 2: \(f(x) = 3x^2 - 2x\) at \(x = 1\)

1
Difference quotient: \(\dfrac{3(1+h)^2 - 2(1+h) - [3(1)^2 - 2(1)]}{h}\)
2
Expand: \(\dfrac{3(1+2h+h^2) - 2 - 2h - 1}{h} = \dfrac{3+6h+3h^2-2-2h-1}{h}\)
3
Simplify: \(\dfrac{4h+3h^2}{h} = 4 + 3h\)
4
Limit: \(\displaystyle\lim_{h \to 0}(4 + 3h) = \) \(4\)

Example 3: \(f(x) = \dfrac{1}{x}\) at \(x = 2\)

1
Difference quotient: \(\dfrac{\frac{1}{2+h} - \frac{1}{2}}{h}\)
2
Combine fractions: \(\dfrac{\frac{2 - (2+h)}{2(2+h)}}{h} = \dfrac{-h}{2h(2+h)}\)
3
Cancel \(h\): \(\dfrac{-1}{2(2+h)}\)
4
Limit: \(\displaystyle\lim_{h \to 0}\dfrac{-1}{2(2+h)} = \dfrac{-1}{4}\) = \(-\tfrac{1}{4}\)

Alternative Limit Form

There is an equivalent way to write the derivative using a different variable. Instead of \(h\), let \(x\) approach \(a\) directly:

\(h\)-form
\(\displaystyle\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}\)
\(x\)-form
\(\displaystyle\lim_{x \to a}\frac{f(x)-f(a)}{x-a}\)

Common Mistakes to Avoid

✕ Plugging in \(h = 0\) too early
Substituting \(h = 0\) before simplifying gives \(\frac{0}{0}\), which is undefined. You must cancel \(h\) first, then take the limit.
✕ Forgetting to expand correctly
A common algebra error: \((a+h)^2 \neq a^2 + h^2\). Remember the cross term: \((a+h)^2 = a^2 + 2ah + h^2\).
✕ Confusing average and instantaneous
The average rate over \([a, b]\) does not require a limit. The instantaneous rate at a point always does.

Summary: Key Concepts

Concept Definition Geometric Meaning
Average Rate of Change \(\dfrac{f(b)-f(a)}{b-a}\) Slope of secant line
Difference Quotient \(\dfrac{f(a+h)-f(a)}{h}\) Slope of secant (variable \(h\))
Derivative at \(a\) \(\displaystyle\lim_{h\to 0}\dfrac{f(a+h)-f(a)}{h}\) Slope of tangent line
Instantaneous Rate Same as derivative Rate of change at a single point

Question 1 of 5

The expression \(\displaystyle\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}\) represents which of the following?

Question 2 of 5

Use the limit definition to find \(f'(2)\) for \(f(x) = x^2 + 1\).

Question 3 of 5

Geometrically, the derivative \(f'(a)\) represents the slope of the:

Question 4 of 5

When computing \(\displaystyle\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}\), why can't you simply substitute \(h = 0\) right away?

Question 5 of 5

If the derivative \(f'(a)\) is negative, what does that tell us about \(f\) at \(x = a\)?