← Calculus syllabus

United StatesCalculus

Unit 5: Analytical Applications of Differentiation

12 dot points across 12 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How are the properties of a function, its first derivative, and its second derivative connected, and how do you justify conclusions about one from another?

Topic 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative: justify conclusions about f using f-prime and f-double-prime.
A focused answer to AP Calculus AB Topic 5.9, drawing and justifying conclusions about a function from its first and second derivatives, including extrema and inflection justifications phrased with the correct derivative, with worked examples.
9 min answer →

How does the sign of the second derivative tell you about concavity and points of inflection?

Topic 5.6 Determining Concavity of Functions over Their Domains: use the second derivative to find concavity and points of inflection.
A focused answer to AP Calculus AB Topic 5.6, using the sign of the second derivative to determine concavity and locate points of inflection, with worked sign-chart examples and the required inflection-point justification.
9 min answer →

How does the sign of the first derivative tell you where a function is increasing or decreasing?

Topic 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing: use the sign of the first derivative to find intervals of increase and decrease.
A focused answer to AP Calculus AB Topic 5.3, using the sign of the first derivative on a sign chart to determine where a function is increasing or decreasing, with worked sign-chart examples and the correct justification language.
9 min answer →

How do you find extrema and analyze the behavior of a curve defined implicitly?

Topic 5.12 Exploring Behaviors of Implicit Relations: analyze extrema and concavity of implicitly defined relations using implicit differentiation.
A focused answer to AP Calculus AB Topic 5.12, applying analytical tools to implicitly defined curves by finding horizontal and vertical tangents and second derivatives through implicit differentiation, with worked examples.
9 min answer →

What guarantees a continuous function has a maximum and minimum, and where can extrema occur?

Topic 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points: identify critical points and distinguish local from global extrema.
A focused answer to AP Calculus AB Topic 5.2, stating the Extreme Value Theorem, defining critical points where the derivative is zero or undefined, and distinguishing global from local extrema, with worked examples.
9 min answer →

How do you turn a real-world maximum or minimum question into a calculus problem?

Topic 5.10 Introduction to Optimization Problems: set up an optimization problem by writing the quantity to be optimized as a function of one variable.
A focused answer to AP Calculus AB Topic 5.10, setting up optimization problems by identifying the quantity to optimize, writing a constraint, and reducing to a single-variable objective function, with worked setups.
8 min answer →

How do you use the first and second derivatives to sketch a function, and read between the graphs of f, f-prime and f-double-prime?

Topic 5.8 Sketching Graphs of Functions and Their Derivatives: use derivative information to sketch a graph and relate the graphs of f, f-prime and f-double-prime.
A focused answer to AP Calculus AB Topic 5.8, combining increasing/decreasing and concavity information to sketch a function and to read across the graphs of f, f-prime and f-double-prime, with worked feature-by-feature analysis.
9 min answer →

How do you solve a full optimization problem and justify that you have found the absolute maximum or minimum?

Topic 5.11 Solving Optimization Problems: solve a complete optimization problem and justify the absolute extremum.
A focused answer to AP Calculus AB Topic 5.11, solving complete optimization problems by differentiating the objective, finding critical points, and justifying the absolute extremum, with worked box and area examples.
10 min answer →

How do you find the absolute maximum and minimum of a function on a closed interval?

Topic 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema: find absolute extrema by comparing values at critical points and endpoints.
A focused answer to AP Calculus AB Topic 5.5, using the candidates test to find absolute extrema on a closed interval by comparing function values at critical points and endpoints, with worked tabulated examples.
8 min answer →

How does a sign change in the first derivative classify a critical point as a local maximum or minimum?

Topic 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema: classify critical points using sign changes in the first derivative.
A focused answer to AP Calculus AB Topic 5.4, using sign changes of the first derivative to classify critical points as relative maxima, relative minima, or neither, with worked sign-chart classifications and the required justification.
8 min answer →

When does the Mean Value Theorem guarantee a point where the instantaneous rate equals the average rate, and how do you use it?

Topic 5.1 Using the Mean Value Theorem: state the hypotheses and conclusion of the MVT and apply it to find a guaranteed point.
A focused answer to AP Calculus AB Topic 5.1, stating the continuity and differentiability hypotheses of the Mean Value Theorem, its geometric meaning, and how to find the guaranteed value of c, with worked examples and hypothesis checks.
9 min answer →

How does the sign of the second derivative at a critical point classify it as a local maximum or minimum?

Topic 5.7 Using the Second Derivative Test to Determine Extrema: classify critical points using the sign of the second derivative.
A focused answer to AP Calculus AB Topic 5.7, using the sign of the second derivative at a critical point to classify it as a relative maximum or minimum, when the test is inconclusive, and how it compares to the first derivative test.
8 min answer →