Taylor Polynomial Calculator

Compute a finite Taylor polynomial through order 10 around a selected expansion point, with domain and approximation checks.

Results are calculated automatically as you enter data.

Function f(x) Use x, powers, parentheses, and functions such as sin, cos, exp, ln, or sqrt. Expansion point a Order 5

Examples

Taylor polynomial

Original function Taylor polynomial

Centered at a = 0

Expanded form

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Assumptions and limitations

Results are educational estimates, not professional engineering, construction, safety, scientific, legal, financial, or academic advice. No guarantee is made that outputs are accurate, complete, suitable, or error-free. Verify units, formulas, assumptions, tolerances, rounding, and limitations independently, and use qualified professional review for technical or safety-sensitive work. This tool computes finite real-valued Taylor polynomials, not an infinite series or rigorous remainder bound. Use of this calculator is subject to our Terms of Use.

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What Is a Taylor Polynomial?

A Taylor polynomial is a polynomial that approximates a function near a chosen point. Instead of trying to work with the original function directly, you build a simpler polynomial that has the same value, slope, curvature, and higher-order derivative behavior at one point.

The chosen point is called the center or expansion point, usually written as $a$. The polynomial is designed to work best near that center. As $x$ moves farther away from $a$, the approximation may become less accurate unless there is a separate convergence or error argument showing that it still behaves well.

Taylor polynomials are useful because polynomials are easier to evaluate, graph, differentiate, integrate, and compare than many functions. A complicated function such as $e^x$, $\sin(x)$, $\ln(1+x)$, or $\sqrt{1+x}$ can often be approximated near a convenient point by a short polynomial.

When the center is $a = 0$, the Taylor polynomial is called a Maclaurin polynomial. Maclaurin polynomials are common because many familiar functions have simple values and derivative patterns at zero.

Why Taylor Polynomial Approximation Matters

Taylor polynomials turn local derivative information into a practical approximation. This is one of the central ideas in calculus: if you know enough about a function at one point, you can often predict how it behaves nearby.

This matters in several common situations:

Calculus study: Taylor polynomials connect derivatives, power series, and approximation error in one method.
Graphing and visualization: Comparing a function with its Taylor polynomial shows how slope, curvature, and higher-order terms shape a curve.
Numerical estimation: A polynomial can estimate function values when the original expression is harder to evaluate by hand.
Model building: Many scientific and engineering models use local approximations to simplify nonlinear behavior near an operating point.
Error awareness: Taylor's theorem shows that approximation is not just guessing; under the right conditions, the omitted remainder can be bounded.

Key Terms to Know

Function $f(x)$: The original single-variable function being approximated.
Center $a$: The point where the Taylor polynomial is built. The approximation is usually most reliable near this point.
Order $n$: The highest power included in the polynomial. A first-order polynomial is linear, a second-order polynomial is quadratic, and so on.
Derivative $f^{(k)}(a)$: The $k$th derivative of the function evaluated at the center.
Factorial $k!$: The product $k \times (k-1) \times \cdots \times 2 \times 1$, with $0! = 1$.
Taylor coefficient $c_k$: The coefficient of the $k$th Taylor term, equal to $\frac{f^{(k)}(a)}{k!}$.
Centered form: A polynomial written in powers of $(x-a)$, such as $1 + 2(x-a) - 3(x-a)^2$.
Expanded form: The same polynomial rewritten in powers of $x$, such as $3x^2 - 4x + 1$.
Remainder: The difference between the original function and the Taylor polynomial: $R_n(x) = f(x) - P_n(x)$.
Interval of convergence: For an infinite Taylor series, the interval where the series converges. A finite Taylor polynomial does not by itself determine that interval.

How Taylor Polynomial Approximation Works

The idea is to make a polynomial match the function's derivative data at the center. The $n$th-order Taylor polynomial for $f(x)$ centered at $a$ is

$$ P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k $$

This means

$$ P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f^{(3)}(a)}{3!}(x-a)^3 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n $$

Where:

$P_n(x)$ is the Taylor polynomial of order $n$.
$a$ is the expansion point.
$f^{(k)}(a)$ is the $k$th derivative evaluated at $a$.
$k!$ is the factorial of $k$.
$(x-a)^k$ measures how far $x$ is from the center, raised to the $k$th power.

The formula starts with the function value. That gives the constant approximation:

$$ P_0(x) = f(a) $$

The first derivative adds the tangent-line approximation:

$$ P_1(x) = f(a) + f'(a)(x-a) $$

The second derivative adds curvature:

$$ P_2(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 $$

Higher derivatives add more local shape information. In many cases, increasing the order improves the approximation near $a$, but it does not automatically guarantee accuracy far from $a$.

Taylor Polynomials and Taylor Series

A Taylor polynomial is finite. It stops at a chosen order $n$. A Taylor series is the infinite series that continues all Taylor terms indefinitely:

$$ \sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!}(x-a)^k $$

The finite polynomial is often useful even when you do not know the full interval of convergence. However, saying that an infinite Taylor series actually equals the original function requires additional conditions. Taylor's theorem explains the missing part through a remainder term.

When the needed derivative exists on an interval around $a$, one common form of the remainder says that, for some number $c$ between $a$ and $x$,

$$ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1} $$

If $|f^{(n+1)}(t)| \le M$ on the interval between $a$ and $x$, then the size of the error can be bounded by

$$ |R_n(x)| \le \frac{M}{(n+1)!}|x-a|^{n+1} $$

This is why Taylor polynomials tend to be strongest near the center: the factor $|x-a|^{n+1}$ is small when $x$ is close to $a$.

Examples of Taylor Polynomial Approximation in Practice

Example 1: Approximating $e^x$ near zero

For $f(x) = e^x$, every derivative is also $e^x$. At $a = 0$, each derivative value is $1$. The third-order Maclaurin polynomial is

$$ P_3(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} $$

To estimate $e^{0.2}$, substitute $x = 0.2$:

$$ P_3(0.2) = 1 + 0.2 + \frac{0.2^2}{2} + \frac{0.2^3}{6} $$

$$ P_3(0.2) = 1 + 0.2 + 0.02 + 0.00133333 \approx 1.22133333 $$

The approximation is close because $0.2$ is near the center $0$. A higher-order polynomial would include more terms and usually improve the estimate near the center.

Example 2: A linear approximation for a square root

Suppose you want to estimate $\sqrt{4.1}$. Use $f(x)=\sqrt{x}$ and center the approximation at $a=4$, because $4$ is close to $4.1$ and $\sqrt{4}=2$ is easy to evaluate.

The derivative is

$$ f'(x) = \frac{1}{2\sqrt{x}} $$

At $a=4$,

$$ f(4)=2 $$

and

$$ f'(4)=\frac{1}{2\sqrt{4}}=\frac{1}{4} $$

The first-order Taylor polynomial is

$$ P_1(x)=2+\frac{1}{4}(x-4) $$

Now substitute $x=4.1$:

$$ P_1(4.1)=2+\frac{1}{4}(0.1)=2.025 $$

This is a local estimate. It works well because $4.1$ is close to the center $4$, not because the tangent line is globally identical to the square-root curve.

Example 3: A nondifferentiable edge case

Taylor polynomials depend on derivatives. If the needed derivative does not exist at the center, the Taylor polynomial of that order is not valid.

For example, $f(x)=|x|$ has a sharp corner at $x=0$. Its zeroth-order approximation at $0$ is simply

$$ P_0(x)=f(0)=0 $$

But a first-order Taylor polynomial at $0$ would require $f'(0)$, and that derivative does not exist. This is why differentiability at the expansion point matters before using a higher-order Taylor polynomial.

How to Interpret the Result

The most important result is the centered Taylor polynomial. It is written in powers of $(x-a)$, which makes the role of the expansion point clear. This form usually shows the Taylor structure most directly because each term is tied to a derivative value at $a$.

The expanded polynomial is the same approximation rewritten in powers of $x$. It may look different, but it represents the same polynomial after algebraic expansion. The centered form is usually better for understanding the approximation; the expanded form is often useful for simplifying, comparing, or graphing polynomial expressions.

The derivative and coefficient table explains where the polynomial comes from. Each row connects a derivative value $f^{(k)}(a)$ to a coefficient

$$ c_k = \frac{f^{(k)}(a)}{k!} $$

and then to the corresponding term

$$ c_k(x-a)^k $$

The graph compares the original function with the Taylor polynomial near the center. When the curves are close near $a$, the polynomial is behaving as a good local approximation. If the curves separate away from $a$, that is a reminder that Taylor polynomials are local tools unless a broader convergence or error result is known.

The sample diagnostics report maximum observed absolute and relative error at $a-2$, $a-1$, $a+1$, and $a+2$, together with the number of points skipped outside the real domain. They should not be read as maximum possible errors over an interval. A rigorous error bound requires a Taylor remainder argument and information about a derivative over the whole interval being considered.

The next-term estimate gives a rough sense of the scale of the omitted next term near $x=a+1$, when the next derivative is finite. It is helpful for intuition, but it is not the same thing as a proof of error.

The order shown in the result is the actual order used. This tool accepts whole-number orders from $0$ through $10$; values outside that range are rejected rather than silently changed.

Common Mistakes and Misconceptions

Expecting accuracy far from the center. Taylor polynomials are built from information at one point. They usually approximate best near that point, and they may become poor approximations farther away.

Confusing a Taylor polynomial with a Taylor series. A Taylor polynomial has finitely many terms. A Taylor series has infinitely many terms and raises separate questions about convergence.

Treating sample error as a guaranteed bound. A few sampled differences can be useful, but they do not replace a formal remainder estimate.

Choosing a center outside the real domain. For example, $\ln(x)$ at $a=0$ or $\sqrt{x}$ at a negative center is not valid in real-valued calculus. The function and the needed derivatives must be finite at the expansion point.

Using a variable other than $x$. This calculator is designed for single-variable functions of $x$. Expressions using another variable will not represent the intended input.

Using unsupported function syntax. One-variable functions such as sin, cos, tan, exp, ln, log, sqrt, cbrt, abs, sec, csc, and cot are supported. Multi-argument functions and unsupported names are not.

Forgetting that trigonometric inputs are in radians. Taylor and derivative formulas for trigonometric functions use radians. For example, $\sin(x)$ means the sine of $x$ radians, not degrees.

Entering a non-integer order. Taylor polynomial order is a whole number. Order $0$ gives a constant approximation, order $1$ gives a linear approximation, and higher whole-number orders add more terms.

Confusing centered and expanded forms. The centered and expanded results can look different, but they are two forms of the same polynomial approximation.

When to Use Taylor Polynomial Approximation

Use Taylor polynomial approximation when you want to:

approximate a function value near a convenient point;
study how derivative values become polynomial coefficients;
compare different polynomial orders visually or numerically;
understand the local shape of a function near an expansion point;
practice Maclaurin polynomials for functions centered at $0$;
estimate simple values of functions such as $e^x$, $\sin(x)$, $\ln(1+x)$, and $\sqrt{1+x}$;
explore why higher-order terms can improve an approximation near the center.

For exact proofs, interval-of-convergence questions, or guaranteed error bounds, use Taylor's theorem and remainder estimates rather than relying only on a computed polynomial or graph.

Limitations and Things to Keep in Mind

A Taylor polynomial is a local approximation, not automatically a global replacement for the original function. A good match near the center does not mean the two expressions stay close everywhere.

The calculator works with real-valued functions of one variable, $x$. It does not support complex-valued Taylor series, multiple variables, or functions outside the supported one-argument function list.

The expansion point must be finite, and the function's required derivative values must be finite at that point. If a derivative is undefined or non-finite, the corresponding Taylor term cannot be formed.

The original function is checked before algebraic simplification so undefined factors are not silently erased. Trigonometric poles and unresolved absolute-value kinks at the center are rejected. Common real-valued smooth identities such as abs(x^2), abs(x)^2, cbrt(x^3), and cbrt(x)^3 are simplified safely.

The supported polynomial order is limited to whole numbers from $0$ through $10$. Orders above $10$ are not computed as true higher-order Taylor polynomials.

The result does not determine a radius of convergence, an interval of convergence, or a rigorous Taylor remainder bound. These require additional analysis beyond the displayed polynomial.

At very large expansion points, conversion from centered powers of $(x-a)$ to powers of $x$ can overflow or suffer severe cancellation. In that case the calculator keeps the valid centered polynomial and marks only the expanded form unavailable. Expression and derivative complexity limits prevent unusually large symbolic inputs from freezing the page.

Finite nonzero coefficients are preserved and displayed with up to 14 significant digits. Scientific notation is used when it keeps very small or very large values readable.

The graph is an adaptive sampled comparison near the center. It breaks curves at detected domain failures and pole-like jumps and uses robust vertical scaling, but it remains a visual guide rather than a proof of continuity or accuracy.

For coursework, proofs, engineering models, scientific calculations, or any situation where the error matters, double-check the polynomial algebra and use an appropriate Taylor remainder bound.

How to Use This Calculator

Enter a function of $x$, such as sin(x), exp(x), ln(1+x), 1/(1-x), or sqrt(1+x).
Enter the expansion point $a$ where the Taylor polynomial should be centered.
Choose a whole-number Taylor polynomial order from $0$ to $10$.
Use an example button if you want to autofill a common function, center, and order.
Review the centered polynomial, expanded polynomial, derivative/coefficient table, graph, and insight cards.
Use the clear action to reset the inputs.
Use the graph download option, if available, to save the comparison graph.

Frequently Asked Questions

What is the difference between a Taylor polynomial and a Maclaurin polynomial?

A Maclaurin polynomial is a Taylor polynomial centered at $a=0$. Every Maclaurin polynomial is a Taylor polynomial, but not every Taylor polynomial is a Maclaurin polynomial.

Does a higher-order Taylor polynomial always give a better approximation?

A higher order often improves accuracy near the center when the required derivatives exist and behave well. It does not guarantee better accuracy at every point, especially far from the center or near discontinuities, corners, or domain restrictions.

Why does the expansion point matter?

The expansion point is where the polynomial matches the function's derivative data. The approximation is usually strongest near that point, so choosing a center close to the $x$-value of interest is often important.

What does order $0$ mean?

Order $0$ uses only the function value at the center. The result is the constant polynomial $P_0(x)=f(a)$, which is the simplest possible Taylor approximation.

Is the sample error the same as a Taylor remainder bound?

No. A sample error is based on selected finite points and gives a quick comparison. A Taylor remainder bound requires a theorem-based estimate using derivative behavior over an interval.

Why might a function fail at the expansion point?

A Taylor polynomial needs the function and the required derivatives to be defined and finite at the center. Functions such as $\ln(x)$, $\sqrt{x}$, $\tan(x)$, or $|x|$ can fail at certain centers because of domain restrictions, vertical asymptotes, or nondifferentiable corners.

Sources and References

Books and Open Textbooks

Gilbert Strang and Edwin “Jed” Herman. Calculus Volume 2. OpenStax, 2016. Sections 6.3 “Taylor and Maclaurin Series” and 6.4 “Working with Taylor Series.” Section 6.3, Section 6.4, accessed July 4, 2026.
Joel Feldman, Andrew Rechnitzer, and Elyse Yeager. CLP-1 Differential Calculus. University of British Columbia, 2016–2024. Section 3.4 “Approximating Functions Near a Specified Point — Taylor Polynomials.” LibreTexts edition, UBC PDF edition, accessed July 4, 2026.
David Guichard. Calculus. Whitman College / Mathematics LibreTexts. Section 11.12 “Taylor's Theorem.” Taylor's Theorem, accessed July 4, 2026.