Taylor Polynomial Calculator

The Taylor Polynomial Calculator allows users to compute Taylor polynomials for selected functions, evaluate their values and errors at specific points, and analyze the approximation accuracy by providing key data inputs such as function, center point, evaluation point, and polynomial order.

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Guide to Using the Taylor Polynomial Calculator

Introduction

This guide explains the process of using the Taylor Polynomial Calculator to compute the Taylor polynomial for selected functions at a specified order, and evaluate it at a certain point. The calculator will also provide the actual function value, approximation error, and the relative error percentage.

Step-by-Step Instructions

Step 1: Select the Function

Begin by selecting the function for which you want to calculate the Taylor polynomial. The available options are:

  • sin(x)
  • cos(x)
  • e^x
  • ln(x)

This selection is mandatory to proceed with the calculation.

Step 2: Specify the Center Point (a)

Enter the center point, also known as the point of approximation, denoted as a. This is where the Taylor series is centered. Please make sure to enter the value accurately as a number, with increments allowed up to 0.0001. This field is required.

Step 3: Enter the Evaluation Point (x)

Input the evaluation point x, where you want the Taylor polynomial to be evaluated. As with the center point, the evaluation point must be a number, with precision allowed up to 0.0001. This field is also required.

Step 4: Determine the Order of the Taylor Polynomial (n)

The order of the Taylor polynomial dictates the degree of the polynomial. Enter a value between 0 and 5, inclusive. The order is required and should be entered in whole numbers with a step of 1.

Step 5: Calculate the Results

Upon entering all the required fields, the calculator will compute the following:

  • Taylor Polynomial: The approximated polynomial value at the evaluation point.
  • Actual Function Value: The precise value of the function at the evaluation point.
  • Approximation Error: The absolute difference between the actual function value and the Taylor polynomial estimation.
  • Relative Error (%): The percentage error of the approximation relative to the actual function value, formatted as a percentage.

Conclusion

Utilizing these steps, you can effectively calculate the Taylor polynomial and assess the accuracy of the approximation for your chosen function. Ensure all inputs are correct to receive the most accurate results.