AP Syllabus focus:
‘Written explanations should clearly refer to f, f′, and f″ by name, describing how features like extrema, inflection points, and concavity appear in each representation.’
This section explains how to express relationships among f, f′, and f″ in clear mathematical language, emphasizing how slopes, extrema, and concavity correspond across different representations.
Explaining Relationships in Words
A key skill in AP Calculus AB is articulating how function behavior, first derivatives, and second derivatives connect. This means referencing each graph or expression by name and explaining how features such as extrema, slopes, and concavity relate.
Understanding the Role of f, f′, and f″
When describing relationships, it is essential to refer to f, f′, and f″ explicitly rather than vaguely stating that “the graph” or “the derivative” does something. This helps communicate the mathematical reasoning clearly and demonstrates understanding of how derivatives encode information about the original function.
Using Function Behavior
The graph of f shows the overall behavior of the function: where its values rise or fall, where it reaches high or low points, and how sharply it bends. When explaining these behaviors, link them directly to the corresponding features of the derivatives.
Using the First Derivative
The first derivative f′ describes the instantaneous rate of change of f.
Instantaneous Rate of Change: The value of f′(x), representing how quickly f is changing at a specific x-value.
Because f′ gives slope information, verbal explanations should highlight how its sign and magnitude correspond to features in the graph of f:
Positive f′ means f is increasing.
Negative f′ means f is decreasing.
Zero values of f′ correspond to horizontal tangents on f, which may signal local maxima, local minima, or neither.
A complete explanation clearly states which intervals show these sign relationships, rather than simply labeling points.
Using the Second Derivative
The second derivative f″ provides information about concavity, describing how the graph of f bends.
Concavity: The direction in which the graph of f curves; f is concave up when f″ > 0 and concave down when f″ < 0.
Descriptions of concavity must directly address the behavior of f″, emphasizing sign changes and what they reveal about the shape of f.
Describing Extrema Across Representations
When describing extrema in words, explanations should connect the point’s behavior on the graph of f with the corresponding behavior in f′ and f″.
Local Extrema
A local maximum or minimum occurs at a point where the slope changes direction. In verbal explanations:
Refer to the horizontal tangent on f.
Indicate that f′(x) = 0 at that point.
Explain how f′ changes sign around the point, depending on whether the extremum is a maximum or a minimum.
Local maxima of f occur at x-values where f′(x) changes from positive to negative, while local minima occur where f′(x) changes from negative to positive.
The graph shows f(x) and f′(x) together, with vertical lines linking a local extremum of f to the corresponding zero of f′. This illustrates how changes in the sign of the derivative determine maxima and minima. The highlighted points reinforce the verbal reasoning about critical points and slope behavior. Source.
Concavity and Extrema
The relationship between concavity and the nature of extrema is also important to express:
When f″(c) > 0, f is concave up near c, supporting the conclusion that the point is a local minimum.
When f″(c) < 0, f is concave down near c, supporting the conclusion that the point is a local maximum.
These statements must explicitly mention f″ when discussing concavity, rather than relying on generic descriptions of the graph’s shape.
Describing Inflection Points in Words
An inflection point is a location where the concavity of f changes.
Point of Inflection: A point where f changes concavity, provided f is defined near the point.
In written explanations, students should specify:
That f″ changes sign at the x-value.
How the graph of f transitions from concave up to concave down or vice versa.
That f′ reflects this change through a shift in increasing or decreasing behavior of its own graph.
Connecting Verbal Statements Across Representations
A strong explanation includes all three perspectives—function, first derivative, and second derivative—woven into a coherent description. Effective relationships include:
Identifying intervals where f′ is above or below the x-axis and linking these to increases or decreases in f.
Noting intervals where f″ is positive or negative and connecting these to concave behavior.
Referring to zeros of f′ as critical points and stating how changes in signs of f′ or values of f″ determine the type of extremum.
Describing how inflection points in f appear as local extrema in f′, while corresponding to zeros or undefined values in f″ where sign changes occur.
When f″(x) is positive on an interval, we describe the graph of f as concave up on that interval and say that the slopes of the tangent lines are increasing.
The curve bends upward while the tangent lines grow steeper as x increases. This visual demonstrates how increasing slopes correspond to f′ increasing and f″ > 0. The diagram directly supports the verbal explanation of concavity. Source.
When f″(x) is negative on an interval, we describe the graph of f as concave down on that interval and say that the slopes of the tangent lines are decreasing.
The tangent lines become flatter as x increases, reflecting decreasing slope values. This demonstrates the relationship between concave-down behavior, decreasing f′, and f″ < 0. It visually mirrors the text’s explanation of how concavity is identified. Source.
These explanations must flow logically, making explicit connections rather than assuming the reader infers relationships. The goal is clarity: stating not only what happens, but also why it happens, using terminology such as slope, rate of change, concavity, critical point, and inflection point appropriately and precisely.
Structuring Effective Explanations
When writing about relationships among f, f′, and f″, students may improve clarity by:
Naming each function explicitly in every statement.
Using phrases like “because f′ is positive,” “since f′ changes sign,” or “as f″ becomes negative.”
Demonstrating cause-and-effect reasoning instead of listing observations.
Connecting graphs, tables, and algebraic expressions consistently by pointing out matching features across them.
A well-crafted explanation uses accurate calculus vocabulary while clearly linking behaviors in f, f′, and f″ to create a complete picture of how a function behaves.
FAQ
A complete explanation explicitly mentions all three functions when relevant, not just one.
It should identify the mathematical cause behind the behaviour, not merely state observations.
A strong check is to ask:
• Have I stated what f is doing?
• Have I stated what f′ is doing that causes it?
• Have I stated how f″ contributes (if concavity is relevant)?
If any of these links is missing, the explanation is incomplete.
Avoid vague phrases such as “the graph goes up here” or “the line curves” without naming the function and derivative involved.
Instead of writing “the slope changes”, specify whether f′ changes sign, becomes positive, or becomes negative.
Clear writing identifies the function, quotes the derivative behaviour, and links cause to effect.
You should state both the sign before and after the point of interest.
For example:
• Rather than saying “f′ changes sign”, say “f′ changes from positive to negative at x = a”.
This precision strengthens the reasoning and shows understanding of how slope behaviour determines maxima or minima.
Similarly, for concavity, describe the direction of change in f″ rather than simply saying it “switches”.
Use structured phrases that explicitly connect the sign of f″ with the geometric shape of the graph.
For improved clarity:
• State the sign of f″.
• State whether the graph is concave up or concave down.
• State how this affects tangent-line slopes or the bending of the curve.
This layered structure ensures the explanation remains logical and easy to follow.
Diagrams show the shape visually, but exam marking relies on clear verbal reasoning.
A written explanation demonstrates that you know:
• f must be defined near the point,
• f″ must change sign, and
• The concavity of f changes as a result.
Using words forces you to articulate each requirement, reducing ambiguity and avoiding incorrect assumptions that might be made from a sketch alone.
Practice Questions
Question 1 (1–3 marks)
The graph of a differentiable function f is shown. At x = 2, the tangent to the graph is horizontal. Explain, in words, what this indicates about the value of f′(2) and how this feature would appear on the graph of f′.
Question 1
• 1 mark: States that a horizontal tangent means f′(2) = 0.
• 1 mark: Explains that the graph of f′ would cross or touch the x-axis at x = 2.
• 1 mark: States clearly that this is because f′ represents the slope of f.
Award 1–3 marks depending on completeness and clarity of explanation.
Question 2 (4–6 marks)
A function f is twice differentiable on the interval 0 < x < 6.
You are told the following information about its derivatives:
• f′(x) is positive for 0 < x < 2, equal to 0 at x = 2, negative for 2 < x < 4, equal to 0 at x = 4, and positive for 4 < x < 6.
• f″(x) is negative for 0 < x < 1, positive for 1 < x < 3, negative for 3 < x < 5, and positive for 5 < x < 6.
Using clear verbal descriptions relating f, f′, and f″, answer the following:
(a) Describe the behaviour of f on each interval in terms of increasing and decreasing.
(b) Identify and classify the points x = 2 and x = 4 as local maxima, minima, or neither, explaining your reasoning with reference to f′.
(c) Describe how the concavity of f changes across the interval and identify all possible points of inflection, justifying your answer with reference to f″.
Question 2
(a) Increasing/decreasing behaviour (1–2 marks)
• 1 mark: Correctly states that f is increasing on 0 < x < 2 and 4 < x < 6.
• 1 mark: Correctly states that f is decreasing on 2 < x < 4.
(b) Classification of extrema (1–2 marks)
• 1 mark: Correctly identifies x = 2 as a local maximum because f′ changes from positive to negative.
• 1 mark: Correctly identifies x = 4 as a local minimum because f′ changes from negative to positive.
(c) Concavity and inflection points (1–2 marks)
• 1 mark: Correctly describes concavity changes according to the sign of f″: concave down on 0 < x < 1, up on 1 < x < 3, down on 3 < x < 5, up on 5 < x < 6.
• 1 mark: Identifies possible points of inflection at x = 1, 3, and 5, with justification that each corresponds to a change in sign of f″.
Award 4–6 marks total depending on thoroughness, accuracy, and clarity of reasoning.
