Free Response · The 'Inflow vs. Outflow' dynamic in real-world contexts

Rate In/Out & Accumulation AP Calculus AB FRQ: Explained with Practice & Scoring Tips

A substance (water, people, snow, gravel) enters and leaves a system at given rates. You are typically given $R_{in}(t)$ and $R_{out}(t)$ . Your task is to track the total amount, find the maximum/minimum amount, and interpret the meaning of integrals in context.

The Amount Function $A(t)$ (The Master Formula) $A(t) = A(0) + \int_{0}^{t} [R_{in}(x) - R_{out}(x)] \, dx$
When to useCrucial: Always add the initial amount $A(0)$ . The integral represents the 'net change' over time.
Net Rate of Change $A'(t)$ $A'(t) = R_{in}(t) - R_{out}(t)$
When to useThe derivative of the amount is simply the 'In-rate' minus the 'Out-rate'. Use this to find critical points.
Total Amount Entered/Left $\int_{a}^{b} R_{in}(t) \, dt \quad \text{or} \quad \int_{a}^{b} R_{out}(t) \, dt$
When to useThe definite integral of a rate gives the total accumulation of that specific flow over $[a, b]$ .
Average Rate of Change of Amount $\frac{A(b) - A(a)}{b - a}$
When to useUse when asked for the 'average rate' at which the amount is changing over a period.
Average Value of a Rate $\frac{1}{b - a} \int_{a}^{b} R_{in}(t) \, dt$
When to useUse to find the 'average rate' of entry or exit (Mean Value Theorem for Integrals).

Step-by-Step SOP

1
Define the Net Rate function
Identify $R_{in}(t)$ and $R_{out}(t)$ . Define $N(t) = R_{in}(t) - R_{out}(t)$ . This is the rate at which the total amount is changing.
2
Find the Absolute Extrema (Min/Max Amount)
Use the Candidates Test. (1) Set $A'(t) = 0$ to find critical points. (2) Evaluate $A(t)$ at the endpoints and at all critical points. (3) The largest value is the maximum; smallest is the minimum.
3
Determine if the amount is increasing or decreasing
Check the sign of $A'(t) = R_{in}(t) - R_{out}(t)$ . If $R_{in} > R_{out}$ , the amount is increasing. If $R_{out} > R_{in}$ , the amount is decreasing.
4
Interpret integrals in context
When asked 'What does $\int R(t) dt$ mean?', always include: (1) Total amount of [substance], (2) Units (e.g., gallons), (3) Time interval (e.g., from $t=0$ to $t=8$ ).

Practice Questions

1 2

Question 1

2025 AP Calculus AB Exam — FRQ 1

Hard

An invasive species of plant appears in a fruit grove at time

t=0

and begins to spread. The function

C

defined by

C(t) = 7.6 \arctan(0.2t)

models the number of acres in the fruit grove affected by the species

t

weeks after the species appears. It can be shown that

C'(t) = \frac{38}{25+t^2}

. (Note: Your calculator should be in radian mode.)

APart A

Easy

Find the average number of acres affected by the invasive species from time

t=0

to time

t=4

weeks. Show the setup for your calculations.

Need a Hint?

This asks for the 'Average Value of a Function'. Use the formula

\frac{1}{b-a} \int_a^b f(t)\,dt

. You are given

C(t)

, so integrate it over the interval

[0, 4]

Show Solution

Write the Average Value formula

\text{Average value} = \frac{1}{4-0} \int_{0}^{4} C(t)\,dt

Evaluate using a calculator

= \frac{1}{4} \int_{0}^{4} 7.6 \arctan(0.2t)\,dt \approx 2.778

State the final answer with units

From time

t=0

t=4

weeks, the average number of acres affected by the invasive species was

2.778

acres.

AP Scoring Note — P1 + P2

P1: Earned for the correct integral expression including the division by 4.

P2: Earned for the correct numerical answer accurate to three decimal places (

2.778

BPart B

Medium

Find the time

t

when the instantaneous rate of change of

C

equals the average rate of change of

C

over the time interval

0 \leq t \leq 4

. Show the setup for your calculations.

Need a Hint?

This is a Mean Value Theorem (MVT) application. 'Instantaneous rate' is

C'(t)

. 'Average rate' is the slope formula

\frac{C(4)-C(0)}{4-0}

. Set them equal and solve for

t

Show Solution

Calculate the average rate of change

\text{Avg rate} = \frac{C(4) - C(0)}{4 - 0} = \frac{5.1280 - 0}{4} = 1.282008

Set the instantaneous rate equal to the average rate

C'(t) = \frac{38}{25 + t^2} = 1.282008

Solve for t using a calculator

t \approx 2.154

weeks

AP Scoring Note — P3 + P4

P3: Earned for showing the average rate of change expression or value (

1.282

).

P4: Earned for the correct value of

t

supported by an equation setting

C'(t)

equal to the average rate.

CPart C

Medium

Assume that the invasive species continues to spread according to the given model for all times

t > 0

. Write a limit expression that describes the end behavior of the rate of change in the number of acres affected by the species. Evaluate this limit expression.

Need a Hint?

End behavior means as

t \to \infty

. The 'rate of change' is

C'(t)

. You need to find

\lim_{t \to \infty} C'(t)

Show Solution

Write the limit expression

\lim_{t \to \infty} C'(t) = \lim_{t \to \infty} \frac{38}{25 + t^2}

Evaluate the limit

t

approaches infinity, the denominator

25 + t^2

becomes infinitely large, so the fraction approaches 0.

\lim_{t \to \infty} C'(t) = 0

AP Scoring Note — P5 + P6

P5: Earned for the correct limit notation using

C'(t)

.

P6: Earned for the correct value 0. Note: Arithmetic with infinity (e.g.,

\frac{38}{\infty}

) is treated as scratch work and does not earn P6.

DPart D

Hard

At time

t = 4

weeks after the invasive species appears in the fruit grove, measures are taken to counter the spread of the species. The function

A

, defined by

A(t) = C(t) - \int_{4}^{t} 0.1 \ln(x)\,dx

, models the number of acres affected over

4 \leq t \leq 36

. At what time

t

does

A

attain its maximum value? Justify your answer.

Need a Hint?

To find the maximum on a closed interval, use the Candidates Test. (1) Find critical points where

A'(t) = 0

. (2) Test the endpoints (

t=4, 36

) and the critical point in the function

A(t)

Show Solution

Find the derivative A′(t)

By the Second Fundamental Theorem of Calculus:

A'(t) = C'(t) - 0.1 \ln t

Find the critical point

Set

A'(t) = 0

\frac{38}{25 + t^2} - 0.1 \ln t = 0 \implies t \approx 11.4417

Perform the Candidates Test

Evaluate

A(t)

at endpoints and critical points:
-

A(4) = C(4) - \int_{4}^{4} \ldots \approx 5.128

A(11.442) \approx 7.317

(Maximum)
-

A(36) \approx 1.743

State the final conclusion

The number of acres affected by the species is a maximum at time

t = 11.442

weeks.

AP Scoring Note — P7 + P8 + P9

P7: Earned for setting

A'(t) = 0

.

P8: Earned for the justification, which requires evaluating

A(t)

at all three candidates (Candidates Test) or using a global derivative sign argument.

P9: Earned for the correct time

t = 11.442

Question 2

2023 AP Calculus AB Exam — FRQ 1

Hard

A customer at a gas station is pumping gasoline into a gas tank. The rate of flow of gasoline is modeled by a differentiable function

f

, where

f(t)

is measured in gallons per second and

t

is measured in seconds since pumping began. Selected values of

f(t)

are given in the table shown.

Values of f(t)

t (seconds)	0	60	90	120	135	150
f(t) (gallons per second)	0	0.1	0.15	0.1	0.05	0

APart A

Easy

Using correct units, interpret the meaning of

\int_{60}^{135} f(t)\,dt

in the context of the problem. Use a right Riemann sum with the three subintervals

[60, 90]

[90, 120]

, and

[120, 135]

to approximate the value of

\int_{60}^{135} f(t)\,dt

. Show the work that leads to your answer.

Need a Hint?

For the interpretation: what does integrating a rate give you? State the quantity, units, and time interval. For the Riemann sum: a right sum uses the right endpoint of each subinterval. The subintervals have widths 30, 30, and 15 — they are NOT equal.

Show Solution

Interpret the integral

\int_{60}^{135} f(t)\,dt

represents the total number of gallons of gasoline pumped into the gas tank from time

t = 60

seconds to time

t = 135

seconds.

Set up the right Riemann sum

\int_{60}^{135} f(t)\,dt \approx f(90)\cdot(90-60) + f(120)\cdot(120-90) + f(135)\cdot(135-120)

= (0.15)(30) + (0.1)(30) + (0.05)(15)

Evaluate

= 4.5 + 3.0 + 0.75 = 8.25

gallons

AP Scoring Note — P1 + P2 + P3

**P1 (Interpretation)**: Must mention gallons of gasoline pumped AND the time interval

t = 60

t = 135

. Missing either loses this point.

**P2 (Form)**: At least 5 of the 6 factors in the Riemann sum must be correct. Writing only the products without factors (e.g.,

4.5 + 3.0 + 0.75

) earns P3 but NOT P2.

**P3 (Answer)**: Requires P2 to be earned. Any error in the Riemann sum forfeits P3.

BPart B

Medium

Must there exist a value of

c

, for

60 < c < 120

, such that

f'(c) = 0

? Justify your answer.

Need a Hint?

This is an MVT (or Rolle's Theorem) question — NOT IVT. You need

f'(c) = 0

, which is a derivative condition. Check whether

f(60) = f(120)

, then cite the appropriate theorem.

Show Solution

Establish continuity and differentiability

f

is differentiable on

(0, 150)

, which implies

f

is continuous on

[60, 120]

Check the endpoint values

f(60) = 0.1

and

f(120) = 0.1

, so

f(120) - f(60) = 0.1 - 0.1 = 0

.

The average rate of change of

f

[60, 120]

equals

\frac{f(120) - f(60)}{120 - 60} = \frac{0}{60} = 0

Apply the Mean Value Theorem

By the Mean Value Theorem (or Rolle's Theorem), there must exist at least one value

c

with

60 < c < 120

such that

f'(c) = 0

. **Yes**, such a value must exist.

AP Scoring Note — P1 + P2

**P1**: Must present

f(120) - f(60) = 0

, or equivalently state

f(60) = f(120)

, or show

0.1 - 0.1 = 0

as a numerator. This is required before any theorem can apply.

**P2**: Requires P1, PLUS stating

f

is continuous because

f

is differentiable, PLUS answering yes. You may cite MVT or Rolle's Theorem — but citing IVT here earns zero for P2.

CPart C

Medium

The rate of flow of gasoline can also be modeled by

g(t) = \frac{t}{500}\cos\!\left(\left(\frac{t}{120}\right)^2\right)

for

0 \leq t \leq 150

. Using this model, find the average rate of flow of gasoline over the time interval

0 \leq t \leq 150

. Show the setup for your calculations.

Need a Hint?

Use the average value formula:

f_{\text{avg}} = \frac{1}{b-a}\int_a^b g(t)\,dt

. You need a calculator to evaluate the integral — show the setup first before writing the numerical answer.

Show Solution

Write the average value formula

\text{Average rate} = \frac{1}{150 - 0}\int_0^{150} g(t)\,dt = \frac{1}{150}\int_0^{150} \frac{t}{500}\cos\!\left(\left(\frac{t}{120}\right)^2\right)dt

Evaluate with a calculator

= \frac{1}{150} \cdot 14.3995\ldots \approx 0.0960

gallons per second

AP Scoring Note — P1 + P2

**P1**: Earned by writing the average value formula

\frac{1}{150-0}\int_0^{150} g(t)\,dt

with the correct integrand. The formula may be shown in one or two steps.

**P2**: The correct answer

\approx 0.096

(or

0.095

). Writing only

\int_0^{150} g(t)\,dt = 0.096

without the

\frac{1}{150}

factor earns neither point.

DPart D

Medium

Using the model

g

defined in part (C), find the value of

g'(140)

. Interpret the meaning of your answer in the context of the problem.

Need a Hint?

Use your calculator to differentiate

g(t)

t = 140

. For the interpretation:

g'(140)

is the rate of change of a rate — state what is changing, at what rate, and at what time. Do NOT use the words 'acceleration' or 'velocity' (those are for motion problems).

Show Solution

Compute g′(140) with a calculator

g'(140) \approx -0.00491 \approx -0.005

gallons per second per second

Interpret in context

At time

t = 140

seconds, the rate at which gasoline is flowing into the tank is **decreasing** at a rate of approximately

0.005

gallon per second per second.

AP Scoring Note — P1 + P2

**P1**: Earned by the correct numerical value

g'(140) \approx -0.005

(or

-0.004

). The value may appear inside the interpretation sentence.

**P2**: The interpretation must include (1) the rate of flow is changing, (2) at the declared value of

g'(140)

, and (3) at

t = 140

seconds.

⚠ Writing 'decreasing at a rate of

-0.005

' loses P2 — the rate itself is

0.005

; 'decreasing' already conveys the direction. Also, using the words 'acceleration' or 'velocity' here loses P2 since this is not a motion problem.

Timed Practice

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Common Pitfalls

⚠
Forgetting the Initial ConditionThe most common error. If 100 gallons are already in the tank, $A(t)$ must start with 100. Don't just integrate the rates!
⚠
Confusing 'Rate' with 'Amount'Remember: $R(t)$ is a rate (e.g., tons/hr). To get an amount (tons), you must integrate. If the question asks 'Is the amount increasing?', look at the sign of $R_{in} - R_{out}$ , not the derivative of $R$ .
⚠
Incorrect Upper Limit on IntegralsIf you are finding the amount at $t=5$ , the integral must be $\int_{0}^{5}$ . Students often use the full interval (e.g., $\int_{0}^{12}$ ) by mistake.
⚠
Calculator Entry ErrorsSince these are often 'Calculator Active', store $R_{in}(t)$ as $Y_1$ and $R_{out}(t)$ as $Y_2$ in your calculator to avoid typing long decimals multiple times and making typos.
⚠
Misinterpreting 'Rate of the Rate'If asked 'Is the rate of entry increasing?', you need to check $R_{in}'(t)$ (the derivative of the rate), not just $R_{in}(t)$ .

On this page

Define the Net Rate function

Find the Absolute Extrema (Min/Max Amount)

Determine if the amount is increasing or decreasing

Interpret integrals in context

Practice Questions

Write the Average Value formula

Evaluate using a calculator

State the final answer with units

AP Scoring Note — P1 + P2

Calculate the average rate of change

Set the instantaneous rate equal to the average rate

Solve for t using a calculator

AP Scoring Note — P3 + P4

Write the limit expression

Evaluate the limit

AP Scoring Note — P5 + P6

Find the derivative A′(t)

Find the critical point

Perform the Candidates Test

State the final conclusion

AP Scoring Note — P7 + P8 + P9

Interpret the integral

Set up the right Riemann sum

Evaluate

AP Scoring Note — P1 + P2 + P3

Establish continuity and differentiability

Check the endpoint values

Apply the Mean Value Theorem

AP Scoring Note — P1 + P2

Write the average value formula

Evaluate with a calculator

AP Scoring Note — P1 + P2

Compute g′(140) with a calculator

Interpret in context

AP Scoring Note — P1 + P2

Ready to challenge yourself?