AP Calculus AB and BC

Unit 5: Analytical Applications of Differentiation

Exam weighting: AB 15-20% · BC 10-15%

Unit 5 uses the first derivative to find where a function increases, decreases, and reaches extrema, and the second derivative for concavity and inflection points. It is the highest-weighted differentiation unit on the AB exam (15-18%), and the core skill is picking the right test: First, Second, or Candidates.

The whole unit rests on one move: the sign of a derivative is information about the function one level below it. Where f′(x)>0f'(x) > 0 the graph of ff rises, where f′(x)<0f'(x) < 0 it falls, and local extrema hide at critical points where f′(x)=0f'(x) = 0 or f′f' is undefined (Topic 5.2). One level up, where f′′(x)>0f''(x) > 0 the graph is concave up, where f′′(x)<0f''(x) < 0 it is concave down, and concavity flips at points of inflection (Topic 5.6). Every technique in Unit 5 is a way of reading those signs.

Topics 5.1 and 5.2 open the unit with two guarantees. The Extreme Value Theorem says a function continuous on a closed interval [a,b][a, b] must attain both a maximum and a minimum somewhere on it. The Mean Value Theorem says a function continuous on [a,b][a, b] and differentiable on (a,b)(a, b) must, at some point cc inside, have an instantaneous rate of change equal to its average rate of change across the interval. These theorems promise that a point exists without telling you where, which is exactly what a justification-style exam question tests.

f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}
TestReach for it whenWhat it tells you
First Derivative Test (5.4)You know the sign of f′f' on both sides of a critical pointLocal max if f′f' changes from positive to negative; local min if f′f' changes from negative to positive; neither if no sign change
Second Derivative Test (5.7)f′′f'' is quick to evaluate at the critical point and is nonzero thereLocal max if f′′<0f'' < 0; local min if f′′>0f'' > 0; no conclusion if f′′=0f'' = 0
Candidates Test (5.5)You need the absolute max or min on a closed interval [a,b][a, b]Compare the value of ff at every critical point and both endpoints; largest and smallest win
  • f′f' positive means ff is increasing; f′f' negative means ff is decreasing (Topic 5.3).
  • f′f' increasing means ff is concave up; f′f' decreasing means ff is concave down. This is the trap: f′f' positive and f′f' increasing say two different things.
  • A sign change in f′f' locates a local extremum of ff; a sign change in f′′f'' locates a point of inflection of ff (Topics 5.4, 5.6).
  • Given only a graph of f′f', you can describe the shape features of ff: its extrema, its concavity, and its inflection points (Topic 5.9).

Topics 5.10 and 5.11 apply the extrema machinery to word problems: translate the scenario into a function of one variable, restrict the domain to what the context allows, then find the absolute max or min, usually with the Candidates Test. Topic 5.12 extends the same critical-point analysis to implicitly defined relations, where you locate points with dydx=0\frac{dy}{dx} = 0 or undefined and may need d2ydx2\frac{d^2y}{dx^2} written in terms of xx, yy, and dydx\frac{dy}{dx}.

What the exam asks

Unit 5 carries 15-18% of the AB multiple-choice section and 8-11% of BC, making it the heaviest of the differentiation units on AB. Free-response questions here are graded on justification, not just the answer: you must name the reason, such as "f′f' changes from positive to negative at x=cx = c, so ff has a local maximum there." A correct extremum with no sign-based reasoning earns the point for the value but loses the justification point.

Topics in this unit

Topic numbers and titles from the College Board Course and Exam Description.

  • 5.1Using the Mean Value Theorem
  • 5.2Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
  • 5.3Determining Intervals on Which a Function Is Increasing or Decreasing
  • 5.4Using the First Derivative Test to Determine Relative (Local) Extrema
  • 5.5Using the Candidates Test to Determine Absolute (Global) Extrema
  • 5.6Determining Concavity of Functions over Their Domains
  • 5.7Using the Second Derivative Test to Determine Extrema
  • 5.8Sketching Graphs of Functions and Their Derivatives
  • 5.9Connecting a Function, Its First Derivative, and Its Second Derivative
  • 5.10Introduction to Optimization Problems
  • 5.11Solving Optimization Problems
  • 5.12Exploring Behaviors of Implicit Relations

How to study this unit

  • When an FRQ asks you to justify a local extremum, state the test and the sign change in words: "$f'$ changes from positive to negative at $x = c$" (Topic 5.4). A bare answer with no sign reasoning earns no justification credit.
  • Reach for the Candidates Test (Topic 5.5) the instant you see "absolute" or "on the closed interval $[a, b]$." Evaluate $f$, not $f'$, at every critical point and both endpoints. The endpoints are the step students forget.
  • Use the Second Derivative Test (Topic 5.7) only when $f''$ is fast to compute and nonzero at the critical point. If $f'' = 0$ the test is silent, so fall back to the First Derivative Test.
  • Drill reading a graph of $f'$ to describe $f$ (Topics 5.8-5.9). Where $f'$ is positive $f$ rises; where $f'$ is increasing $f$ is concave up. Confusing "$f'$ positive" with "$f'$ increasing" is the single most common error.
  • Check hypotheses before invoking an existence theorem (Topics 5.1-5.2): the Mean Value Theorem needs continuity on $[a, b]$ and differentiability on $(a, b)$, while the Extreme Value Theorem needs only continuity on $[a, b]$.

Guides for this unit

Tools and tables for this unit