Review AP Calculus AB/BC Unit 5 to build the analytical toolkit for describing function behavior using derivatives. This unit covers the Mean Value Theorem, critical points, extrema tests, concavity, optimization, and implicit relations, all of which carry 15-18% of the AP exam.
Use the topic guides, practice questions, FRQ practice, and AP score calculator available for this unit to focus your review on justification and reasoning skills.
Unit 5 is where calculus shifts from computing derivatives to using them as evidence. Every major skill in this unit asks you to read the sign or behavior of f' or f'' and draw a justified conclusion about f itself.
Unit 5 covers how to use derivatives analytically: applying the MVT and EVT, finding and classifying critical points, determining increasing/decreasing intervals and concavity, sketching graphs of f from f' and f'', solving optimization problems, and analyzing implicit relations.
The Mean Value Theorem (5.1) and Extreme Value Theorem (5.2) are not just facts to memorize. AP questions ask you to verify their conditions (continuity on [a,b], differentiability on (a,b)) and then state what the theorem guarantees. MVT guarantees a c where f'(c) equals the average rate of change; EVT guarantees at least one absolute max and one absolute min on a closed interval.
Topics 5.3 through 5.7 all rely on reading the sign of f' or f''. A positive f' means f is increasing; a negative f' means decreasing. A sign change in f' at a critical point identifies a local extremum via the First Derivative Test. The sign of f'' tells you concavity, and a sign change in f'' locates an inflection point. The Second Derivative Test uses f''(c) to classify a critical point without a full sign chart.
Optimization (5.10-5.11) asks you to write an objective function, use a constraint to reduce it to one variable, find critical points, and confirm whether the result is a max or min using the First or Second Derivative Test. Topic 5.12 extends all of this to implicit relations: you find dy/dx through implicit differentiation, locate critical points where dy/dx equals zero or is undefined, and use the second derivative to describe concavity.
The central idea of Unit 5 is that the sign and behavior of f' and f'' are evidence for conclusions about f. Every topic in this unit, from the MVT to implicit relations, asks you to state a conclusion about a function and justify it using derivative information. That justification skill is what AP free-response questions in this unit are testing.
If f is continuous on [a,b] and differentiable on (a,b), the MVT guarantees a point c where f'(c) equals the average rate of change (f(b)-f(a))/(b-a). AP questions require you to verify conditions and state what the theorem guarantees.
The EVT guarantees absolute extrema for continuous functions on closed intervals. Critical points occur where f'(x) = 0 or f' is undefined. All local extrema are at critical points, but not all critical points are extrema.
Use a sign chart for f': positive f' means increasing, negative f' means decreasing. Partition the domain at critical points and domain breaks, then test the sign of f' in each interval.
At a critical point, if f' changes from positive to negative, f has a local max; from negative to positive, a local min; no sign change means no extremum. Always justify with the sign change of f'.
On a closed interval, collect all critical points and endpoints, evaluate f at each, and compare. The largest output is the absolute max; the smallest is the absolute min.
f''(x) > 0 means concave up; f''(x) < 0 means concave down. Inflection points occur where f'' changes sign. Build a sign chart for f'' to find concavity intervals.
At a critical point c where f'(c) = 0: f''(c) > 0 gives a local min, f''(c) < 0 gives a local max. If f''(c) = 0 or is undefined, the test is inconclusive; use the First Derivative Test instead.
Use the signs of f' and f'' to sketch f, or read a graph of f' to identify increasing/decreasing intervals, extrema, and inflection points of f. Corners and cusps on f appear as discontinuities on f'.
The graphs of f, f', and f'' are linked: zeros of f' with sign changes are extrema of f; zeros of f'' with sign changes are inflection points of f; local extrema of f' are inflection points of f.
Optimization uses derivatives to find maximum or minimum values of a quantity. Write an objective function, use a constraint to reduce it to one variable, find critical points, and confirm the extremum type.
Apply the full optimization process in applied contexts: define variables, write and reduce the objective function, find and confirm critical points, check endpoints on closed intervals, and interpret the answer with units.
Use implicit differentiation to find dy/dx for curves defined by F(x,y) = 0. Horizontal tangents occur where dy/dx = 0; vertical tangents where dy/dx is undefined. The second derivative, expressed in x, y, and dy/dx, determines concavity.
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Review Extreme Value Theorem, Global vs. Local Extrema, and Critical Points with attention to how the concept appears in AP-style source and evidence questions.
Review Connecting a Function, Its First Derivative, and Its Second Derivative with attention to how the concept appears in AP-style source and evidence questions.
Review Using the Mean Value Theorem with attention to how the concept appears in AP-style source and evidence questions.
Review Using the First Derivative Test to Determine Relative (Local) Extrema with attention to how the concept appears in AP-style source and evidence questions.
The MVT states that if f is continuous on [a,b] and differentiable on (a,b), then there exists at least one c in (a,b) such that f'(c) = (f(b) - f(a)) / (b - a). Geometrically, the tangent line at c is parallel to the secant line through (a, f(a)) and (b, f(b)). On the AP exam, you must verify both conditions before invoking the theorem.
Given f(x) = x^3 - 3x on [0,2], verify MVT conditions and find the value of c guaranteed by the theorem.
| Theorem | Conditions | Conclusion |
|---|---|---|
| Mean Value Theorem | Continuous on [a,b], differentiable on (a,b) | There exists c in (a,b) where f'(c) = (f(b)-f(a))/(b-a) |
| Rolle's Theorem | Continuous on [a,b], differentiable on (a,b), f(a)=f(b) | There exists c in (a,b) where f'(c) = 0 |
| Extreme Value Theorem | Continuous on [a,b] | f has at least one absolute max and one absolute min on [a,b] |
The EVT guarantees that a function continuous on a closed interval [a,b] has at least one absolute maximum and at least one absolute minimum on that interval. Critical points are where f'(x) = 0 or f'(x) is undefined. All local extrema occur at critical points, but not every critical point is a local extremum. A critical point with no sign change in f' is not an extremum.
Explain why f(x) = 1/x on [-1,1] is not guaranteed an absolute maximum by the EVT.
| Type of Extremum | Where it can occur | How to confirm |
|---|---|---|
| Local maximum | Critical points only | f' changes from positive to negative |
| Local minimum | Critical points only | f' changes from negative to positive |
| Absolute maximum on [a,b] | Critical points or endpoints | Compare all candidate values |
| Absolute minimum on [a,b] | Critical points or endpoints | Compare all candidate values |
To find where f is increasing or decreasing, locate all critical points and domain breaks, then test the sign of f' in each resulting interval. Where f'(x) > 0, f is increasing; where f'(x) < 0, f is decreasing. The First Derivative Test classifies each critical point: if f' changes from positive to negative, f has a local max; if f' changes from negative to positive, f has a local min; if the sign does not change, there is no extremum.
For f(x) = x^4 - 4x^3, find all critical points, determine intervals of increase and decrease, and classify each critical point using the First Derivative Test.
| Sign of f' before c | Sign of f' after c | Conclusion at c |
|---|---|---|
| Positive | Negative | Local maximum |
| Negative | Positive | Local minimum |
| Positive | Positive | No extremum (increasing through c) |
| Negative | Negative | No extremum (decreasing through c) |
On a closed interval [a,b], the absolute maximum and minimum can only occur at critical points or at the endpoints. The Candidates Test collects all critical points in (a,b) plus the two endpoints, evaluates f at each, and compares the output values. The largest output is the absolute maximum; the smallest is the absolute minimum. You compare f-values, not f'-values.
Find the absolute maximum and minimum of f(x) = 2x^3 - 9x^2 + 12x on [0,3] using the Candidates Test.
| Step | Action |
|---|---|
| 1 | Find f'(x) and solve f'(x) = 0 or identify where f' is undefined |
| 2 | List all critical points in (a,b) plus endpoints a and b |
| 3 | Evaluate f at each candidate |
| 4 | Compare values: largest is absolute max, smallest is absolute min |
The sign of f'' determines concavity: f''(x) > 0 means f is concave up on that interval (f' is increasing); f''(x) < 0 means f is concave down (f' is decreasing). An inflection point occurs where f'' changes sign. The Second Derivative Test classifies a critical point c: if f'(c) = 0 and f''(c) > 0, then f has a local min at c; if f''(c) < 0, then f has a local max at c. If f''(c) = 0 or is undefined, the test is inconclusive and you must use the First Derivative Test instead.
For f(x) = x^4, find all critical points and explain why the Second Derivative Test is inconclusive at x = 0. Then use the First Derivative Test to classify the point.
| Test | What you need | Conclusion | When it fails |
|---|---|---|---|
| First Derivative Test | Sign of f' on both sides of c | Local max, min, or neither | Never fails, always gives an answer |
| Second Derivative Test | f'(c) = 0 and f''(c) defined and nonzero | Local max (f''<0) or local min (f''>0) | f''(c) = 0 or f''(c) undefined |
Topics 5.8 and 5.9 ask you to move between the graphs of f, f', and f'' in both directions. Where f has a local max or min, f' crosses zero with a sign change. Where f has an inflection point, f'' crosses zero with a sign change, and f' has a local extremum. Where f is increasing, f' is positive; where f is concave up, f'' is positive. A corner or cusp on f appears as a discontinuity or undefined point on f'.
Given a graph of f', identify all intervals where f is increasing, all local extrema of f, and all inflection points of f.
| Feature of f | What it looks like on f' | What it looks like on f'' |
|---|---|---|
| Increasing | f' > 0 | No direct requirement |
| Local maximum | f' crosses zero from + to - | f'' < 0 (if Second Derivative Test applies) |
| Local minimum | f' crosses zero from - to + | f'' > 0 (if Second Derivative Test applies) |
| Inflection point | f' has a local extremum | f'' changes sign |
| Concave up | f' is increasing | f'' > 0 |
Optimization problems ask you to find the maximum or minimum value of a quantity in a real context. The process: define variables and state what you are optimizing, write an objective function, use a constraint equation to reduce the objective function to one variable, determine the feasible domain, find critical points using f', and confirm whether each critical point is a max or min using the First or Second Derivative Test. Always check endpoints if the domain is a closed interval. Interpret the answer with correct units and in the context of the problem.
A farmer has 200 meters of fencing to enclose a rectangular field against a barn wall (one side needs no fencing). Write the objective function, find the dimensions that maximize area, and confirm it is a maximum.
| Step | What to do |
|---|---|
| 1. Define variables | Name the quantity to optimize and any other relevant quantities with units |
| 2. Write objective function | Express the quantity to optimize as a function of your variables |
| 3. Apply constraint | Use the given relationship to reduce the objective function to one variable |
| 4. Find critical points | Differentiate and solve f'(x) = 0; note where f' is undefined |
| 5. Confirm and interpret | Use First or Second Derivative Test; state the answer with units in context |
For a curve defined by an equation F(x,y) = 0, you find dy/dx through implicit differentiation. Critical points occur where dy/dx = 0 (horizontal tangent) or where dy/dx is undefined (vertical tangent or cusp). To find the second derivative d2y/dx2 for an implicit relation, differentiate dy/dx implicitly again; the result is typically expressed in terms of x, y, and dy/dx. Use the sign of d2y/dx2 to determine concavity at a point, substituting the coordinates and the value of dy/dx at that point.
For x^2 + y^2 = 25, find all points with horizontal tangents, find all points with vertical tangents, and determine the concavity at (3,4) using the second derivative.
| Feature | Condition on dy/dx | How to find it |
|---|---|---|
| Horizontal tangent | dy/dx = 0 | Set numerator of dy/dx = 0, check denominator is nonzero |
| Vertical tangent | dy/dx undefined | Set denominator of dy/dx = 0, check numerator is nonzero |
| Concavity | Sign of d2y/dx2 | Differentiate dy/dx implicitly; substitute x, y, and dy/dx at the point |
Try AP-style multiple-choice questions and written prompts after you review the notes.
| Term | Definition |
|---|---|
| Critical Points | An x-value where f'(x) = 0 or f'(x) does not exist; a necessary but not sufficient condition for a local extremum of f. |
| Absolute Extrema | The highest and lowest values a function reaches over a given interval; found using the Candidates Test on closed intervals. |
| Local Extrema | The highest or lowest values of f within a small neighborhood of a point; all local extrema occur at critical points. |
| Candidates Test | The procedure for finding absolute extrema on a closed interval: evaluate f at all critical points in (a,b) and at the endpoints a and b, then compare output values. |
| monotonicity | The property of a function being entirely increasing (f' > 0) or entirely decreasing (f' < 0) on an interval, determined by the sign of the first derivative. |
| Concave Up | A function is concave up on an interval where f''(x) > 0, meaning f' is increasing and the graph curves upward. |
| Concave Down | A function is concave down on an interval where f''(x) < 0, meaning f' is decreasing and the graph curves downward. |
| Inflection Point | A point on the graph of f where f'' changes sign, indicating a change in concavity from up to down or down to up. |
| Rolle's Theorem | A special case of the MVT: if f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists c in (a,b) where f'(c) = 0. |
| Average Rate of Change | The slope of the secant line between two points on f, equal to (f(b)-f(a))/(b-a); the value that f'(c) equals at the MVT-guaranteed point c. |
| Instantaneous Rate of Change | The value of f'(c) at a specific point c; geometrically, the slope of the tangent line to f at that point. |
| Reasoning with derivatives | Using the sign and behavior of f' and f'' as evidence to draw justified conclusions about the behavior of f, such as increasing/decreasing intervals, extrema, and concavity. |
| implicit relation | An equation relating x and y that is not solved explicitly for y; analyzed using implicit differentiation to find dy/dx and d2y/dx2. |
| Horizontal Tangent Line | A tangent line with slope zero, occurring where f'(x) = 0 for explicit functions or where dy/dx = 0 for implicit relations. |
| Points of Inflection | Locations on a graph where concavity changes; require f'' to change sign, not just equal zero. |
Applying the MVT or EVT without first confirming continuity (and differentiability for MVT) is an incomplete justification. Always check and state the conditions before citing the theorem's conclusion.
A critical point where f' does not change sign is not a local extremum. For example, f(x) = x^3 has f'(0) = 0, but x = 0 is not a local max or min because f' stays positive on both sides.
When comparing candidates for absolute extrema, you must evaluate and compare f(x) values, not f'(x) values. The largest f-value is the absolute max; the smallest is the absolute min.
An inflection point requires f'' to change sign, not just equal zero. For f(x) = x^4, f''(0) = 0 but f'' does not change sign, so x = 0 is not an inflection point.
The second derivative of an implicit relation is expressed in terms of x, y, and dy/dx. When evaluating concavity at a specific point, you must substitute the coordinates and the value of dy/dx at that point, not just x and y.
AP Calculus free-response questions in this unit frequently ask you to justify a conclusion rather than just compute an answer. A complete response names the theorem or test being used (MVT, EVT, First Derivative Test, Second Derivative Test), states the condition or sign observed, and writes the conclusion it supports. Partial credit is often lost when students state a correct answer without the required reasoning.
Multiple-choice and free-response questions often present information about f through a graph of f' or a table of values, then ask about features of f such as extrema, concavity, or inflection points. Practicing the translation between f, f', and f'' in both directions is essential for these tasks.
Free-response questions may present a real-world scenario requiring you to set up and solve an optimization problem, including writing the objective function, applying a constraint, and interpreting the result with units. Implicit differentiation questions may ask for the equation of a tangent line, the location of horizontal or vertical tangents, or the concavity of a curve at a specific point.
For both theorems, identify whether the function is continuous on [a,b] and, for MVT, differentiable on (a,b). State what each theorem guarantees before applying it.
Set f'(x) = 0 and identify where f'(x) is undefined. Include domain restrictions. Remember that a critical point is not automatically an extremum.
Partition the domain at critical points and domain breaks. Test the sign of f' in each interval for increasing/decreasing behavior, and the sign of f'' for concavity. Use these charts to classify extrema and inflection points.
Use the Candidates Test for absolute extrema on closed intervals. Use the First Derivative Test for local extrema when the Second Derivative Test is inconclusive (f''(c) = 0 or undefined).
Define variables with units, write the objective function, substitute the constraint to get one variable, determine the feasible domain, and confirm whether the critical point gives a max or min.
Find dy/dx by implicit differentiation, locate horizontal and vertical tangents, and compute d2y/dx2 by differentiating dy/dx implicitly. Substitute x, y, and dy/dx at a specific point to evaluate concavity.
AP free-response scoring rewards explicit justification. Name the theorem or test you are using, state the sign or condition you observed, and write the conclusion it supports.
Read the MVT and EVT topic guides and practice stating each theorem's conditions and conclusion in your own words. Work through two or three examples where you verify conditions, apply the theorem, and write a complete justification sentence.
Practice constructing sign charts for f' from scratch: find critical points, partition the domain, test signs, and write increasing/decreasing intervals. Then apply the First Derivative Test and Candidates Test on the same function to classify both local and absolute extrema.
Build sign charts for f'' to find concavity intervals and inflection points. Practice the Second Derivative Test and identify cases where it is inconclusive. Compare the First and Second Derivative Tests on the same critical point to see when each is more efficient.
Use the topic guides for 5.8 and 5.9 to practice translating between graphs of f, f', and f''. Given a graph of f', identify all features of f. Given a graph of f, sketch f'. Focus on the correspondence between zeros, sign changes, and extrema across all three graphs.
Use the optimization topic guides to practice the full five-step process on classic problems (maximum area, minimum surface area, shortest distance). Then work through 5.12 examples finding horizontal and vertical tangents on implicit curves and computing d2y/dx2 at a specific point. Use available FRQ practice to rehearse written justification under exam conditions.
Open the individual guides for Unit 5 when you want a closer review of one topic.
Use AP-style practice after you review the notes so you can check what you understand.
Practice free-response reasoning and compare your answer with scoring guidance.
Watch past review streams filtered to Unit 5 when you want a video walkthrough.
Open the Fiveable one-page unit review, then explore visual cheatsheets for a quick refresher.
Estimate your broader AP score goal after you review the course and exam format.
AP Calc Unit 5 covers 12 topics focused on using derivatives to analyze function behavior. Topics include the Mean Value Theorem, the Extreme Value Theorem, critical points, intervals of increase and decrease, the First and Second Derivative Tests, concavity, sketching graphs of functions and their derivatives, and optimization problems. Implicit relations are covered in Topic 5.12. Here's the full topic list: - 5.1 Using the Mean Value Theorem - 5.2 Extreme Value Theorem, Global vs. Local Extrema, and Critical Points - 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing - 5.4 Using the First Derivative Test to Determine Relative Extrema - 5.5 Using the Candidates Test to Determine Absolute Extrema - 5.6 Determining Concavity of Functions over Their Domains - 5.7 Using the Second Derivative Test to Determine Extrema - 5.8 Sketching Graphs of Functions and Their Derivatives - 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative - 5.10 Introduction to Optimization Problems - 5.11 Solving Optimization Problems - 5.12 Exploring Behaviors of Implicit Relations See AP Calc Unit 5 for notes and practice on each topic.
AP Calc Unit 5 makes up 15-18% of the AP exam, making it one of the more heavily tested units. It covers analytical applications of differentiation, including critical points, the First and Second Derivative Tests, concavity, optimization problems, and graph sketching. Expect several multiple-choice questions and at least one FRQ that draws on these skills.
The AP Calc Unit 5 progress check in AP Classroom includes both MCQ and FRQ parts drawn from the unit's 12 topics. The MCQ section tests skills like applying the Mean Value Theorem, identifying critical points, and using the First and Second Derivative Tests. The FRQ part asks you to justify conclusions about increasing and decreasing intervals, concavity, absolute and relative extrema, and optimization. The progress check also covers graph sketching and connecting a function to its first and second derivatives (Topics 5.8 and 5.9). Practicing these topics before the progress check is the best prep, and you can find matched practice at AP Calc Unit 5.
AP Calc Unit 5 FRQs most often ask you to analyze a function using its first and second derivatives, justify extrema with the First or Second Derivative Test, determine concavity, or solve an optimization problem. To practice, work through problems that require written justification, not just a numerical answer, because the AP exam awards points specifically for correct reasoning. Focus on Topics 5.3-5.7 for sign-chart justifications and Topics 5.10-5.11 for optimization setups. You can find Unit 5 FRQ-style practice at AP Calc Unit 5.
For AP Calc Unit 5 practice questions, including multiple-choice and practice test problems, head to AP Calc Unit 5. You'll find topic-level MCQ practice covering the Mean Value Theorem, critical points, derivative tests, concavity, graph sketching, and optimization. Mixing MCQ practice with FRQ-style justification questions is the most effective way to prepare for the full range of question types this unit appears in on the exam.
Start AP Calc Unit 5 by making sure you understand what a derivative tells you about a function, because every topic in this unit builds on that idea. Work through the topics in order: the Mean Value Theorem (5.1), then critical points and the Extreme Value Theorem (5.2), then increasing and decreasing intervals (5.3), then the First and Second Derivative Tests (5.4 and 5.7). Once those feel solid, move to concavity (5.6), graph sketching (5.8-5.9), and optimization (5.10-5.11). A few concrete steps that help: - Draw sign charts for f' and f'' on every practice problem. Writing out the reasoning is exactly what earns points on FRQs. - Practice justifying your answers in full sentences, not just circling a value. - For optimization, always define your variables and write the constraint equation before taking a derivative. - Review Topic 5.12 on implicit relations if you're in AP Calc BC or want extra challenge. All 12 topics with practice are at AP Calc Unit 5.