{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.


√1.06

Verified step by step guidance

1. Identify the function to approximate: here, we want to approximate \( f(x) = \sqrt{x} \) near a point where the function and its derivatives are easy to compute. A good choice is \( a = 1 \) because \( \sqrt{1} = 1 \).
2. Write the Taylor polynomial of degree 3 for \( f(x) \) centered at \( a = 1 \). The general formula is: \[ T_3(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 \] Calculate the first, second, and third derivatives of \( f(x) = \sqrt{x} = x^{1/2} \).
3. Evaluate each derivative at \( x = 1 \) to find \( f(1), f'(1), f''(1), \) and \( f'''(1) \). Substitute these values into the Taylor polynomial formula.
4. Substitute \( x = 1.06 \) into the Taylor polynomial \( T_3(x) \) to approximate \( \sqrt{1.06} \). This gives the approximate value using the third-degree Taylor polynomial.
5. To find the absolute error, calculate the exact value of \( \sqrt{1.06} \) using a calculator, then subtract the Taylor polynomial approximation from this exact value. The absolute error is: \[ \text{Absolute Error} = |\sqrt{1.06} - T_3(1.06)| \]

Key Concepts

Taylor Polynomials

Taylor polynomials approximate functions near a point by using derivatives at that point. For n=3, the polynomial includes terms up to the cubic degree, providing a close estimate of the function's value near the expansion point.

Error Estimation in Approximations

The absolute error measures the difference between the exact value and the approximation. Calculating this helps assess the accuracy of the Taylor polynomial approximation.

Function Expansion Point and Domain

Choosing the expansion point (often near the value to approximate) is crucial for accuracy. Understanding the domain and behavior of the function, like √x near x=1, ensures the Taylor polynomial converges well.