How many points is each Unit 5 free-response question worth?

Every AP Calculus free-response question is scored out of 9 points, distributed across its parts. Setup, correct computation, and justification each carry separate points, which is why writing your reasoning matters even when your final number is wrong.

When do I use the First Derivative Test versus the Second Derivative Test?

Both classify extrema. Use the First Derivative Test when you have a sign chart of f' or when f'' is hard to compute. Use the Second Derivative Test when f'' is easy and you already know f'(c)=0. If the second derivative equals zero at the point, the Second Derivative Test is inconclusive and you must fall back on the First Derivative Test.

What is the difference between where f' equals zero and where f has an inflection point?

Where f'=0 you have a candidate for a local extremum of f. Inflection points come from f'' changing sign, which corresponds to where the graph of f' has a local max or min (its slope changes sign). These are different features, and confusing them is a frequent cause of lost points.

Do I need to show work if I can find the answer on a calculator?

Yes. On calculator-active FRQs you must present the mathematical setup — for example the equation you solved or the integral you evaluated — before the numerical answer. A bare answer with no setup typically earns no credit even if it is correct, and answers should generally be given to three decimal places when a decimal is required.

AP-CALCBC-5-FRQ

U5 FRQ Practice

Master AP Calculus BC Unit 5 free-response questions: connect f, f', f'', justify extrema and concavity, and earn every analysis point.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on U5 FRQ Practice, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

Unit 5 is where the AP exam rewards you for reasoning, not just computing. The free-response questions in this unit rarely ask "what is the derivative?" Instead they ask you to interpret sign charts, justify why a point is a maximum, connect a function to its first and second derivatives, and translate calculus facts into complete English sentences.

This guide is about strategy: how to read a multi-part Unit 5 FRQ, how to structure justifications that graders can award points to, and how to avoid the classic mistakes that cost partial credit. You already know the theorems from U5.1 through U5.12 — here you learn to deploy them under exam conditions.

Anatomy of a Unit 5 FRQ

Unit 5 free-response prompts are almost always multi-part, and the parts build a mini-analysis of one function. A typical structure looks like this.

Part	What it asks	What earns points
(a)	Find critical points or intervals of increase/decrease	Set $f'(x)=0$ , show sign analysis
(b)	Classify local extrema	State the test AND the sign change
(c)	Discuss concavity / inflection	Use $f''$ sign change, not just $f''=0$
(d)	Justify or interpret in context	A complete written justification

The defining feature is that later parts depend on earlier work. If part (a) gives you critical numbers, part (b) expects you to use them. Graders read for specific tokens: the correct derivative, the correct sign behavior, and an explicit conclusion.

A common misconception is that showing

f'(c)=0

is enough to claim a maximum. It is not. The exam wants evidence that

f'

changes from positive to negative (first derivative test) or that

f''(c)<0

(second derivative test). Reading the prompt carefully tells you which analytical tool the writers expect — often the phrasing "justify your answer" is the signal that a bare number will score zero.

Writing justifications that score

The single biggest source of lost points in Unit 5 is weak justification. A justification is a logical chain: a calculus fact, applied to specific numbers, leading to a conclusion.

Compare a weak and a strong answer for "Does

f

have a local minimum at

x=2

?"

Weak: "Yes, because

f'(2)=0

." This earns little — many points where

f'=0

are not minima.

Strong: "

f'(x)

changes from negative to positive at

x=2

, so by the First Derivative Test

f

has a local minimum at

x=2

." This names the sign change, the location, and the theorem.

Use this template for every classification question: state what the derivative does (sign, or value of

f''

), then name the test, then state the conclusion. Keep it to one sentence if possible.

Avoid these traps. Do not confuse

f

f'

, and

f''

— if the graph shown is

f'

, then increasing intervals of

f

come from where the graph is above the axis, and concavity of

f

comes from where the

f'

graph is increasing. Do not say "there is an inflection point because

f''=0

"; you must confirm

f''

actually changes sign. And always answer the exact question: if asked for the

x

-value, give the

x

-value; if asked for a point, give an ordered pair.

When the graph of f' is given

A signature Unit 5 FRQ hands you the graph of

f'

(not

f

) and asks questions about

f

. This tests whether you truly understand the connection between a function and its derivatives, the skill developed in U5.8.

Translate carefully using this dictionary.

Feature of the graph of $f'$	Conclusion about $f$
$f'>0$ (above axis)	$f$ is increasing
$f'<0$ (below axis)	$f$ is decreasing
$f'$ crosses axis $+\to-$	$f$ has a local max
$f'$ crosses axis $-\to+$	$f$ has a local min
$f'$ increasing (slope $>0$ )	$f$ is concave up
$f'$ decreasing (slope $<0$ )	$f$ is concave down
$f'$ has a local extremum	$f$ has an inflection point

The most-missed row is the last: a maximum or minimum of the

f'

graph corresponds to an inflection point of

f

, because that is where

f'

changes from increasing to decreasing (so

f''

changes sign). Students routinely report inflection points where

f'

crosses the axis — that is a wrong translation.

When the FRQ gives

f'

and asks for a value of

f

, you often integrate:

f(b) = f(a) + \int_a^b f'(x)\,dx

, reading the integral as area from the graph. This blends Unit 5 analysis with accumulation, a frequent BC crossover.

Optimization and implicit FRQs under time pressure

Unit 5 optimization prompts (U5.10) and implicit-relation prompts (U5.12) show up in free response with their own structures. For optimization, the grader looks for four things: a primary equation to optimize, a constraint used to reduce to one variable, the derivative set equal to zero, and a justification that your critical point is truly the max or min. Never skip the justification — a candidate-test argument or a sign analysis of the derivative is required for the final point.

For implicit relations, expect a part that asks for

\frac{dy}{dx}

by implicit differentiation, then a part asking for a tangent line, and sometimes a part asking for

\frac{d^2y}{dx^2}

or where the tangent is horizontal or vertical. Horizontal tangents occur where the numerator of

\frac{dy}{dx}

is zero (and denominator nonzero); vertical tangents where the denominator is zero. State this explicitly.

Manage time: each FRQ is worth 9 points and should take roughly 15 minutes. Do not leave a part blank because you are stuck earlier — parts are often independent enough to attempt. Write down setups even if you cannot finish, because setup points are real. Show the substitution before you evaluate, and box or clearly label each final answer so the reader can find it.

Key terms

First Derivative Test.: A method that classifies a critical point by checking whether $f'$ changes sign: $+\to-$ gives a local max, $-\to+$ gives a local min.
Second Derivative Test.: At a critical point where $f'(c)=0$ : if $f''(c)<0$ the point is a local max, if $f''(c)>0$ a local min; inconclusive if $f''(c)=0$ .
Inflection point.: A point where the concavity of $f$ changes, requiring $f''$ to change sign — not merely $f''=0$ .
Critical point.: An $x$ -value in the domain where $f'(x)=0$ or $f'(x)$ does not exist; candidates for local extrema.
Justification.: A written argument connecting a calculus fact to specific values and a named theorem, required for full credit on analysis FRQs.
Candidates test.: For a closed interval, comparing $f$ -values at critical points and endpoints to locate the absolute extrema.
Implicit differentiation.: Differentiating an equation in $x$ and $y$ with respect to $x$ , treating $y$ as a function of $x$ and applying the chain rule to $y$ terms.

Worked example

Let

f

be a function with

f'(x) = (x-1)(x-4)^2

. The domain of

f

is all real numbers. (a) Find all

x

-values where

f

has a local extremum and classify each. (b) Find all

x

-values where

f

has an inflection point. (c) On what intervals is

f

concave up?

Part (a): Set

f'(x)=(x-1)(x-4)^2=0

, giving critical numbers

x=1

and

x=4

.

Check sign of

f'

around each. For

x<1

, say

x=0

(0-1)(0-4)^2=(-1)(16)<0

. For

1<x<4

, say

x=2

(2-1)(2-4)^2=(1)(4)>0

. So

f'

changes

-\to+

x=1

: local minimum at

x=1

by the First Derivative Test.

At

x=4

: for

1<x<4

we found

f'>0

, and for

x>4

, say

x=5

(5-1)(5-4)^2=(4)(1)>0

. No sign change, so

x=4

is not an extremum.

Part (b): Differentiate.

f'(x)=(x-1)(x-4)^2

. Using the product rule,

f''(x)=(x-4)^2+(x-1)\cdot 2(x-4)=(x-4)[(x-4)+2(x-1)]=(x-4)(3x-6)=3(x-4)(x-2)

.

Set

f''=0

x=2

and

x=4

. Test sign changes. For

x<2

3(-)(-)>0

. For

2<x<4

3(-)(+)<0

. For

x>4

3(+)(+)>0

. Sign changes at both

x=2

and

x=4

, so inflection points at

x=2

and

x=4

.

Part (c):

f''>0

(-\infty,2)

and

(4,\infty)

, so

f

is concave up on those intervals. Note

x=4

was not an extremum but IS an inflection point — a common surprise worth confirming with the sign chart.

Practice questions

The graph of

f'

, the derivative of

f

, passes through the origin, has a local maximum at

x=-1

, crosses the

x

-axis at

x=2

going from positive to negative, and has a local minimum at

x=3

. At which

x

-value does

f

have a local maximum?

$x=-1$
$x=2$
$x=3$
$x=0$

Answer: $x=2$

A local maximum of

f

occurs where

f'

changes from positive to negative. The prompt says

f'

crosses the axis at

x=2

going positive to negative, so

f

has a local max there. The points

x=-1

and

x=3

are extrema of the

f'

graph, which correspond to inflection points of

f

, not extrema. This is the most common translation error on graph-of-

f'

questions.

A curve is defined implicitly by

x^2 + xy + y^2 = 7

. Find

\frac{dy}{dx}

, and determine the

x

-coordinate(s) where the tangent line to the curve is horizontal. Justify.

Answer: $\frac{dy}{dx}=-\frac{2x+y}{x+2y}$ ; horizontal tangents occur where $y=-2x$ , giving $x=\pm\sqrt{7}/\sqrt{3}$ region points found by substitution.

Differentiate implicitly:

2x + y + x\frac{dy}{dx} + 2y\frac{dy}{dx}=0

, so

\frac{dy}{dx}(x+2y)=-(2x+y)

and

\frac{dy}{dx}=-\frac{2x+y}{x+2y}

. A horizontal tangent needs the numerator zero and denominator nonzero:

2x+y=0

, i.e.

y=-2x

. Substitute into the original equation:

x^2 + x(-2x)+(-2x)^2 = x^2 -2x^2+4x^2=3x^2=7

, so

x=\pm\sqrt{7/3}

. Both give nonzero denominators, confirming genuine horizontal tangents. The key exam skill is stating the numerator-zero condition explicitly.

Explain why finding

f''(c)=0

alone does not prove that

f

has an inflection point at

x=c

. Give a supporting example.

Answer: Because an inflection point requires $f''$ to change sign at $c$ , and $f''(c)=0$ can occur without a sign change.

Concavity changes only when

f''

actually switches sign. Consider

f(x)=x^4

f''(x)=12x^2

, which equals

0

x=0

but is positive on both sides, so the graph is concave up throughout and has no inflection point at the origin. On an FRQ you must show the sign change (via a sign chart or test values) to claim an inflection point, or you lose the justification point.

FAQ

How many points is each Unit 5 free-response question worth?: Every AP Calculus free-response question is scored out of 9 points, distributed across its parts. Setup, correct computation, and justification each carry separate points, which is why writing your reasoning matters even when your final number is wrong.
When do I use the First Derivative Test versus the Second Derivative Test?: Both classify extrema. Use the First Derivative Test when you have a sign chart of $f'$ or when $f''$ is hard to compute. Use the Second Derivative Test when $f''$ is easy and you already know $f'(c)=0$ . If the second derivative equals zero at the point, the Second Derivative Test is inconclusive and you must fall back on the First Derivative Test.
What is the difference between where f' equals zero and where f has an inflection point?: Where $f'=0$ you have a candidate for a local extremum of $f$ . Inflection points come from $f''$ changing sign, which corresponds to where the graph of $f'$ has a local max or min (its slope changes sign). These are different features, and confusing them is a frequent cause of lost points.
Do I need to show work if I can find the answer on a calculator?: Yes. On calculator-active FRQs you must present the mathematical setup — for example the equation you solved or the integral you evaluated — before the numerical answer. A bare answer with no setup typically earns no credit even if it is correct, and answers should generally be given to three decimal places when a decimal is required.

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