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Introducing Calculus
Can Change Occur at an Instant?

Explore the foundational question of calculus: how we measure instantaneous change using limits, average rates, and the leap from secant to tangent lines.

AP Calculus AB

What Is Change?

Every time you check the speedometer of a car, track the temperature over a day, or watch a stock price move, you are observing change. But a crucial question arises: the speedometer shows your speed right now, at a single frozen moment. How is that possible?

Average Rate of Change

We start with what we can measure: the average rate of change over an interval. For a function \( f(x) \), the average rate of change from \( x = a \) to \( x = b \) is:

Think of driving 120 miles in 2 hours: your average speed is 60 mph. But you weren't going exactly 60 mph every second: you sped up, slowed down, maybe stopped. The average hides the details.

The Secant Line

\(x\)
\(y\)
\(a\)
\(b\)
 secant line
\(f(x)\)
\(b - a\)
\(f(b)-f(a)\)

From Secant to Tangent

Now comes the key idea: what happens if we slide point \( b \) closer and closer to point \( a \)? The secant line rotates and approaches a special line that just touches the curve at a single point.

tangent line
\(f(x)\)
\( b \to a \)

The Limit Makes It Rigorous

Calculus formalizes this "sliding closer" process using the concept of a limit. Instead of letting \( b = a \) (which gives \( \frac{0}{0} \)), we ask: what value does the ratio approach as \( b \) gets arbitrarily close to \( a \)?

This limit, when it exists, is called the derivative of \( f \) at \( x = a \). It is the precise answer to "Can change occur at an instant?" Yes: it is the limit of average rates of change.

Why Does This Matter?

Physics
Velocity is the instantaneous rate of change of position. Acceleration is the instantaneous rate of change of velocity.
Biology
Population growth rate at a specific moment tells us if a species is thriving or declining right now, not just on average.
Economics
Marginal cost is the instantaneous rate of change of total cost with respect to quantity produced.

Average vs Instantaneous

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

A falling ball seen through a drop-tower cutaway

One orange ball falls within a museum drop tower beside a computed height-time curve,tangent and signed velocity/speed readouts.For height s(t) = 100 − 4.9t² meters, equal time intervals produce increasingly large drops. The model ends at ground impact, about 4.52 seconds.

Equal times, unequal distances

For height s(t) = 100 − 4.9t² meters, equal time intervals produce increasingly large drops. The model ends at ground impact, about 4.52 seconds.For height s(t) = 100 − 4.9t² meters, equal time intervals produce increasingly large drops. The model ends at ground impact, about 4.52 seconds.

A negative slope means downward velocity

A decreasing height-time curve ends at ground impact around 4.52 seconds. At 2 seconds, a tangent touches the point at 80.4 m with slope −19.6 m/s.For s(t) = 100 − 4.9 t² meters, the point at t = 2 seconds has height 80.4 m and tangent slope −19.6 m/s. The negative velocity means downward motion; speed is 19.6 m/s.Open full diagram

How to Find the Instantaneous Rate

1
Identify the function \( f(x) \) and the point \( x = a \) where you want the rate.
2
Write the difference quotient: \( \frac{f(a + h) - f(a)}{h} \).
3
Simplify the expression algebraically. Cancel the \( h \) in the denominator.
4
Take the limit as \( h \to 0 \). The result is the instantaneous rate of change at \( x = a \).

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

1
Difference quotient: \( \frac{(3+h)^2 - 3^2}{h} = \frac{9 + 6h + h^2 - 9}{h} \)
2
Simplify: \( \frac{6h + h^2}{h} = \frac{h(6 + h)}{h} = 6 + h \)
3
Take the limit: \( \displaystyle\lim_{h \to 0}(6 + h) = \) 6

Visualizing \( f(x) = x^2 \) at \( x = 3 \)

\(x\)
\(y\)
\(1\)
\(2\)
\(3\)
\(4\)
\(5\)
\((3, 9)\)
\(\text{slope} = 6\)

Example 2: \( g(t) = 18t - 5t^2 \) at \( t = 1 \)

A ball is thrown upward. Its height in meters after \( t \) seconds is \( g(t) = 18t - 5t^2 \). Find the instantaneous velocity at \( t = 1 \).

1
Difference quotient: \( \frac{g(1+h) - g(1)}{h} = \frac{(18(1+h) - 5(1+h)^2) - 13}{h} \)
2
Expand: \( \frac{18 + 18h - 5(1+2h+h^2) - 13}{h} = \frac{8h - 5h^2}{h} = 8 - 5h \)
3
Limit: \( \displaystyle\lim_{h \to 0}(8 - 5h) = \) 8 m/s

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

1
Difference quotient: \( \frac{\frac{1}{2+h} - \frac{1}{2}}{h} = \frac{\frac{2 - (2+h)}{2(2+h)}}{h} = \frac{-h}{2h(2+h)} \)
2
Simplify: \( \frac{-1}{2(2+h)} \)
3
Limit: \( \displaystyle\lim_{h \to 0} \frac{-1}{2(2+h)} = \frac{-1}{4} = \) \( -0.25 \)

Estimating with Shrinking Intervals

When you don't have a formula, you can estimate the instantaneous rate by computing average rates over smaller and smaller intervals and watching the values converge.

Interval \( h \) \( \frac{f(3+h)-f(3)}{h} \) for \( f(x)=x^2 \)
\( [3, 4] \) \( 1 \) \( 7 \)
\( [3, 3.1] \) \( 0.1 \) \( 6.1 \)
\( [3, 3.01] \) \( 0.01 \) \( 6.01 \)
\( [3, 3.001] \) \( 0.001 \) \( 6.001 \)

Common Mistakes

✕ Plugging in \( h = 0 \) first
Substituting \( h = 0 \) before simplifying gives \( \frac{0}{0} \), which is undefined. You must cancel \( h \) algebraically first, then take the limit.
✕ Confusing average and instantaneous
The average rate over \( [a, b] \) uses two distinct points. The instantaneous rate uses a limit as the interval shrinks to zero: they are different values.
✕ Forgetting to expand correctly
Errors in expanding \( (a+h)^2 \) or distributing negatives are the most common algebra mistakes. Double-check every term.

Question 1 of 5

What does the instantaneous rate of change of a function at a point represent geometrically?

Question 2 of 5

What is \( \displaystyle\lim_{h \to 0} \frac{(5+h)^2 - 25}{h} \)?

Question 3 of 5

A car travels \( s(t) = 4t^2 \) meters in \( t \) seconds. What is its instantaneous velocity at \( t = 3 \)?

Question 4 of 5

The expression \( \displaystyle\frac{f(a+h) - f(a)}{h} \) is called the:

Question 5 of 5

As the interval \( [a, b] \) shrinks (i.e., \( b \to a \)), the secant line approaches the: