Sketching Graphs Using the Derivative
Big Idea:
The derivative $f'(x)$ tells us how a function $f(x)$ is changing at every point. By analyzing $f'(x)$, we can determine where a function is increasing, decreasing, and where it has peaks (local maxima) or valleys (local minima) — all without plotting hundreds of points!
📈 Key Concept 1: Increasing & Decreasing
The sign of the derivative tells us the direction of the function:
| Derivative | Function Behavior | Slope |
|---|---|---|
| $f'(x) > 0$ | $f(x)$ is increasing | Positive (uphill ↗) |
| $f'(x) < 0$ | $f(x)$ is decreasing | Negative (downhill ↘) |
| $f'(x) = 0$ | $f(x)$ has a horizontal tangent | Zero (flat —) |
🎯 Key Concept 2: Critical Points
Definition: A critical point of $f(x)$ occurs where $f'(x) = 0$ or $f'(x)$ is undefined.
Critical points are candidates for local maxima and minima. But not every critical point is a max or min!
- Local Maximum: $f'(x)$ changes from $+$ to $-$
- Local Minimum: $f'(x)$ changes from $-$ to $+$
- Neither: $f'(x)$ does not change sign (inflection point)