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

Worked Example

Let $F(x) = \int_0^x (2t - 4)\,dt$. Find $F(3)$ and $F'(3)$.

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

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

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

1. $F(x) = \int_a^x f(t)\,dt$ accumulates starting from $a$
2. $F(a) = 0$ always
3. $F' = f$ connects the accumulation function to the original function

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

• 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

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

Accumulation Functions

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

Chain Rule Variant

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

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

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

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:

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

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

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

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

Accumulation Functions

Part 5 of 7 — Average Value of a Function

Average Value Formula

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

Mean Value Theorem for Integrals

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

Worked Example

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

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

Accumulation Functions

Part 6 of 7 — Practice Workshop

Mixed Accumulation Problems 🎯

Workshop Complete!

Accumulation Functions — Review

Part 7 of 7 — Comprehensive Assessment

Final Assessment 🎯

Accumulation Functions — Complete! ✅