AP Syllabus focus:
‘The first derivative tells us where a function is increasing or decreasing: if f′(x) is positive on an interval, f is increasing there; if f′(x) is negative, f is decreasing.’
A function’s first derivative provides essential information about how the function behaves, allowing us to determine whether it rises or falls on specific intervals of its domain.
Understanding How the Sign of the First Derivative Controls Behavior
The central idea of this subsubtopic is that the sign of the first derivative directly indicates whether a function is increasing or decreasing on an interval.
This diagram compares intervals where a function is increasing with f'(x) > 0 and decreasing with f'(x) < 0. Each panel pairs the graph of with the sign of its derivative on the same interval. It emphasizes that a consistently positive or negative derivative produces a consistent upward or downward trend in the graph of . Source.
When discussing derivative-based behavior, we must understand the term derivative before using it further.
Derivative: The derivative of a function at a point represents the instantaneous rate of change of the function, or the slope of the tangent line at that point.
This idea leads to a deeper understanding of how the sign of the derivative influences the movement of the graph.
Interpreting Positive and Negative First Derivatives
A function’s behavior on an interval depends entirely on whether its derivative is positive, negative, or zero at points within that interval.
Because the derivative measures slope, its sign reveals the graph’s direction of change.
Positive Derivative and Increasing Behavior
When the derivative takes positive values across an interval, the function moves upward as x increases. This means the outputs get larger as the inputs move along the interval.
• A positive derivative corresponds to an upward-sloping tangent line.
• On these intervals, the function exhibits increasing behavior, meaning each successive value of the function is greater than the previous one.
Negative Derivative and Decreasing Behavior
A negative derivative signals that the function is trending downward as x increases.
• A negative derivative corresponds to a downward-sloping tangent line.
• The function demonstrates decreasing behavior, meaning its outputs shrink as the inputs move forward.
In both situations, it is the consistent sign of the derivative across the interval—not momentary values—that determines the overall trend. At this point, it is helpful to specify what an interval is in this context.
Interval: A connected subset of the real number line on which a function may exhibit consistent behavior such as increasing or decreasing.
This allows us to frame increasing or decreasing behavior as something that must hold across a stretch of x-values, not just isolated points.
Special Role of Points Where the Derivative Is Zero
Although this subsubtopic focuses on increasing and decreasing behavior, it is important to recognize that the derivative’s sign can only be interpreted meaningfully when we understand what occurs where the derivative equals zero.
Critical Point: A point in the domain of a function where the first derivative is zero or does not exist.
Critical points are important because they may mark boundaries where the derivative changes sign. A change from positive to negative indicates the function switches from increasing to decreasing, while a change from negative to positive indicates the reverse. Even though the full classification of critical points belongs to other subtopics, recognizing their role helps establish why sign analysis typically involves examining intervals separated by these points.
Using Derivative Sign Information to Describe Function Behavior
Clear communication about increasing and decreasing intervals requires stating explicitly why a function behaves the way it does. In AP Calculus AB, such justifications must refer directly to the sign of the derivative.
Expressing Increasing Behavior
When describing increasing behavior, emphasize that the derivative is positive on the interval:
• “The function is increasing on the interval because f′(x) > 0 for all x in that interval.”
Expressing Decreasing Behavior
Likewise, for decreasing intervals:
• “The function is decreasing on the interval because f′(x) < 0 for all x in that interval.”
These verbal justifications reinforce the fundamental link between rates of change and the direction of the function’s graph.
Connecting Tangent Lines and Graph Shape
The derivative’s sign is closely related to the geometry of the graph.
The upper graph shows a cubic function with horizontal tangents and directional changes, while the lower graph displays with zeros aligned to those points. Dashed vertical lines connect the two graphs, illustrating how features of correspond to values of . This figure also displays a point of inflection and concavity details that extend slightly beyond the emphasis on derivative sign but remain consistent with calculus interpretation. Source.
Tangent lines provide a visual interpretation of the derivative’s sign.
• A positive slope tangent (above the x-axis when plotted as f′) reflects an increasing portion of the graph.
• A negative slope tangent reflects a decreasing portion of the graph.
Tangent Line: A line that touches a curve at a point and has the same instantaneous slope as the curve does at that point.
Understanding tangent lines gives students a concrete geometric representation of derivative signs.
Organizing Information About Derivative Signs
To systematically analyze behavior, students often use a sign chart, though constructing one belongs to the next subsubtopic. For this subsubtopic, note that sign information must be interpreted in a structured way.
• Identify intervals separated by critical points.
• Determine the sign of the derivative on each interval.
• Use these signs to state where the function is increasing or decreasing.
• Support each conclusion using explicit derivative inequalities.
These steps ensure clear, consistent reasoning aligned with AP expectations.
Importance of Interval-Based Reasoning
The derivative at a single point does not determine overall behavior. Instead, students must focus on derivative signs across entire intervals. This reinforces why connected intervals, critical points, and slope interpretations all contribute to understanding how a function behaves.
Overall, the sign of the first derivative provides a powerful tool for describing a function’s increasing or decreasing behavior and forms the foundation for deeper analysis across differential calculus.
FAQ
If the derivative changes sign gradually, the graph transitions smoothly between increasing and decreasing behaviour.
If the derivative is undefined at a point where the function changes direction, the graph may form a sharp corner or cusp rather than a smooth turning point.
A smooth change in direction typically occurs when the derivative passes through zero. A non-smooth change often happens when the derivative does not exist.
Yes. A function may still be increasing on an interval even if the derivative is zero at single points, provided the derivative remains positive everywhere else.
For example, a derivative could be positive on an interval except for a single point where it is zero. The presence of these isolated zero-derivative points does not interrupt overall increasing behaviour.
Only if the derivative is zero on an entire subinterval does the function remain constant there.
This indicates that the function oscillates, switching rapidly between increasing and decreasing behaviour.
Such behaviour often produces a curve with many small peaks and valleys, even if the oscillations eventually smooth out.
From an analytical standpoint:
• The function has many short intervals of alternating behaviour.
• Identifying these requires careful evaluation of the derivative’s sign across all relevant intervals.
The behaviour of a function depends on the derivative’s sign across a connected stretch of x-values, not just at single points.
A derivative might be positive at one point and negative at the next, but without identifying the boundaries of the intervals, the overall behaviour is unclear.
Interval reasoning ensures consistent classification and prevents incorrect assumptions based on incomplete information.
Yes. Knowing only whether the derivative is positive or negative does not fully determine the shape of the function.
Two functions may both be increasing on the same interval yet look entirely different due to differing rates of increase.
Derivative sign information controls direction of change, but not steepness, curvature, or vertical position.
Practice Questions
Question 1 (1–3 marks)
The derivative of a function f is given by the following information:
f'(x) > 0 for all x < 2
f'(x) < 0 for all x > 2
State the intervals on which f is increasing and decreasing.
Question 1
• 1 mark: Identifies that f is increasing for x < 2.
• 1 mark: Identifies that f is decreasing for x > 2.
• 1 mark: Uses correct interval notation (e.g., increasing on (–∞, 2) and decreasing on (2, ∞)).
Question 2 (4–6 marks)
The graph of the derivative f'(x) of a continuous function f is shown below (description in words):
• f'(x) is positive on the interval (–4, –1)
• f'(x) is zero at x = –1 and x = 3
• f'(x) is negative on (–1, 3)
• f'(x) is positive on (3, 6)
(a) Determine all intervals on which f is increasing.
(b) Determine all intervals on which f is decreasing.
(c) Explain how the sign of f'(x) supports your conclusions.
Question 2
(a) Increasing intervals
• 1 mark: States f is increasing on (–4, –1).
• 1 mark: States f is increasing on (3, 6).
(b) Decreasing intervals
• 1 mark: States f is decreasing on (–1, 3).
(c) Explanation
• 1 mark: States that f is increasing where f'(x) > 0.
• 1 mark: States that f is decreasing where f'(x) < 0.
• 1 mark: Connects the sign of f'(x) to the direction of change of f (e.g., positive derivative implies rising function, negative derivative implies falling function).
