Taylor Expansion Calculator

The Taylor Series Expansion Calculator helps users approximate and analyze mathematical functions through Taylor series, providing results with actual values, absolute error, and relative error calculations.

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How to Use the Taylor Series Expansion Calculator

The Taylor Series Expansion Calculator is a tool designed to approximate the value of a function using a finite number of terms of its Taylor series expansion. Follow the steps below to effectively use the calculator:

Step 1: Select the Function

Begin by selecting the function you wish to approximate. Options available include:

  • e^x – The exponential function.
  • sin(x) – The sine function.
  • cos(x) – The cosine function.
  • ln(1+x) – The natural logarithm of 1 plus x.

Select the function from the dropdown menu provided. This step is mandatory to proceed further.

Step 2: Enter the x Value

Enter the value of x at which you want to evaluate the Taylor series. This is a crucial input, as the function will be approximated around this point. The value should be a number between -10 and 10. A placeholder is provided to guide you to enter the value of x. The value must follow a precision of up to two decimal places (step of 0.01).

Step 3: Define the Center Point (a)

Specify the center point (a) around which the Taylor series is expanded. Like the x value, the center point should also be a number between -10 and 10, with a step precision of 0.01. Enter this value in the provided input field. This input is required to compute the series expansion.

Step 4: Specify the Number of Terms

Input the number of terms (n) to include in the Taylor series expansion. The number of terms can range from 1 to 10. Increasing the number of terms increases the accuracy of the approximation, but keep in mind that computation time might increase as well.

Step 5: View the Results

Once all inputs are provided, the calculator will compute the results automatically. The following values will be displayed:

  • Taylor Series Approximation: The calculated value of the Taylor series up to the specified number of terms, formatted to six decimal places.
  • Actual Function Value: The exact value of the function at the specified x value, also formatted to six decimal places.
  • Absolute Error: The absolute difference between the Taylor series approximation and the actual function value, with precision to six decimals.
  • Relative Error (%): The percentage error relative to the actual function value, displayed with four decimal places of precision.

Make use of these results to understand how well the Taylor series approximates the function for the given inputs.