# Interpreting behavior of accumulation functions

> AP Calculus BC · Unit 6: Integration and Accumulation of Change
> Source: https://www.owlsprep.com/study/ap-calculus-bc-u6-interpreting-behavior-of-accumulation-functions/

This subtopic uses the First Fundamental Theorem of Calculus to analyze the behavior of accumulation functions, including identifying intervals of increase/decrease, extrema, concavity, and inflection points directly from the integrand, no explicit integration required.

**Prerequisites:** Computing derivatives of basic functions; Identifying increasing/decreasing intervals and concavity from a derivative; [Basic definite integral notation](https://www.owlsprep.com/study/ap-calculus-bc-u5-definite-integrals/)

## Learning objectives

- Differentiate accumulation functions with constant and variable limits using the First Fundamental Theorem of Calculus
- Identify intervals of increase/decrease and extrema of accumulation functions directly from the integrand
- Determine concavity and inflection points of accumulation functions directly from the integrand
- Write correct justifications of accumulation function behavior for AP free-response questions

## Introduction to Accumulation Functions

An accumulation function is any function defined by a definite integral with a variable upper limit, outputting a net area that depends on the input value of the upper limit. Unlike standard definite integrals that produce numerical values, accumulation functions are dynamic functions that require full behavior analysis, just like any other function.

**Accumulation Function** — A function that gives the net signed area under the curve of an integrand $f(t)$ from a constant lower limit $a$ to a variable upper limit $x$, where the output is a function of $x$ rather than a constant.

*Notation:* A(x) = \int_{a}^{x} f(t) dt

*Example:* $A(x) = \int_{0}^{x} \sin t dt$

This topic focuses on using the inherent relationship between the accumulation function $A(x)$ and its integrand $f(t)$ to analyze $A(x)$’s monotonicity, extrema, concavity, and inflection points, often without ever computing the explicit antiderivative of $f(t)$. Per the AP Calculus CED, this subtopic accounts for roughly 2-4% of total exam points, appearing in both multiple-choice and free-response questions.

## Differentiating Accumulation Functions

The entire topic of interpreting accumulation function behavior relies on the First Fundamental Theorem of Calculus (FFTC), which links the derivative of an accumulation function directly to its integrand. For a basic accumulation function with a constant lower limit and variable upper limit, the theorem states:

$$\frac{d}{dx} \left( \int_{a}^{x} f(t) dt \right) = f(x)$$

Intuitively, this means the rate of change of the accumulated area under $f(t)$ from $a$ to $x$ is exactly equal to the height of $f$ at $x$. When the upper limit is a differentiable function $u(x)$, we must apply the chain rule to get the extended derivative rule:

$$\frac{d}{dx} \left( \int_{a}^{u(x)} f(t) dt \right) = f(u(x)) \cdot u'(x)$$

If both the upper and lower limits are variable, we split the integral around a constant $a$, using the property $\int_{v(x)}^{u(x)} f(t) dt = -\int_{a}^{v(x)} f(t) dt + \int_{a}^{u(x)} f(t) dt$. The full derivative becomes:

$$\frac{d}{dx} \left( \int_{v(x)}^{u(x)} f(t) dt \right) = f(u(x)) u'(x) - f(v(x)) v'(x)$$

**Worked example:** Find $\frac{d}{dx} \left( \int_{\cos x}^{x^2} e^{t^2} dt \right)$

1. This is an accumulation function with two variable limits, so we use the full extended FFTC rule.
2. Identify components: $f(t) = e^{t^2}$, $u(x) = x^2$ (upper limit), $v(x) = \cos x$ (lower limit). Compute derivatives of the limits:
3. $$u'(x) = 2x, \quad v'(x) = -\sin x$$
4. Substitute the upper limit term into the rule:
5. $$f(u(x)) u'(x) = e^{(x^2)^2} \cdot 2x = 2x e^{x^4}$$
6. Substitute the lower limit term, remembering the negative sign from the rule:
7. $$-f(v(x)) v'(x) = - e^{(\cos x)^2} \cdot (-\sin x) = e^{\cos^2 x} \sin x$$
8. Combine terms to get the final result:
9. $$\frac{d}{dx} \left( \int_{\cos x}^{x^2} e^{t^2} dt \right) = 2x e^{x^4} + (\sin x) e^{\cos^2 x}$$

> **tip**
>
> Always remember the negative sign for the variable lower limit term. Double-check this step before moving on, as missed negative signs are the most common error on this type of problem.

> **Exam tip:** Always check for variable limits and apply the chain rule when needed

## Monotonicity and Extrema of Accumulation Functions

Once we know the derivative of an accumulation function $A(x) = \int_{a}^{x} f(t) dt$ is $A'(x) = f(x)$, we can apply all standard derivative behavior rules to analyze $A(x)$ directly from $f(x)$, no integration required:

- $A(x)$ is increasing on an interval if and only if $f(x) > 0$ on that interval
- $A(x)$ is decreasing on an interval if and only if $f(x) < 0$ on that interval
- Critical points of $A(x)$ occur where $f(x) = 0$ or $f(x)$ is undefined
- Classify critical points using the First Derivative Test: negative to positive = local minimum; positive to negative = local maximum

**Worked example:** Let $A(x) = \int_{0}^{x} f(t) dt$, where $f(t)$ is continuous on $(0,5)$, negative on $(0,1)$, positive on $(1,3)$, and negative on $(3,5)$, crossing the $t$-axis only at $t=1$ and $t=3$. Identify all local minima and maxima of $A(x)$ on $(0,5)$

1. By FFTC, $A'(x) = f(x)$, so critical points of $A$ are at $x=1$ and $x=3$, where $f(x)=0$.
2. At $x=1$: $f(x)$ changes from negative (left of 1) to positive (right of 1). By the First Derivative Test, this means $A(x)$ has a local minimum at $x=1$.
3. At $x=3$: $f(x)$ changes from positive (left of 3) to negative (right of 3). By the First Derivative Test, this means $A(x)$ has a local maximum at $x=3$.
4. Final answer: Local minimum at $x=1$, local maximum at $x=3$.

> **tip**
>
> Extrema and critical points for $A(x)$ are in terms of $x$ (the input to $A$), not $t$ (the dummy variable in the integrand). Don’t use the wrong variable in your answer on the FRQ section.

## Concavity and Inflection Points of Accumulation Functions

To find concavity and inflection points of $A(x)$, we need the second derivative of $A(x)$. Since $A'(x) = f(x)$, the second derivative is $A''(x) = f'(x)$, meaning the concavity of $A(x)$ depends entirely on the derivative of the integrand $f$, not the value of $f$ itself:

- $A(x)$ is concave up on an interval if and only if $f(x)$ is increasing on that interval ($f'(x) > 0$)
- $A(x)$ is concave down on an interval if and only if $f(x)$ is decreasing on that interval ($f'(x) < 0$)
- An inflection point of $A(x)$ occurs where $f(x)$ changes from increasing to decreasing (or vice versa), which is exactly at a local extremum of $f(x)$

**Worked example:** Let $A(x) = \int_{2}^{x} f(t) dt$. $f(t)$ has a local maximum at $t=4$ and a local minimum at $t=7$, with no other critical points on $[2, 10]$. Does $A(x)$ have inflection points at $x=4$ and $x=7$? Justify your answer.

1. For $A(x)$, $A''(x) = f'(x)$, so inflection points occur where $f'(x)$ changes sign.
2. At $x=4$: $f$ has a local maximum, so by definition $f'(x)$ changes from positive (left of 4) to negative (right of 4). This means $A''(x)$ changes sign, so concavity of $A$ changes at $x=4$.
3. At $x=7$: $f$ has a local minimum, so by definition $f'(x)$ changes from negative (left of 7) to positive (right of 7). This means $A''(x)$ also changes sign at $x=7$.
4. Final answer: Yes, $A(x)$ has inflection points at both $x=4$ and $x=7$.

> **tip**
>
> Don’t confuse inflection points of $A(x)$ with zeros of $f(x)$. Zeros of $f$ give extrema of $A$; extrema of $f$ give inflection points of $A$. This is the most commonly tested distinction on multiple-choice questions.

## AP-Style Worked Practice

**Worked example:** Let $A(x) = \int_{-2}^{x^3} f(t) dt$, where $f(t)$ is continuous for all real $t$, and $f(8) = 3$. What is $A'(2)$?  
(A) $3$  
(B) $12$  
(C) $36$  
(D) $72$

1. We use the extended First Fundamental Theorem of Calculus for variable upper limits, which requires the chain rule:
2. $$A'(x) = f(x^3) \cdot \frac{d}{dx}(x^3) = 3x^2 f(x^3)$$
3. Substitute $x=2$:
4. $$A'(2) = 3(2^2) f(2^3) = 12 f(8) = 12(3) = 36$$
5. The correct answer is (C).

**Worked example:** Let $f(t)$ be continuous on $[0,6]$, with the following properties:  
- $f(t) < 0$ on $(0,1)$, $f(t) > 0$ on $(1,4)$, $f(t) < 0$ on $(4,6)$  
- $f'(t) > 0$ on $(0,2)$, $f'(t) < 0$ on $(2,5)$, $f'(t) > 0$ on $(5,6)$  

Let $A(x) = \int_{0}^{x} f(t) dt$ for $0 \leq x \leq 6$.  
(a) Identify all $x$ on $(0,6)$ where $A(x)$ has a local minimum. Justify your answer.  
(b) Identify all $x$ on $(0,6)$ where $A(x)$ has an inflection point. Justify your answer.  
(c) On what intervals is $A(x)$ both decreasing and concave up?

1. (a) $A'(x) = f(x)$, so critical points of $A$ are at $x=1$ and $x=4$ where $f(x)=0$. At $x=1$, $f(x)$ changes from negative to positive, so by the First Derivative Test, $A(x)$ has a local minimum at $x=1$. At $x=4$, $f(x)$ changes from positive to negative, so $A$ has a local maximum there. Final answer: $x=1$.
2. (b) $A''(x) = f'(x)$, so inflection points occur where $f'(x)$ changes sign. $f'(x)$ changes from positive to negative at $x=2$, and from negative to positive at $x=5$, so $A''(x)$ changes sign at both points. Final answer: inflection points at $x=2$ and $x=5$.
3. (c) $A(x)$ is decreasing when $f(x) < 0$, on intervals $(0,1)$ and $(4,6)$. $A(x)$ is concave up when $f'(x) > 0$, on intervals $(0,2)$ and $(5,6)$. The intersection of these sets is $(0,1)$ and $(5,6)$. Final answer: $(0,1) \cup (5,6)$.

**Worked example:** The rate of change of the number of fish in a lake $t$ months after a conservation project begins is given by $r(t)$ fish per month, where $r(t) > 0$ when the number of fish is increasing, $r(t) < 0$ when decreasing. Let $N(h) = \int_{0}^{h} r(t) dt$ be the net change in the number of fish after $h$ months. At $h=6$ months, $r(t)$ changes from increasing to decreasing. Is the rate of change of the fish population increasing or decreasing at 6 months? Justify your answer, and interpret the result in context.

1. $N(h)$ is the net change in fish population, so the rate of change of the population is $N'(h) = r(h)$. We need to find if the rate of change itself is increasing or decreasing, which depends on the derivative of the rate, $N''(h) = r'(h)$.
2. At $h=6$, $r(t)$ changes from increasing to decreasing, so $r'(h)$ changes from positive to negative, meaning $r'(6) < 0$.
3. Therefore, the rate of change of the fish population is decreasing at 6 months. Interpretation: At 6 months, the fish population may still be growing (if $r(6) > 0$), but the rate at which new fish are added to the lake is slowing down.

## Common pitfalls

- **Wrong:** For $A(x) = \int_{a}^{x} f(t) dt$, find an inflection point of $A$ where $f(x) = 0$
  - Why it fails: Students confuse first and second derivative relationships, mixing up conditions for extrema vs inflection points
  - Correct: Find inflection points of $A$ where $f(x)$ changes from increasing to decreasing (i.e., where $f$ has an extremum), not where $f(x) = 0$
- **Wrong:** When computing $\frac{d}{dx} \left( \int_{x}^{7} e^{\sin t} dt \right)$, write the result as $e^{\sin 7} - e^{\sin x}$
  - Why it fails: Students mistakenly treat the constant upper limit as a variable term
  - Correct: Recognize that $\int_{x}^{7} e^{\sin t} dt = C - \int_{7}^{x} e^{\sin t} dt$, so the derivative is just $-e^{\sin x}$
- **Wrong:** When differentiating $\int_{a}^{2x} f(t) dt$, write the derivative as $f(2x)$ and forget the chain rule term
  - Why it fails: Students remember the basic FFTC but overlook the chain rule when the upper limit is a simple linear function
  - Correct: Always check if either limit is a non-constant function of $x$, and multiply by the derivative of the variable limit every time
- **Wrong:** When asked for the absolute maximum *value* of $A(x)$ on $[0,4]$, give the $x$-coordinate of the maximum
  - Why it fails: Students get used to identifying locations of extrema and miss that the question asks for the function value
  - Correct: If asked for the value of the extremum, compute the net area from $a$ to the $x$-coordinate of the extremum to get $A(x)$
- **Wrong:** Claim $A(x)$ is concave up when $f(x) > 0$
  - Why it fails: Students mix up conditions for increasing vs concave up, since both rely on a positive derivative
  - Correct: Remember $A$ is concave up when $f(x)$ is increasing, regardless of whether $f(x)$ itself is positive or negative

## Cheatsheet

| Category | Rule/Formula | Notes |
| --- | --- | --- |
| Basic accumulation derivative | $\frac{d}{dx} \int_a^x f(t) dt = f(x)$ | Applies when $f$ is continuous, $a$ is constant |
| Variable upper limit derivative | $\frac{d}{dx} \int_a^{u(x)} f(t) dt = f(u(x)) u'(x)$ | Requires chain rule for upper limit $u(x)$ |
| Two variable limits derivative | $\frac{d}{dx} \int_{v(x)}^{u(x)} f(t) dt = f(u(x)) u'(x) - f(v(x)) v'(x)$ | Negative sign comes from swapping variable lower limit |
| Monotonicity of $A(x)$ | $A \uparrow$ when $f(x) > 0$; $A \downarrow$ when $f(x) < 0$ | Uses $A'(x) = f(x)$ |
| Extrema of $A(x)$ | Local extrema at $x$ where $f(x)$ changes sign | Positive $\to$ negative = local max; negative $\to$ positive = local min |
| Concavity of $A(x)$ | $A$ concave up when $f(x)$ increasing; concave down when $f(x)$ decreasing | Uses $A''(x) = f'(x)$, depends on derivative of $f$ |
| Inflection points of $A(x)$ | Inflection at $x$ where $f(x)$ changes increasing/decreasing | Occurs at local extrema of $f(x)$, not zeros of $f(x)$ |

## What's next

This topic is the foundation for all further applications of integration that connect a rate function to an accumulated total function, which you will use extensively in the remaining units of AP Calculus BC. Mastery of the relationships between the accumulation function, its first derivative, and its second derivative is critical for writing complete, correct justifications for AP free-response questions, which make up half your total exam score. Immediately after this, you will apply accumulation function behavior to analyze motion problems and solve separable differential equations, where solutions are often interpreted as accumulation functions. This topic also feeds into larger core concepts including area between curves and volumes of revolution, all tested heavily on the BC exam.

- [Integration by Substitution](https://www.owlsprep.com/study/ap-calculus-bc-u6-integration-by-substitution/)
- [Properties of definite integrals](https://www.owlsprep.com/study/ap-calculus-bc-u6-properties-of-definite-integrals/)
- [FTC and Definite Integrals](https://www.owlsprep.com/study/ap-calculus-bc-u6-ftc-and-definite-integrals/)

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