{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=64; approximate ∛60.

Verified step by step guidance

1. Identify the function and the point about which to expand: here, the function is \(f(x) = \sqrt[3]{x} = x^{1/3}\) and the expansion point is \(a = 64\).
2. Compute the derivatives of \(f(x)\) up to the third order, 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 \(f(x) = x^{1/3}\), the first derivative is \(f'(x) = \frac{1}{3} x^{-2/3}\), and so on.
3. Evaluate each derivative at the point \(x = 64\). This means calculating \(f(64)\), \(f'(64)\), \(f''(64)\), and \(f'''(64)\).
4. Write the Taylor series expansion formula centered at \(a=64\): \[ T_3(x) = f(64) + f'(64)(x - 64) + \frac{f''(64)}{2!}(x - 64)^2 + \frac{f'''(64)}{3!}(x - 64)^3 \] Substitute the values of the derivatives evaluated at 64 into this formula.
5. Use the Taylor polynomial \(T_3(x)\) to approximate \(f(60) = \sqrt[3]{60}\) by plugging in \(x = 60\) into the polynomial and simplifying the expression (without calculating the final numeric value).

Key Concepts

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. It approximates the function near that point using polynomial terms, making complex functions easier to estimate.

Derivatives of Functions

Derivatives measure the rate of change of a function and are essential for finding the coefficients in a Taylor series. Calculating successive derivatives at the expansion point provides the terms needed for the polynomial approximation.

Approximation Using Partial Sums

Using only the first few terms of a Taylor series (partial sums) provides an approximation of the function near the expansion point. The accuracy depends on the number of terms and the distance from the point a; here, the first four terms approximate ∛60 near a=64.