f(x) = cos 2x - sin x; degree 4, about x = 0.

P_4(x) = 1 - x - 2x^2 + x^3/6 + 2x^4/3

f(x) = sqrt(x + 5); degree 3, about x = -4.

P_3(x) = 1 + 1/2(x + 4) - 1/8(x + 4)^2 + 1/16(x + 4)^3

f(x) = (1)/(x - 1); degree 3, about x = 4.

P_3(x) = 1/3 - 1/9(x - 4) + 1/27(x - 4)^2 - 1/81(x - 4)^3

f(x) = ln (x - 3); degree 3, about x = 4.

P_3(x) = (x - 4) - 1/2(x - 4)^2 + 1/3(x - 4)^3

f(x) = -(6x)/(1 + 2x); degree 4, about x = 0.

P_4(x) = -6x + 12x^2 - 24x^3 + 48x^4

f(x) = (1)/((x + 2)^2); degree 3, about x = 1.

P_3(x) = 1/9 - 2/27(x - 1) + 1/27(x - 1)^2 - 4/243(x - 1)^3

f(x) = e^x - 2; degree 3, about x = 2.

P_3(x) = 1 + (x - 2) + 1/2(x - 2)^2 + 1/6(x - 2)^3

f(x) = 3arctan 2x - 3ln (1 + x); degree 4, about x = 0.

P_4(x) = 3x + 3x^2/2 - 9x^3 + 3x^4/4

f(x) = ln (x + 2); degree 3, about x = -1.

P_3(x) = (x + 1) - 1/2(x + 1)^2 + 1/3(x + 1)^3

f(x) = 5xe^3x; degree 4, about x = 0.

P_4(x) = 5x + 15x^2 + 45x^3/2 + 45x^4/2

f(x) = (1)/((x - 3)^2); degree 3, about x = 1.

P_3(x) = 1/4 + 1/4(x - 1) + 3/16(x - 1)^2 + 1/8(x - 1)^3

f(x) = (1)/(x + 5); degree 3, about x = -2.

P_3(x) = 1/3 - 1/9(x + 2) + 1/27(x + 2)^2 - 1/81(x + 2)^3

f(x) = 1/x^2; degree 3, about x = 2.

P_3(x) = 1/4 - 1/4(x - 2) + 3/16(x - 2)^2 - 1/8(x - 2)^3

f(x) = (1)/((x + 3)^2); degree 3, about x = -2.

P_3(x) = 1 - 2(x + 2) + 3(x + 2)^2 - 4(x + 2)^3

f(x) = -(2x)/(1 + x); degree 4, about x = 0.

P_4(x) = -2x + 2x^2 - 2x^3 + 2x^4

f(x) = 2e^-x - cos 2x; degree 4, about x = 0.

P_4(x) = 1 - 2x + 3x^2 - x^3/3 - 7x^4/12

Worksheets · Calculus

Taylor polynomials worksheet

The degree-n Taylor polynomial about x = a matches f and its first n derivatives at a. Each term is the kth derivative at a, divided by k factorial, times (x − a) to the k.

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Sigma Prep · sigmaprep.io/worksheets Name Taylor Polynomials Date Period Write the Taylor polynomial of the given degree about the given center. $f (x) = 6 x^{2} \ln (1 + 3 x)$ ; degree 4, about $x = 0$ . $f (x) = \cos 2 x - \sin x$ ; degree 4, about $x = 0$ . $f (x) = \sqrt{x + 5}$ ; degree 3, about $x = - 4$ . $f (x) = \frac{1}{x - 1}$ ; degree 3, about $x = 4$ . $f (x) = \ln (x - 3)$ ; degree 3, about $x = 4$ . $f (x) = - \frac{6 x}{1 + 2 x}$ ; degree 4, about $x = 0$ . $f (x) = \frac{1}{{(x + 2)}^{2}}$ ; degree 3, about $x = 1$ . $f (x) = x^{2} \sin 4 x$ ; degree 4, about $x = 0$ . $f (x) = e^{x - 2}$ ; degree 3, about $x = 2$ . $f (x) = 3 \arctan 2 x - 3 \ln (1 + x)$ ; degree 4, about $x = 0$ . $f (x) = 3 x^{2} \arctan 4 x$ ; degree 4, about $x = 0$ . $f (x) = \ln (x + 2)$ ; degree 3, about $x = - 1$ . $f (x) = 5 x e^{3 x}$ ; degree 4, about $x = 0$ . $f (x) = \frac{1}{{(x - 3)}^{2}}$ ; degree 3, about $x = 1$ . $f (x) = - x^{2} e^{3 x}$ ; degree 4, about $x = 0$ . $f (x) = \frac{1}{x + 5}$ ; degree 3, about $x = - 2$ . $f (x) = \frac{1}{x^{2}}$ ; degree 3, about $x = 2$ . $f (x) = \frac{1}{{(x + 3)}^{2}}$ ; degree 3, about $x = - 2$ . $f (x) = - \frac{2 x}{1 + x}$ ; degree 4, about $x = 0$ . $f (x) = 2 e^{- x} - \cos 2 x$ ; degree 4, about $x = 0$ . Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Taylor Polynomials

Date Period

Write the Taylor polynomial of the given degree about the given center.

$f (x) = 6 x^{2} \ln (1 + 3 x)$ ; degree 4, about $x = 0$ .
$f (x) = \cos 2 x - \sin x$ ; degree 4, about $x = 0$ .
$f (x) = \sqrt{x + 5}$ ; degree 3, about $x = - 4$ .
$f (x) = \frac{1}{x - 1}$ ; degree 3, about $x = 4$ .
$f (x) = \ln (x - 3)$ ; degree 3, about $x = 4$ .
$f (x) = - \frac{6 x}{1 + 2 x}$ ; degree 4, about $x = 0$ .
$f (x) = \frac{1}{{(x + 2)}^{2}}$ ; degree 3, about $x = 1$ .
$f (x) = x^{2} \sin 4 x$ ; degree 4, about $x = 0$ .
$f (x) = e^{x - 2}$ ; degree 3, about $x = 2$ .
$f (x) = 3 \arctan 2 x - 3 \ln (1 + x)$ ; degree 4, about $x = 0$ .
$f (x) = 3 x^{2} \arctan 4 x$ ; degree 4, about $x = 0$ .
$f (x) = \ln (x + 2)$ ; degree 3, about $x = - 1$ .
$f (x) = 5 x e^{3 x}$ ; degree 4, about $x = 0$ .
$f (x) = \frac{1}{{(x - 3)}^{2}}$ ; degree 3, about $x = 1$ .
$f (x) = - x^{2} e^{3 x}$ ; degree 4, about $x = 0$ .
$f (x) = \frac{1}{x + 5}$ ; degree 3, about $x = - 2$ .
$f (x) = \frac{1}{x^{2}}$ ; degree 3, about $x = 2$ .
$f (x) = \frac{1}{{(x + 3)}^{2}}$ ; degree 3, about $x = - 2$ .
$f (x) = - \frac{2 x}{1 + x}$ ; degree 4, about $x = 0$ .
$f (x) = 2 e^{- x} - \cos 2 x$ ; degree 4, about $x = 0$ .



Sigma Prep · sigmaprep.io/worksheets Name Taylor PolynomialsAnswer keyVersion 1 Date Period Write the Taylor polynomial of the given degree about the given center. $f (x) = 6 x^{2} \ln (1 + 3 x)$ ; degree 4, about $x = 0$ . $P_{4} (x) = 18 x^{3} - 27 x^{4}$ $f (x) = \cos 2 x - \sin x$ ; degree 4, about $x = 0$ . $P_{4} (x) = 1 - x - 2 x^{2} + \frac{x^{3}}{6} + \frac{2 x^{4}}{3}$ $f (x) = \sqrt{x + 5}$ ; degree 3, about $x = - 4$ . $P_{3} (x) = 1 +$ $\frac{1}{2} (x + 4) -$ $\frac{1}{8} {(x + 4)}^{2} +$ $\frac{1}{16} {(x + 4)}^{3}$ $f (x) = \frac{1}{x - 1}$ ; degree 3, about $x = 4$ . $P_{3} (x) = \frac{1}{3} -$ $\frac{1}{9} (x - 4) +$ $\frac{1}{27} {(x - 4)}^{2} -$ $\frac{1}{81} {(x - 4)}^{3}$ $f (x) = \ln (x - 3)$ ; degree 3, about $x = 4$ . $P_{3} (x) = (x - 4) -$ $\frac{1}{2} {(x - 4)}^{2} +$ $\frac{1}{3} {(x - 4)}^{3}$ $f (x) = - \frac{6 x}{1 + 2 x}$ ; degree 4, about $x = 0$ . $P_{4} (x) = - 6 x + 12 x^{2} - 24 x^{3} + 48 x^{4}$ $f (x) = \frac{1}{{(x + 2)}^{2}}$ ; degree 3, about $x = 1$ . $P_{3} (x) = \frac{1}{9} -$ $\frac{2}{27} (x - 1) +$ $\frac{1}{27} {(x - 1)}^{2} -$ $\frac{4}{243} {(x - 1)}^{3}$ $f (x) = x^{2} \sin 4 x$ ; degree 4, about $x = 0$ . $P_{4} (x) = 4 x^{3}$ $f (x) = e^{x - 2}$ ; degree 3, about $x = 2$ . $P_{3} (x) = 1 +$ $(x - 2) +$ $\frac{1}{2} {(x - 2)}^{2} +$ $\frac{1}{6} {(x - 2)}^{3}$ $f (x) = 3 \arctan 2 x - 3 \ln (1 + x)$ ; degree 4, about $x = 0$ . $P_{4} (x) = 3 x + \frac{3 x^{2}}{2} - 9 x^{3} + \frac{3 x^{4}}{4}$ $f (x) = 3 x^{2} \arctan 4 x$ ; degree 4, about $x = 0$ . $P_{4} (x) = 12 x^{3}$ $f (x) = \ln (x + 2)$ ; degree 3, about $x = - 1$ . $P_{3} (x) = (x + 1) -$ $\frac{1}{2} {(x + 1)}^{2} +$ $\frac{1}{3} {(x + 1)}^{3}$ $f (x) = 5 x e^{3 x}$ ; degree 4, about $x = 0$ . $P_{4} (x) = 5 x + 15 x^{2} + \frac{45 x^{3}}{2} + \frac{45 x^{4}}{2}$ $f (x) = \frac{1}{{(x - 3)}^{2}}$ ; degree 3, about $x = 1$ . $P_{3} (x) = \frac{1}{4} +$ $\frac{1}{4} (x - 1) +$ $\frac{3}{16} {(x - 1)}^{2} +$ $\frac{1}{8} {(x - 1)}^{3}$ $f (x) = - x^{2} e^{3 x}$ ; degree 4, about $x = 0$ . $P_{4} (x) = - x^{2} - 3 x^{3} - \frac{9 x^{4}}{2}$ $f (x) = \frac{1}{x + 5}$ ; degree 3, about $x = - 2$ . $P_{3} (x) = \frac{1}{3} -$ $\frac{1}{9} (x + 2) +$ $\frac{1}{27} {(x + 2)}^{2} -$ $\frac{1}{81} {(x + 2)}^{3}$ $f (x) = \frac{1}{x^{2}}$ ; degree 3, about $x = 2$ . $P_{3} (x) = \frac{1}{4} -$ $\frac{1}{4} (x - 2) +$ $\frac{3}{16} {(x - 2)}^{2} -$ $\frac{1}{8} {(x - 2)}^{3}$ $f (x) = \frac{1}{{(x + 3)}^{2}}$ ; degree 3, about $x = - 2$ . $P_{3} (x) = 1 - 2 (x + 2) + 3 {(x + 2)}^{2} - 4 {(x + 2)}^{3}$ $f (x) = - \frac{2 x}{1 + x}$ ; degree 4, about $x = 0$ . $P_{4} (x) = - 2 x + 2 x^{2} - 2 x^{3} + 2 x^{4}$ $f (x) = 2 e^{- x} - \cos 2 x$ ; degree 4, about $x = 0$ . $P_{4} (x) = 1 - 2 x + 3 x^{2} - \frac{x^{3}}{3} - \frac{7 x^{4}}{12}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Taylor PolynomialsAnswer keyVersion 1

Date Period

Write the Taylor polynomial of the given degree about the given center.

$f (x) = 6 x^{2} \ln (1 + 3 x)$ ; degree 4, about $x = 0$ . $P_{4} (x) = 18 x^{3} - 27 x^{4}$
$f (x) = \cos 2 x - \sin x$ ; degree 4, about $x = 0$ . $P_{4} (x) = 1 - x - 2 x^{2} + \frac{x^{3}}{6} + \frac{2 x^{4}}{3}$
$f (x) = \sqrt{x + 5}$ ; degree 3, about $x = - 4$ . $P_{3} (x) = 1 +$ $\frac{1}{2} (x + 4) -$ $\frac{1}{8} {(x + 4)}^{2} +$ $\frac{1}{16} {(x + 4)}^{3}$
$f (x) = \frac{1}{x - 1}$ ; degree 3, about $x = 4$ . $P_{3} (x) = \frac{1}{3} -$ $\frac{1}{9} (x - 4) +$ $\frac{1}{27} {(x - 4)}^{2} -$ $\frac{1}{81} {(x - 4)}^{3}$
$f (x) = \ln (x - 3)$ ; degree 3, about $x = 4$ . $P_{3} (x) = (x - 4) -$ $\frac{1}{2} {(x - 4)}^{2} +$ $\frac{1}{3} {(x - 4)}^{3}$
$f (x) = - \frac{6 x}{1 + 2 x}$ ; degree 4, about $x = 0$ . $P_{4} (x) = - 6 x + 12 x^{2} - 24 x^{3} + 48 x^{4}$
$f (x) = \frac{1}{{(x + 2)}^{2}}$ ; degree 3, about $x = 1$ . $P_{3} (x) = \frac{1}{9} -$ $\frac{2}{27} (x - 1) +$ $\frac{1}{27} {(x - 1)}^{2} -$ $\frac{4}{243} {(x - 1)}^{3}$
$f (x) = x^{2} \sin 4 x$ ; degree 4, about $x = 0$ . $P_{4} (x) = 4 x^{3}$
$f (x) = e^{x - 2}$ ; degree 3, about $x = 2$ . $P_{3} (x) = 1 +$ $(x - 2) +$ $\frac{1}{2} {(x - 2)}^{2} +$ $\frac{1}{6} {(x - 2)}^{3}$
$f (x) = 3 \arctan 2 x - 3 \ln (1 + x)$ ; degree 4, about $x = 0$ . $P_{4} (x) = 3 x + \frac{3 x^{2}}{2} - 9 x^{3} + \frac{3 x^{4}}{4}$
$f (x) = 3 x^{2} \arctan 4 x$ ; degree 4, about $x = 0$ . $P_{4} (x) = 12 x^{3}$
$f (x) = \ln (x + 2)$ ; degree 3, about $x = - 1$ . $P_{3} (x) = (x + 1) -$ $\frac{1}{2} {(x + 1)}^{2} +$ $\frac{1}{3} {(x + 1)}^{3}$
$f (x) = 5 x e^{3 x}$ ; degree 4, about $x = 0$ . $P_{4} (x) = 5 x + 15 x^{2} + \frac{45 x^{3}}{2} + \frac{45 x^{4}}{2}$
$f (x) = \frac{1}{{(x - 3)}^{2}}$ ; degree 3, about $x = 1$ . $P_{3} (x) = \frac{1}{4} +$ $\frac{1}{4} (x - 1) +$ $\frac{3}{16} {(x - 1)}^{2} +$ $\frac{1}{8} {(x - 1)}^{3}$
$f (x) = - x^{2} e^{3 x}$ ; degree 4, about $x = 0$ . $P_{4} (x) = - x^{2} - 3 x^{3} - \frac{9 x^{4}}{2}$
$f (x) = \frac{1}{x + 5}$ ; degree 3, about $x = - 2$ . $P_{3} (x) = \frac{1}{3} -$ $\frac{1}{9} (x + 2) +$ $\frac{1}{27} {(x + 2)}^{2} -$ $\frac{1}{81} {(x + 2)}^{3}$
$f (x) = \frac{1}{x^{2}}$ ; degree 3, about $x = 2$ . $P_{3} (x) = \frac{1}{4} -$ $\frac{1}{4} (x - 2) +$ $\frac{3}{16} {(x - 2)}^{2} -$ $\frac{1}{8} {(x - 2)}^{3}$
$f (x) = \frac{1}{{(x + 3)}^{2}}$ ; degree 3, about $x = - 2$ . $P_{3} (x) = 1 - 2 (x + 2) + 3 {(x + 2)}^{2} - 4 {(x + 2)}^{3}$
$f (x) = - \frac{2 x}{1 + x}$ ; degree 4, about $x = 0$ . $P_{4} (x) = - 2 x + 2 x^{2} - 2 x^{3} + 2 x^{4}$
$f (x) = 2 e^{- x} - \cos 2 x$ ; degree 4, about $x = 0$ . $P_{4} (x) = 1 - 2 x + 3 x^{2} - \frac{x^{3}}{3} - \frac{7 x^{4}}{12}$

How to do these

$f (1) = 2$ , $f^{'} (1) = - 3$ , $f^{''} (1) = 4$ . Degree 2 about $x = 1$

The constant term is 2, the linear term −3(x − 1).
The quadratic term is 4/2! = 2 times (x − 1)².

The answer is $P_{2} (x) = 2 - 3 (x - 1) + 2 {(x - 1)}^{2}$ .

Where students go wrong

Forgetting the factorials. The coefficient of (x − a)^k is the kth derivative divided by k!, not the derivative itself.

Also called Maclaurin polynomial, Taylor series or Taylor approximation.

Questions about these worksheets

Yes. Every taylor polynomials 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 taylor polynomials 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.

Forgetting the factorials. The coefficient of (x − a)^k is the kth derivative divided by k!, not the derivative itself.

Taylor polynomials worksheet

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