{Use of Tech} Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number.


f(x) = ∜x with a=16; approximate ∜13.

Verified step by step guidance

1. Identify the function and the point about which to expand: here, the function is \(f(x) = \sqrt[4]{x} = x^{\frac{1}{4}}\), and the expansion point is \(a = 16\).
2. Compute the derivatives of \(f(x)\) up to the third derivative, since we want the first four terms of the Taylor series (which includes the function value and the first three derivatives). Use the power rule for derivatives: for \(f(x) = x^{n}\), \(f^{(k)}(x) = n (n-1) \cdots (n-k+1) x^{n-k}\).
3. Evaluate each derivative at the point \(x = 16\). This means calculating \(f(16)\), \(f'(16)\), \(f''(16)\), and \(f'''(16)\).
4. Write the Taylor series expansion of \(f(x)\) about \(a=16\) up to the third derivative term using the formula: \[ T_3(x) = f(16) + f'(16)(x-16) + \frac{f''(16)}{2!}(x-16)^2 + \frac{f'''(16)}{3!}(x-16)^3 \]
5. Substitute \(x = 13\) into the Taylor polynomial \(T_3(x)\) to approximate \(\sqrt[4]{13}\). This gives an approximation using the first four terms of the Taylor series.

Key Concepts

Taylor Series Expansion

The Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a specific point a. It approximates the function near a by using polynomial terms, making it easier to estimate values close to a.

Derivatives of Root Functions

To find the Taylor series coefficients, you need to compute derivatives of the function f(x) = ∜x (the fourth root of x). Understanding how to differentiate root functions and simplify the results is essential for accurate coefficient calculation.

Polynomial Approximation and Evaluation

Using the first four terms of the Taylor series creates a polynomial that approximates the function near the point a. Evaluating this polynomial at the desired value (x=13) provides an approximate value for ∜13, demonstrating practical use of series expansions.