🌌 Grade 11 · Lesson 12 of 12
Secant Explorer: An Introduction to Derivatives
Slide one point of a secant line toward the other and watch the average rate of change tighten into the instantaneous slope — the derivative.
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The big idea: The slope of a line between two points on a curve — a secant — measures the average rate of change over that interval. Squeeze the two points together and that average becomes an instantaneous rate: the slope of the tangent line at a single point. This limiting slope is the derivative, and the secant line is the bridge from the algebra of slopes you already know to the calculus of change.
Average rate of change is just slope
You already know how to find the slope between two points: rise over run. When those two points sit on a curve y = f(x), the line joining them is called a secant line, and its slope is the average rate of change of the function across that interval.
If a car's position over two hours goes from 100 km to 220 km, the secant slope — 60 km per hour — is its average speed, even though its speedometer varied the whole time.
\[ \text{secant slope} = \dfrac{f(b) - f(a)}{b - a} \]
The slope of the line through (a, f(a)) and (b, f(b)) — the average rate of change of f from a to b.
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The difference quotient
It is often handy to write the second point as a small step h away from the first: b = a + h. The secant slope becomes the difference quotient, [f(a + h) − f(a)] / h. As h shrinks, the two points close in on each other.
🎮 Secant Explorer LIVE
Slide the second point toward the first; the secant slope approaches the derivative.
Closing the gap: from average to instantaneous
Now slide the second point toward the first, making h smaller and smaller. The secant line pivots, and its slope homes in on a single limiting value — the slope of the tangent line that just grazes the curve at the point.
That limiting slope is the instantaneous rate of change: how fast the function is changing at that exact instant, not averaged over an interval. It is the central object of differential calculus.
\[ f'(a) = \lim_{h \to 0} \dfrac{f(a + h) - f(a)}{h} \]
The derivative at a: the value the secant slope approaches as the step h shrinks to zero.
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The derivative is the tangent slope
The derivative f′(a) is exactly the slope of the tangent line to the curve at x = a. Where f′ is positive the curve is rising; where it is negative the curve is falling; where it is zero the curve has a level (horizontal) tangent — a peak, a valley, or a flat spot.
📝 Worked example: Find the average rate of change of f(x) = x² over the interval [1, 3].
- Evaluate the endpoints: \( f(1) = 1 \) and \( f(3) = 9 \).
- Apply the secant-slope formula: \( \dfrac{f(3) - f(1)}{3 - 1} = \dfrac{9 - 1}{2} \).
- Simplify: \( \dfrac{8}{2} = 4 \).
✓ The average rate of change is <strong>4</strong> — the secant line from (1, 1) to (3, 9) has slope 4.
📝 Worked example: Use the difference quotient to find the instantaneous rate of change of f(x) = x² at x = 2.
- Build the difference quotient at \( a = 2 \): \( \dfrac{f(2+h) - f(2)}{h} = \dfrac{(2+h)^2 - 4}{h} \).
- Expand the top: \( (2+h)^2 - 4 = 4 + 4h + h^2 - 4 = 4h + h^2 \).
- Divide by h: \( \dfrac{4h + h^2}{h} = 4 + h \). Now let \( h \to 0 \).
✓ As h → 0 the slope approaches <strong>4</strong>, so f′(2) = 4 — the tangent at x = 2 has slope 4.
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Average over an interval is not instantaneous at a point
A secant slope describes the whole interval between two points; the derivative describes a single instant. They agree only in the limit as the interval shrinks to nothing. Reporting a two-hour average speed as the speed “right now” is the same error as confusing a secant with a tangent.
Check your understanding
1. The slope of a line joining two points on a curve is called the…
A line through two points of a curve is a secant, and its slope is the average rate of change.
2. What is the average rate of change of f(x) = x² over [2, 5]?
(f(5) − f(2))/(5 − 2) = (25 − 4)/3 = 21/3 = 7.
3. As h → 0 in the difference quotient, the secant slope approaches the…
Shrinking h brings the two points together, so the secant slope becomes the tangent slope — the derivative.
4. For f(x) = x², the difference quotient at x = 2 simplifies to 4 + h. What is f′(2)?
Taking h → 0 in 4 + h gives 4, so the derivative at x = 2 is 4.
5. If f′(a) is negative, the curve at x = a is…
A negative derivative means a negative tangent slope, so the function is decreasing there.
✅ Key takeaways
- A secant line joins two points on a curve; its slope is the average rate of change over that interval.
- The difference quotient [f(a + h) − f(a)] / h is the secant slope written with a step h.
- As h → 0 the secant slope approaches the tangent slope at the point.
- That limiting slope is the derivative f′(a) — the instantaneous rate of change.
- A positive derivative means the curve is rising, negative means falling, and zero means a horizontal tangent.