Taylor Approximation Calculator

Reviewed by: David Chen, Ph.D. in Applied Mathematics

Use the Taylor Approximation Calculator to estimate the value of a function at a specific point, given a center point and the desired approximation order. This tool specifically implements the Maclaurin series for $\sin(x)$.

Taylor Approximation Calculator

Approximate Result: Enter values and click Calculate. Actual value: N/A

Taylor Approximation Formula

The general formula for the Taylor series approximation of a function $f(x)$ centered at a point $a$ is:

$$P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x-a)^k$$ Where: $P_n(x)$ is the Taylor polynomial of degree $n$, $f^{(k)}(a)$ is the $k$-th derivative of $f(x)$ evaluated at $a$, and $k!$ is the factorial of $k$.

Formula Source: Wikipedia – Taylor Series, Wolfram MathWorld

Variables

The calculator uses the following variables to compute the Taylor approximation for $f(x) = \sin(x)$ centered at $a=0$ (Maclaurin series):

• Target Value (x): The point where the approximation is desired, in radians. This value is substituted into the Taylor polynomial.
• Approximation Order (n): The highest degree of the derivative (and power of $x$) included in the polynomial. A higher order generally results in a better approximation.
• Center Point (a): For the $\sin(x)$ Maclaurin series used in this calculator, the center point $a$ is fixed at $0$.

Related Calculators

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• Derivative Calculator
• Riemann Sum Estimator
• Definite Integral Solver
• Newton’s Method Calculator

What is Taylor Approximation?

Taylor approximation is a powerful mathematical technique used to represent a complicated function as an infinite sum of terms, where each term is calculated from the function’s derivatives at a single point. This process simplifies the function into a polynomial—a Taylor polynomial—which is much easier to evaluate, differentiate, and integrate.

The utility of Taylor approximation lies in its ability to closely model the behavior of a function near the ‘center point’ ($a$). By increasing the order of the approximation ($n$), the Taylor polynomial becomes a progressively more accurate representation of the original function over a larger interval. In practice, these approximations are fundamental in physics, engineering, and computer science for tasks like numerical analysis and algorithm design where precise function evaluation is computationally expensive.

When the center point $a$ is zero, the Taylor series is specifically called the Maclaurin series, which is the type of series employed by this calculator for the $\sin(x)$ function.

How to Calculate Taylor Approximation (Example)

Let’s approximate $\sin(0.5)$ using a 5th-order Maclaurin series ($x=0.5, n=5$). The Maclaurin series for $\sin(x)$ only includes odd powers:

1. Identify the Series Terms: The terms up to the 5th order are the 1st, 3rd, and 5th degree terms: $$P_5(x) = x – \frac{x^3}{3!} + \frac{x^5}{5!}$$
2. Substitute the Target Value: Substitute $x=0.5$ into the polynomial: $$P_5(0.5) = 0.5 – \frac{(0.5)^3}{3!} + \frac{(0.5)^5}{5!}$$
3. Calculate Factorials and Powers: $$3! = 6$$ $$5! = 120$$ $$P_5(0.5) = 0.5 – \frac{0.125}{6} + \frac{0.03125}{120}$$
4. Calculate Term Values: $$P_5(0.5) \approx 0.5 – 0.02083333 + 0.00026041$$
5. Sum the Terms: $$P_5(0.5) \approx 0.47942708$$ (The actual value of $\sin(0.5)$ is approximately $0.47942553$).

Frequently Asked Questions (FAQ)

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

The Maclaurin series is a special case of the Taylor series where the center point of the approximation ($a$) is specifically set to zero. Both are used to approximate functions using polynomials based on derivatives.

How does the approximation order (n) affect the result?

Increasing the order $n$ generally adds more terms to the polynomial, making the approximation more accurate. However, the calculation complexity also increases, and beyond a certain point, the improvement in accuracy diminishes.

Can I use this calculator for any function?

This specific calculator is pre-programmed to approximate the $\sin(x)$ function using its Maclaurin series. To approximate other functions, the terms and derivatives would need to be re-evaluated for that specific function.

Is Taylor approximation used in computers?

Yes, many standard mathematical functions ($\sin$, $\cos$, $e^x$, $\ln$) implemented in computer libraries and microprocessors rely on Taylor or related polynomial approximations (like Chebyshev polynomials) to provide fast and accurate results.

Approximation for $f(x) = \\sin(x)$ centered at $a=0$ up to Order ${n}

Target $x = ${bep_fmtNum(x, 4)}$

Maclaurin series for $\\sin(x)$: $$P_n(x) = \\sum_{k=0}^{n} \\frac{f^{(k)}(0)}{k!} x^k$$ Only odd powers contribute: $$P_n(x) = x – \\frac{x^3}{3!} + \\frac{x^5}{5!} – \\dots$$

Calculation Terms:

`; let termIndex = 0; // Tracks the power for the odd terms (1, 3, 5, …) // The Maclaurin series for sin(x) only uses odd powers. // The loop calculates the terms for k = 1, 3, 5, … up to the order n. for (let k = 1; k 0 ? ‘+’ : ‘-‘; const termFormula = `$\\frac{x^{${power}}}{${power}!}$`; stepsHTML += `
1. Order ${power}: ${signStr} ${termFormula} (${bep_fmtNum(x_power, 6)} / ${fact})`; stepsHTML += ` = ${signStr} ${bep_fmtNum(Math.abs(termValue), 8)}`; stepsHTML += ` (Current Sum: ${bep_fmtNum(approximation, 8)})
`; termIndex++; } stepsHTML += ‘

Final Approximation (P_${n}(${bep_fmtNum(x, 4)})) = ${bep_fmtNum(approximation, 10)}