Textbook Question
Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.
ƒ(x) = sin 2x, n = 3, a = 0
Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 11, Problem 11.R.43
Write out the first three terms of the Maclaurin series for the following functions.
ƒ(x) = (1 + x)^(1/3)"
Verified step by step guidance1
Recall that the Maclaurin series is the Taylor series expansion of a function about \(x = 0\). It is given by the formula:
\[f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots\]
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.
Calculate the first derivative using the power rule for fractional exponents:
\[f'(x) = \frac{1}{3}(1 + x)^{-\frac{2}{3}}\]
Calculate the second derivative by differentiating \(f'(x)\) again:
\[f''(x) = \frac{1}{3} \times \left(-\frac{2}{3}\right)(1 + x)^{-\frac{5}{3}} = -\frac{2}{9}(1 + x)^{-\frac{5}{3}}\]
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:
\[f(0) + f'(0)x + \frac{f''(0)}{2!}x^2\]
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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Convergence of Taylor & Maclaurin Series
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.
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06:00Geometric Series
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.
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07:32Representing Functions as Power Series
Related Practice
Textbook Question
Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
ƒ(x) = 1/(4 + x²), a = 0
Textbook Question
Definite integrals by power series Use a Taylor series to approximate the following definite integrals. Retain as many terms as necessary to ensure the error is less than 10⁻³.
∫₀1 x cos x dx
Textbook Question
Binomial series Write out the first three terms of the Maclaurin series for the following functions.
ƒ(x) = (1 + 2x)^(-5)
Textbook Question
Convergence Write the remainder term Rₙ(x) for the Taylor series for the following functions centered at the given point a. Then show that lim ₙ → ∞ |Rₙ(x)| = 0, for all x in the given interval.
ƒ(x) = sinh x + cosh x, a = 0, - ∞ < x < ∞
Textbook Question
Approximating ln 2 Consider the following three ways to approximate
ln 2.
a. Use the Taylor series for ln (1 + x) centered at 0 and evaluate it at x = 1 (convergence was asserted in Table 11.5). Write the resulting infinite series.