Calculus I · 2B · lesson
Increasing and Decreasing Intervals
Learn increasing and decreasing intervals with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Section overview
Theorems, extrema, and curve shapeWhat this section is building
Learn increasing and decreasing intervals with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Critical numbers divide the domain into testable intervals; endpoints and discontinuities keep local evidence from becoming an unjustified global claim.
List the domain and candidates, test derivative signs, compare endpoint values, and verify each theorem's hypotheses explicitly.
Calling every point with f-prime zero an extremum or every point with f-double-prime zero an inflection point.
Learning objectives
Determine intervals of increase and decrease from a derivative sign chart.
Use the Sign of
Before the formulas
In Increasing and Decreasing Intervals, derivatives become evidence about shape. A zero or undefined derivative creates a candidate, a sign chart determines rising and falling behavior, and the second derivative describes how slopes change. No single fact tells the entire graph story.
State intervals, not just isolated points. Explain why a sign has the consequence you claim. When a theorem is used, verify its hypotheses explicitly; continuity and differentiability are not ceremonial phrases but the conditions that make the conclusion reliable.
The sign of controls direction
A positive derivative means nearby outputs rise as the input increases. A negative derivative means they fall. Critical numbers and points where is undefined divide the domain into intervals on which the sign can be tested.
Use the derivative's sign, not the sign of the function. A function can be below the axis and increasing, or above the axis and decreasing.
The sign of converts local slope information into interval behavior. Positive derivative means the function rises as increases; negative derivative means it falls. A sign chart organizes this information around critical numbers and domain breaks.
Test intervals, not isolated points. One sample point works only because the derivative's sign is known not to change inside the interval under consideration.
If on an interval, is increasing there. If , is decreasing.
First-derivative sign chart
• Find critical numbers and domain breaks. • Place them on a number line. • Determine the sign of on each interval. • Translate positive signs to increasing and negative signs to decreasing.
Analyze a cubic
For
find intervals of increase and decrease.
Worked solution
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increase-01For , where is increasing?
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Make a sign chart for x(x-2).
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