The graph of f is shown and g(x) = integral of_0^x f(t) dt. Find g(-2), g'(3) and g''(6). — the graph of f on [-2, 7], made of straight pieces

g(-2) = 6; g'(3) = 0; g''(6) = 0

The graph of f is shown and g(x) = integral of_3^x f(t) dt. Find g(5), g'(2) and g''(4). — the graph of f on [-1, 6], made of straight pieces

g(5) = 3; g'(2) = 0; g''(4) = -1/2

The graph of f is shown and g(x) = integral of_3^x f(t) dt. Find g(-3), g'(3) and g''(-1). — the graph of f on [-4, 6], made of straight pieces

g(-3) = 21/2; g'(3) = -3; g''(-1) = -2/3

The graph of f is shown and g(x) = integral of_-1^x f(t) dt. Find g(6), g'(8) and g''(5). — the graph of f on [-1, 8], made of straight pieces

g(6) = -3/2; g'(8) = -1; g''(5) = 3/2

The graph of f is shown and g(x) = integral of_-3^x f(t) dt. Find g(1), g'(-3) and g''(2). — the graph of f on [-3, 4], made of straight pieces

g(1) = -7; g'(-3) = -1; g''(2) = 1/2

The graph of f is shown and g(x) = integral of_-2^x f(t) dt. Find g(-1), g'(-3) and g''(2). — the graph of f on [-3, 3], made of straight pieces

g(-1) = -1; g'(-3) = -2; g''(2) = 1/3

The graph of f is shown and g(x) = integral of_0^x f(t) dt. Find g(8), g'(-2) and g''(-1). — the graph of f on [-2, 8], made of straight pieces

g(8) = 1; g'(-2) = 3; g''(-1) = -3/2

Worksheets · Calculus

Accumulation functions worksheet

g(x) = ∫ from a to x of f(t) dt is the signed area under f so far. Its derivative is f itself, so where f is positive g climbs, and where f changes sign g turns.

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Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Accumulation Functions Date Period Answer each question about the accumulation function. $g (x) = \int_{1}^{x} (- 6 t^{2} + 8 t + 2) d t$ . Find $g^{''} (4)$ The graph of $f$ is shown and $g (x) = \int_{0}^{x} f (t) d t$ . Find $g (- 2)$ , $g^{'} (3)$ and $g^{''} (6)$ . The graph of $f$ is shown and $g (x) = \int_{- 1}^{x} f (t) d t$ . Find $g^{''} (0)$ $g (x) = \int_{2}^{x} (- 3 t^{2} + 2 t + 1) d t$ . Find $g (4)$ The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . Find $g (5)$ , $g^{'} (2)$ and $g^{''} (4)$ . The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . Find $g (5)$ $h (x) = 4 + \int_{1}^{x} (3 t^{2} + 2 t - 5) d t$ . Find $h (4)$ The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . Find $g (- 3)$ , $g^{'} (3)$ and $g^{''} (- 1)$ . The graph of $f$ is shown and $g (x) = \int_{1}^{x} f (t) d t$ . Find $g^{''} (- 1)$ $h (x) = - 3 + \int_{2}^{x} (3 t^{2} - 6 t + 2) d t$ . Find $h (4)$ The graph of $f$ is shown and $g (x) = \int_{- 1}^{x} f (t) d t$ . Find $g (6)$ , $g^{'} (8)$ and $g^{''} (5)$ . The graph of $f$ is shown and $g (x) = \int_{- 2}^{x} f (t) d t$ . On what open intervals is $g$ increasing? $g (x) = \int_{1}^{x} (9 t^{2} - 8 t - 6) d t$ . Find $g^{''} (2)$ The graph of $f$ is shown and $g (x) = \int_{- 3}^{x} f (t) d t$ . Find $g (1)$ , $g^{'} (- 3)$ and $g^{''} (2)$ . The graph of $f$ is shown and $g (x) = \int_{2}^{x} f (t) d t$ . On what open intervals is $g$ increasing? $g (x) = \int_{1}^{x} (3 t^{2} - 8 t + 6) d t$ . Find $g (3)$ The graph of $f$ is shown and $g (x) = \int_{- 2}^{x} f (t) d t$ . Find $g (- 1)$ , $g^{'} (- 3)$ and $g^{''} (2)$ . The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . Find $g^{''} (1)$ $g (x) = \int_{2}^{x} (- 3 t^{2} - 6 t + 5) d t$ . Find $g (4)$ The graph of $f$ is shown and $g (x) = \int_{0}^{x} f (t) d t$ . Find $g (8)$ , $g^{'} (- 2)$ and $g^{''} (- 1)$ . Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Accumulation Functions

Date Period

Answer each question about the accumulation function.

$g (x) = \int_{1}^{x} (- 6 t^{2} + 8 t + 2) d t$ . Find $g^{''} (4)$
The graph of $f$ is shown and $g (x) = \int_{0}^{x} f (t) d t$ . Find $g (- 2)$ , $g^{'} (3)$ and $g^{''} (6)$ .
The graph of $f$ is shown and $g (x) = \int_{- 1}^{x} f (t) d t$ . Find $g^{''} (0)$
$g (x) = \int_{2}^{x} (- 3 t^{2} + 2 t + 1) d t$ . Find $g (4)$
The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . Find $g (5)$ , $g^{'} (2)$ and $g^{''} (4)$ .
The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . Find $g (5)$
$h (x) = 4 + \int_{1}^{x} (3 t^{2} + 2 t - 5) d t$ . Find $h (4)$
The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . Find $g (- 3)$ , $g^{'} (3)$ and $g^{''} (- 1)$ .
The graph of $f$ is shown and $g (x) = \int_{1}^{x} f (t) d t$ . Find $g^{''} (- 1)$
$h (x) = - 3 + \int_{2}^{x} (3 t^{2} - 6 t + 2) d t$ . Find $h (4)$
The graph of $f$ is shown and $g (x) = \int_{- 1}^{x} f (t) d t$ . Find $g (6)$ , $g^{'} (8)$ and $g^{''} (5)$ .
The graph of $f$ is shown and $g (x) = \int_{- 2}^{x} f (t) d t$ . On what open intervals is $g$ increasing?
$g (x) = \int_{1}^{x} (9 t^{2} - 8 t - 6) d t$ . Find $g^{''} (2)$
The graph of $f$ is shown and $g (x) = \int_{- 3}^{x} f (t) d t$ . Find $g (1)$ , $g^{'} (- 3)$ and $g^{''} (2)$ .
The graph of $f$ is shown and $g (x) = \int_{2}^{x} f (t) d t$ . On what open intervals is $g$ increasing?
$g (x) = \int_{1}^{x} (3 t^{2} - 8 t + 6) d t$ . Find $g (3)$
The graph of $f$ is shown and $g (x) = \int_{- 2}^{x} f (t) d t$ . Find $g (- 1)$ , $g^{'} (- 3)$ and $g^{''} (2)$ .
The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . Find $g^{''} (1)$
$g (x) = \int_{2}^{x} (- 3 t^{2} - 6 t + 5) d t$ . Find $g (4)$
The graph of $f$ is shown and $g (x) = \int_{0}^{x} f (t) d t$ . Find $g (8)$ , $g^{'} (- 2)$ and $g^{''} (- 1)$ .



Sigma Prep · sigmaprep.io/worksheets Name Accumulation FunctionsAnswer keyVersion 1 Date Period Answer each question about the accumulation function. $g (x) = \int_{1}^{x} (- 6 t^{2} + 8 t + 2) d t$ . Find $g^{''} (4)$ $- 40$ The graph of $f$ is shown and $g (x) = \int_{0}^{x} f (t) d t$ . Find $g (- 2)$ , $g^{'} (3)$ and $g^{''} (6)$ . $\begin{array}{l} g (- 2) = 6 \\ g^{'} (3) = 0 \\ g^{''} (6) = 0 \end{array}$ The graph of $f$ is shown and $g (x) = \int_{- 1}^{x} f (t) d t$ . Find $g^{''} (0)$ $- \frac{1}{2}$ $g (x) = \int_{2}^{x} (- 3 t^{2} + 2 t + 1) d t$ . Find $g (4)$ $- 42$ The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . Find $g (5)$ , $g^{'} (2)$ and $g^{''} (4)$ . $\begin{array}{l} g (5) = 3 \\ g^{'} (2) = 0 \\ g^{''} (4) = - \frac{1}{2} \end{array}$ The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . Find $g (5)$ $1$ $h (x) = 4 + \int_{1}^{x} (3 t^{2} + 2 t - 5) d t$ . Find $h (4)$ $67$ The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . Find $g (- 3)$ , $g^{'} (3)$ and $g^{''} (- 1)$ . $\begin{array}{l} g (- 3) = \frac{21}{2} \\ g^{'} (3) = - 3 \\ g^{''} (- 1) = - \frac{2}{3} \end{array}$ The graph of $f$ is shown and $g (x) = \int_{1}^{x} f (t) d t$ . Find $g^{''} (- 1)$ $\frac{3}{2}$ $h (x) = - 3 + \int_{2}^{x} (3 t^{2} - 6 t + 2) d t$ . Find $h (4)$ $21$ The graph of $f$ is shown and $g (x) = \int_{- 1}^{x} f (t) d t$ . Find $g (6)$ , $g^{'} (8)$ and $g^{''} (5)$ . $\begin{array}{l} g (6) = - \frac{3}{2} \\ g^{'} (8) = - 1 \\ g^{''} (5) = \frac{3}{2} \end{array}$ The graph of $f$ is shown and $g (x) = \int_{- 2}^{x} f (t) d t$ . On what open intervals is $g$ increasing? $(- 4, - 2)$ $g (x) = \int_{1}^{x} (9 t^{2} - 8 t - 6) d t$ . Find $g^{''} (2)$ $28$ The graph of $f$ is shown and $g (x) = \int_{- 3}^{x} f (t) d t$ . Find $g (1)$ , $g^{'} (- 3)$ and $g^{''} (2)$ . $\begin{array}{l} g (1) = - 7 \\ g^{'} (- 3) = - 1 \\ g^{''} (2) = \frac{1}{2} \end{array}$ The graph of $f$ is shown and $g (x) = \int_{2}^{x} f (t) d t$ . On what open intervals is $g$ increasing? $(0, 3)$ $g (x) = \int_{1}^{x} (3 t^{2} - 8 t + 6) d t$ . Find $g (3)$ $6$ The graph of $f$ is shown and $g (x) = \int_{- 2}^{x} f (t) d t$ . Find $g (- 1)$ , $g^{'} (- 3)$ and $g^{''} (2)$ . $\begin{array}{l} g (- 1) = - 1 \\ g^{'} (- 3) = - 2 \\ g^{''} (2) = \frac{1}{3} \end{array}$ The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . Find $g^{''} (1)$ $\frac{1}{3}$ $g (x) = \int_{2}^{x} (- 3 t^{2} - 6 t + 5) d t$ . Find $g (4)$ $- 82$ The graph of $f$ is shown and $g (x) = \int_{0}^{x} f (t) d t$ . Find $g (8)$ , $g^{'} (- 2)$ and $g^{''} (- 1)$ . $\begin{array}{l} g (8) = 1 \\ g^{'} (- 2) = 3 \\ g^{''} (- 1) = - \frac{3}{2} \end{array}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Accumulation FunctionsAnswer keyVersion 1

Date Period

Answer each question about the accumulation function.

$g (x) = \int_{1}^{x} (- 6 t^{2} + 8 t + 2) d t$ . Find $g^{''} (4)$ $- 40$
The graph of $f$ is shown and $g (x) = \int_{0}^{x} f (t) d t$ . Find $g (- 2)$ , $g^{'} (3)$ and $g^{''} (6)$ . $\begin{array}{l} g (- 2) = 6 \\ g^{'} (3) = 0 \\ g^{''} (6) = 0 \end{array}$
The graph of $f$ is shown and $g (x) = \int_{- 1}^{x} f (t) d t$ . Find $g^{''} (0)$ $- \frac{1}{2}$
$g (x) = \int_{2}^{x} (- 3 t^{2} + 2 t + 1) d t$ . Find $g (4)$ $- 42$
The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . Find $g (5)$ , $g^{'} (2)$ and $g^{''} (4)$ . $\begin{array}{l} g (5) = 3 \\ g^{'} (2) = 0 \\ g^{''} (4) = - \frac{1}{2} \end{array}$
The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . Find $g (5)$ $1$
$h (x) = 4 + \int_{1}^{x} (3 t^{2} + 2 t - 5) d t$ . Find $h (4)$ $67$
The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . Find $g (- 3)$ , $g^{'} (3)$ and $g^{''} (- 1)$ . $\begin{array}{l} g (- 3) = \frac{21}{2} \\ g^{'} (3) = - 3 \\ g^{''} (- 1) = - \frac{2}{3} \end{array}$
The graph of $f$ is shown and $g (x) = \int_{1}^{x} f (t) d t$ . Find $g^{''} (- 1)$ $\frac{3}{2}$
$h (x) = - 3 + \int_{2}^{x} (3 t^{2} - 6 t + 2) d t$ . Find $h (4)$ $21$
The graph of $f$ is shown and $g (x) = \int_{- 1}^{x} f (t) d t$ . Find $g (6)$ , $g^{'} (8)$ and $g^{''} (5)$ . $\begin{array}{l} g (6) = - \frac{3}{2} \\ g^{'} (8) = - 1 \\ g^{''} (5) = \frac{3}{2} \end{array}$
The graph of $f$ is shown and $g (x) = \int_{- 2}^{x} f (t) d t$ . On what open intervals is $g$ increasing? $(- 4, - 2)$
$g (x) = \int_{1}^{x} (9 t^{2} - 8 t - 6) d t$ . Find $g^{''} (2)$ $28$
The graph of $f$ is shown and $g (x) = \int_{- 3}^{x} f (t) d t$ . Find $g (1)$ , $g^{'} (- 3)$ and $g^{''} (2)$ . $\begin{array}{l} g (1) = - 7 \\ g^{'} (- 3) = - 1 \\ g^{''} (2) = \frac{1}{2} \end{array}$
The graph of $f$ is shown and $g (x) = \int_{2}^{x} f (t) d t$ . On what open intervals is $g$ increasing? $(0, 3)$
$g (x) = \int_{1}^{x} (3 t^{2} - 8 t + 6) d t$ . Find $g (3)$ $6$
The graph of $f$ is shown and $g (x) = \int_{- 2}^{x} f (t) d t$ . Find $g (- 1)$ , $g^{'} (- 3)$ and $g^{''} (2)$ . $\begin{array}{l} g (- 1) = - 1 \\ g^{'} (- 3) = - 2 \\ g^{''} (2) = \frac{1}{3} \end{array}$
The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . Find $g^{''} (1)$ $\frac{1}{3}$
$g (x) = \int_{2}^{x} (- 3 t^{2} - 6 t + 5) d t$ . Find $g (4)$ $- 82$
The graph of $f$ is shown and $g (x) = \int_{0}^{x} f (t) d t$ . Find $g (8)$ , $g^{'} (- 2)$ and $g^{''} (- 1)$ . $\begin{array}{l} g (8) = 1 \\ g^{'} (- 2) = 3 \\ g^{''} (- 1) = - \frac{3}{2} \end{array}$

How to do these

$g (x) = \int_{0}^{x} (t^{2} - 4) d t$ . Where does $g$ have a relative minimum?

g'(x) = x^2 - 4, which is zero at x = -2 and x = 2.
g' is negative between -2 and 2 and positive after 2, so g falls and then rises.

The answer is Relative minimum at $x = 2$ .

Where students go wrong

Treating g'(x) as the area. The area is g; its slope at x is just the height of f there.

Also called accumulation function, area function, integral defined function or functions defined by integrals.

What you can put on this worksheet

g(x) = ∫ f(t) dt from a formula: $g (x) = \int_{1}^{x} (5 t - 3) d t$ . Find $g (4)$ → $\frac{57}{2}$
Several questions about one graph: The graph of $f$ is shown and $g (x) = \int_{2}^{x} f (t) d t$ . Find $g (- 3)$ and $g (0)$ . — the graph of f on [-3, 6], made of straight pieces → $\begin{array}{l} g (- 3) = - 6 \\ g (0) = - 2 \end{array}$
g(x) = ∫ f from a graph of f: The graph of $f$ is shown and $g (x) = \int_{6}^{x} f (t) d t$ . Find $g^{'} (2)$ — the graph of f on [-1, 8] → $0$
f(b) from f(a) and f': $f^{'} (x) = - 6 x^{2} + 2 x - 3$ and $f (1) = 4$ . Find $f (3)$ → $- 46$

Questions about these worksheets

Yes. Every accumulation functions sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the accumulation functions questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 4: g(x) = ∫ f(t) dt from a formula, several questions about one graph, g(x) = ∫ f from a graph of f and f(b) from f(a) and f'. Tick as many as you want and set how many of each, or let it spread them evenly.

Treating g'(x) as the area. The area is g; its slope at x is just the height of f there.

Accumulation functions worksheet

Select your difficulty

Accumulation Functions

Accumulation FunctionsAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

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