Tangent Lines and Rates of Change
The two motivating problems of differential calculus: finding the tangent to a curve and determining instantaneous velocity.
Historical Hook: Zeno's Dichotomy Paradox
In the 5th century BCE, the Greek philosopher Zeno of Elea proposed a series of paradoxes designed to prove that motion is an illusion.
"To travel any distance, one must first travel half that distance, then half the remaining distance, and so on ad infinitum."
This requires completing an infinite number of tasks in a finite amount of time. Therefore, motion is logically impossible.
Mathematics lacked the vocabulary to refute him for two millennia—until the concept of the limit was formalized.
The limit allows us to prove that an infinite sum can converge to a finite value:
The Tangent Line Problem
The classical definition of a tangent, derived from Euclid, characterizes it as a line intersecting a curve at a single point without crossing it. But this definition fails for curves with inflection points.
From Euclid to Fermat
In the 1630s, Pierre de Fermat developed the method of adequality—a precursor to the derivative. He compared the value of a function at with its value at where is small.
Tangent as a Limit
The dynamic view of the tangent as the limit of secant lines eventually bridged geometry and calculus. This is the foundation of the derivative.
The Secant Line Approximation
To find the slope of the tangent line at a point , we construct a secant line passing through and a nearby point . As moves along the curve toward , the secant line approximates the tangent.
Worked Example: at
| x (from right) | x (from left) | ||
|---|---|---|---|
| 2 | -6 | 0 | -2 |
| 1.5 | -5 | 0.5 | -3 |
| 1.1 | -4.2 | 0.9 | -3.8 |
| 1.01 | -4.02 | 0.99 | -3.98 |
| 1.001 | -4.002 | 0.999 | -3.998 |
Conclusion: As from either side, the slope . This is the slope of the tangent line.
Interactive: Manipulate Point Q
From Average to Instantaneous Rate
The change over a discrete interval . This corresponds to the slope of the secant line.
The rate at a single moment in time. This corresponds to the slope of the tangent line.
Real-World Applications
Velocity is the instantaneous rate of change of position. Acceleration is the rate of change of velocity.
Marginal Cost is the cost of producing one more unit—the derivative of the cost function.
Population growth models rely on differential equations where growth rate depends on current population.
Common Pitfalls
- •Trying to evaluate directly instead of recognizing the indeterminate form.
- •Confusing the value of the function at a point with the limit as the point approaches.
Key Takeaway
The tangent line and instantaneous velocity are mathematically identical problems—both resolve to the limit of a difference quotient.
Practice Quiz
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