{Use of Tech} Approximations with Taylor polynomials


a. Approximate the given quantities using Taylor polynomials with n = 3.


b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.


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Verified step by step guidance

1. Identify the function to approximate: here, the function is \(f(x) = e^x\), and we want to approximate \(e^{0.12}\) using a Taylor polynomial centered at \(x=0\) (Maclaurin polynomial).
2. Write the Taylor polynomial of degree 3 for \(f(x) = e^x\) centered at 0. Recall that the Maclaurin series for \(e^x\) is \(\sum_{k=0}^\infty \frac{x^k}{k!}\). The degree 3 polynomial is: \(P_3(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}\).
3. Substitute \(x = 0.12\) into the polynomial \(P_3(x)\) to get the approximation: \(P_3(0.12) = 1 + 0.12 + \frac{(0.12)^2}{2} + \frac{(0.12)^3}{6}\).
4. Calculate the exact value of \(e^{0.12}\) using a calculator or software to have a reference for the true value.
5. Compute the absolute error by subtracting the approximation from the exact value: \(\text{Absolute Error} = |e^{0.12} - P_3(0.12)|\).

Key Concepts

Taylor Polynomials

Taylor polynomials approximate functions near a point by using derivatives at that point. For a function f(x), the nth-degree Taylor polynomial at x = a sums terms involving derivatives of f up to order n, scaled by powers of (x - a). This provides a polynomial approximation that becomes more accurate as n increases.

Error and Remainder in Taylor Approximations

The error in a Taylor polynomial approximation is the difference between the actual function value and the polynomial estimate. The remainder term quantifies this error and can be bounded using higher-order derivatives, helping to assess the accuracy of the approximation.

Absolute Error Calculation

Absolute error measures the magnitude of the difference between an approximate value and the exact value. It is calculated as |approximate value - exact value| and provides a straightforward way to evaluate the precision of numerical approximations.