Write out the first three terms of the Maclaurin series for the following functions.
ƒ(x) = (1 + x)^(1/3)"

Verified step by step guidance

1. Recall that the Maclaurin series is the Taylor series expansion of a function about \(x = 0\). It is given by the formula: \[f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots\]
2. Identify the function: \(f(x) = (1 + x)^{\frac{1}{3}}\). We will need to find the first and second derivatives of \(f(x)\) to get the first three terms.
3. Calculate the first derivative using the power rule for fractional exponents: \[f'(x) = \frac{1}{3}(1 + x)^{-\frac{2}{3}}\]
4. Calculate the second derivative by differentiating \(f'(x)\) again: \[f''(x) = \frac{1}{3} \times \left(-\frac{2}{3}\right)(1 + x)^{-\frac{5}{3}} = -\frac{2}{9}(1 + x)^{-\frac{5}{3}}\]
5. Evaluate \(f(0)\), \(f'(0)\), and \(f''(0)\) by substituting \(x = 0\) into each expression, then write the first three terms of the Maclaurin series as: \[f(0) + f'(0)x + \frac{f''(0)}{2!}x^2\]

Key Concepts

Maclaurin Series

The Maclaurin series is a special case of the Taylor series expanded at x = 0. It represents a function as an infinite sum of its derivatives evaluated at zero, multiplied by powers of x. This series helps approximate functions near zero using polynomials.

Binomial Series Expansion

The binomial series generalizes the binomial theorem to any real exponent, allowing expansion of expressions like (1 + x)^r. It uses coefficients derived from generalized binomial coefficients, which involve factorials or the Gamma function for non-integer powers.

Derivatives of Power Functions

To find terms in the Maclaurin series, you compute derivatives of the function at zero. For functions like (1 + x)^(1/3), derivatives involve applying the power rule repeatedly, which helps determine coefficients for each term in the series.