🎯⭐ INTERACTIVE LESSON

Accumulation Functions

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Accumulation Functions - Complete Interactive Lesson

Part 1: The Accumulation Concept

Accumulation Functions

Part 1 of 7 — The Accumulation Concept

What is an Accumulation Function?

$F(x) = \int_a^x f(t)\,dt$

$F(x)$ measures how much has accumulated from $a$ to $x$ .

Key Properties

Property	Explanation
$F(a) = 0$	Nothing has accumulated at the starting point
$F'(x) = f(x)$

Worked Example

Let $F(x) = \int_0^x (2t - 4)\,dt$ . Find $F(3)$ and .

$F(3) = \int_0^3 (2t-4)\,dt = [t^2 - 4t]_0^3 = 9 - 12 = -3$

$F'(3) = f(3) = 2(3) - 4 = 2$

So at $x = 3$ , only $-3$ has accumulated so far, but the rate is $+2$ (accumulating positively).

Accumulation Functions 🎯

Let $g(x) = \int_1^x f(t)\,dt$ where $f$ is continuous.

Key Takeaways — Part 1

$F(x) = \int_a^x f(t)\,dt$ accumulates starting from $a$

Part 2: Reading Graphs of f to Analyze F

Accumulation Functions

Part 2 of 7 — Reading Graphs of $f$ to Analyze $F$

Graph-Based Analysis

Given the graph of $f$ , you can determine everything about $F(x) = \int_a^x f(t)\,dt$ :

Part 3: FTC Part 1 with Chain Rule Review

Accumulation Functions

Part 3 of 7 — FTC Part 1 with Chain Rule Review

Chain Rule Variant

$\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x)) \cdot g'(x)$

Part 4: Net Change Applications

Accumulation Functions

Part 4 of 7 — Net Change Applications

Rate In / Rate Out Problems

If $R_{in}(t)$ = rate in and $R_{out}(t)$ = rate out:

Part 5: Average Value of a Function

Accumulation Functions

Part 5 of 7 — Average Value of a Function

Average Value Formula

$f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx$

Part 6: Practice Workshop

Accumulation Functions

Part 6 of 7 — Practice Workshop

Graph → Accumulation Function Analysis

This is one of the most heavily tested skills on the AP exam. Given the graph of $f$ , you must be able to determine:

To find...	Use...
$g(x) = \int_0^x f(t)\,dt$

Part 7: Comprehensive Assessment

Accumulation Functions — Review

Part 7 of 7 — Comprehensive Assessment

This final assessment tests your ability to analyze accumulation functions from graphs of $f$ , compute values of $g$ , and determine properties of $g$ , $g'$ , and .

F' = f

connects the accumulation function to the original function

Values of $F$ : computed as signed areas under $f$
Where $F$ increases/decreases: where $f$ is positive/negative
Max/min of $F$ : where $f$ changes sign
Concavity of $F$ : $F'' = f'$ , so look at whether $f$ is increasing/decreasing

AP Tip: This is one of the most commonly tested skills on the AP exam!

Graph Analysis 🎯

Suppose $f$ is a piecewise linear function on $[0, 6]$ : $f(0) = 2$ , $f(2) = 2$ , $f(4) = -2$ , $f(6) = 0$ . Let $g(x) = \int_0^x f(t)\,dt$ .

Key Takeaways — Part 2

Read the graph of $f$ to determine the behavior of $g = \int f$
Signed area under $f$ gives the value of $g$
This skill is tested on nearly every AP exam

Both Limits Variable

$\frac{d}{dx}\int_{h(x)}^{g(x)} f(t)\,dt = f(g(x)) \cdot g'(x) - f(h(x)) \cdot h'(x)$

FTC with Chain Rule 🎯

Key Takeaways — Part 3

Both limits variable: subtract the lower limit contribution
Each limit contributes: $f(\text{limit}) \cdot \text{limit derivative}$

$\text{Net change} = \int_a^b [R_{in}(t) - R_{out}(t)]\,dt$

$\text{Amount at time } b = \text{Initial amount} + \int_a^b [R_{in} - R_{out}]\,dt$

These problems appear on nearly every AP exam!

Rate In/Rate Out 🎯

Water flows into a tank at $R_{in}(t) = 10 + 2t$ gallons/hr and leaks out at $R_{out}(t) = 5$ gallons/hr. Initially the tank has 100 gallons.

Key Takeaways — Part 4

Net change = $\int (\text{rate in} - \text{rate out})\,dt$
Current amount = initial + net change
This is one of the most common AP FRQ formats

Mean Value Theorem for Integrals

There exists $c \in [a,b]$ such that $f(c) = f_{\text{avg}}$ .

Find the average value of $f(x) = x^2$ on $[0, 3]$ .

$f_{\text{avg}} = \frac{1}{3-0}\int_0^3 x^2\,dx = \frac{1}{3} \cdot \frac{27}{3} = 3$

Average Value 🎯

Key Takeaways — Part 5

Average value = $\frac{1}{b-a}\int_a^b f(x)\,dx$
MVT for integrals guarantees $f(c) = f_{\text{avg}}$ for some $c$

Strategy: Always write down $g' = f$ and $g'' = f'$ before answering any question.

Piecewise Linear Graph Analysis 🎯

Concavity & Inflection Points 🎯

Computing g from Areas 🎯

Local Extrema from the Graph of f 🎯

Workshop Complete! 🎯

Key Connections:

$g' = f$ → use the values of $f$ to find where $g$ increases/decreases
$g'' = f'$ → use the slope of $f$ to find concavity of $g$
Local extrema of $g$ → where $f$ changes sign
Inflection points of $g$ → where $f$ changes from increasing to decreasing (or vice versa)

AP-Style Graph Analysis 🎯

The graph of $f$ is piecewise linear with vertices at $(0, -1)$ , $(2, 3)$ , $(4, 3)$ , $(6, -1)$ . Let $g(x) = \int_0^x f(t)\,dt$ .

Inequality & Ordering Questions 🎯

Mixed Graph & Accumulation 🎯

Accumulation Functions — Complete! ✅

You have mastered the key skills:

Computing $g(x) = \int_0^x f(t)\,dt$ as signed area
Finding extrema using $g' = f$ (sign changes of $f$ )
Analyzing concavity using $g'' = f'$ (increasing/decreasing behavior of $f$ )
Ordering values of $g$ , $g'$ , and $g''$