AP® Calculus BC · Unit 5 · Topic 5.3
Determining Intervals on Which a Function Is Increasing or Decreasing
A function increases where its derivative is positive and decreases where its derivative is negative, so a sign chart of f' maps out the rising and falling stretches of f.
The derivative f'(x) measures the slope of the tangent line at each point. A positive slope means the curve is climbing as x increases; a negative slope means it's falling. So the sign of f' at a point tells you the direction f is heading right there, and since f' is itself a function, you can track its sign across the whole domain to map out where f rises and falls.
To do this, find f'(x), then find the x-values where f'(x) = 0 or is undefined. These are the only places the sign of f' can switch, so they split the number line into intervals. Pick a test point in each interval, plug it into f', and record whether the result is positive or negative. Wherever f' > 0, f is increasing; wherever f' < 0, f is decreasing. This sign chart is the same tool you'll reuse to locate local max and min values, since a switch from positive to negative in f' marks a local max, and negative to positive marks a local min.
This idea is the foundation for everything else in the unit. Once you can read increasing and decreasing behavior off f', the next step is asking what the second derivative says about concavity, and then combining both to sketch a complete picture of a function or to justify where extrema occur using the first derivative test.
Key terms
- Increasing interval
- A stretch of the domain where f'(x) > 0 and the function's values rise as x increases.
- Decreasing interval
- A stretch of the domain where f'(x) < 0 and the function's values fall as x increases.
- Critical point
- An x-value where f'(x) equals zero or is undefined, marking a possible change in direction.
- Sign chart
- A number line marked with critical points and the sign of f' tested on each resulting interval.
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End of topic.