Maclaurin Polynomial Calculator

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Reviewed by David Chen, CFA

Use this maclaurin polynomial calculator to find the polynomial approximation of a given function (currently supporting $f(x) = e^x$) up to the specified order $n$, and evaluate it at a given point $x$.

Maclaurin Polynomial Calculator

Polynomial Approximation ($P_n(x)$) Value:

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Detailed Calculation Steps


      

    


    
    

      
Maclaurin Polynomial Formula

      

        $$P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(0)}{k!} x^k = f(0) + f'(0)x + \frac{f”(0)}{2!}x^2 + \dots + \frac{f^{(n)}(0)}{n!}x^n$$
      

      

        Source Links:
        Wolfram MathWorld,
        Wikipedia (Taylor Series)
      


      
Variables

      
    
        
  • Function ($f(x)$): The mathematical function being approximated (e.g., $e^x$).
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  • Polynomial Order ($n$): The degree of the resulting polynomial. A higher order generally results in a better approximation near $x=0$.
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  • Evaluation Point ($x$): The specific value at which the resulting polynomial $P_n(x)$ is calculated.
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Related Calculators

      

      
What is a Maclaurin Polynomial?

      

        The Maclaurin Polynomial, a special case of the Taylor Polynomial centered at $a=0$, is a mathematical tool used to approximate the value of a function $f(x)$ using an $n$-degree polynomial. This approximation is highly accurate near the center point ($x=0$) and is fundamental in calculus, physics, and engineering.
      

      

        By matching the function’s value and its derivatives at a single point (the center), the polynomial creates a curve that closely mimics the original function over a certain interval. This process transforms complex, transcendental functions (like $e^x$ or $\sin(x)$) into simpler, manageable polynomial expressions.
      

      

        The ability to represent functions as polynomials allows for easier computation of integrals, limits, and solutions to differential equations, making it an indispensable concept in advanced mathematics and computational modeling.
      


      
How to Calculate the Maclaurin Polynomial (Example: $f(x) = e^x$, $n=3$)

      
    
        
  1. Find Derivatives: Calculate the derivatives of $f(x) = e^x$ up to the 3rd order.
            ($f'(x) = e^x, f”(x) = e^x, f”'(x) = e^x$)
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  2. Evaluate at $x=0$: Substitute $x=0$ into the function and all derivatives.
            ($f(0)=1, f'(0)=1, f”(0)=1, f”'(0)=1$)
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  3. Apply Formula: Substitute these values into the Maclaurin formula:
            $$P_3(x) = f(0) + f'(0)x + \frac{f”(0)}{2!}x^2 + \frac{f”'(0)}{3!}x^3$$
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  4. Resulting Polynomial:
            $$P_3(x) = 1 + 1x + \frac{1}{2}x^2 + \frac{1}{6}x^3$$
    5. 
      


      
Frequently Asked Questions (FAQ)

      

        
What is the difference between a Maclaurin and a Taylor series?
          The Maclaurin series is simply a Taylor series centered at $a=0$. A Taylor series can be centered at any point $a$, but the Maclaurin series is restricted to the origin.

        
Why is the Maclaurin polynomial useful?
          It allows us to approximate the behavior of complex functions using simple polynomials, which are much easier to manipulate computationally for integration, differentiation, and estimation near zero.

        
Does a Maclaurin series always work?
          No. For a series to exist and accurately represent the function, the function must be infinitely differentiable at the center point (or at least up to the required order $n$).

        
What is the remainder term?
          The remainder term (or error) is the difference between the actual function value and the polynomial approximation. It is often described by Taylor’s Theorem and shows how accurate the approximation is.

      

    

  




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