Textbook Question
{Use of Tech} Binomial series
a. Find the first four nonzero terms of the binomial series centered at 0 for the given function.
f(x) = (1+x)⁻²/³; approximate 1.18⁻²/³.
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 11.3.27a
Taylor series
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x)=sin x, a = π/2
Verified step by step guidance1
Recall the definition of the Taylor series of a function \(f(x)\) centered at \(a\):
\[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n,\]
where \(f^{(n)}(a)\) is the \(n\)-th derivative of \(f\) evaluated at \(x = a\).
Identify the function and center: here, \(f(x) = \sin x\) and \(a = \frac{\pi}{2}\). We will need to compute the derivatives of \(\sin x\) and evaluate them at \(x = \frac{\pi}{2}\).
Calculate the first four derivatives of \(f(x) = \sin x\):
- \(f(x) = \sin x\)
- \(f'(x) = \cos x\)
- \(f''(x) = -\sin x\)
- \(f^{(3)}(x) = -\cos x\)
- \(f^{(4)}(x) = \sin x\) (which repeats the cycle).
Evaluate each derivative at \(x = \frac{\pi}{2}\):
- \(f\left(\frac{\pi}{2}\right) = \sin \frac{\pi}{2} = 1\)
- \(f'\left(\frac{\pi}{2}\right) = \cos \frac{\pi}{2} = 0\)
- \(f''\left(\frac{\pi}{2}\right) = -\sin \frac{\pi}{2} = -1\)
- \(f^{(3)}\left(\frac{\pi}{2}\right) = -\cos \frac{\pi}{2} = 0\)
Write the first four nonzero terms of the Taylor series using the formula:
\[f(x) \approx f(a) + \frac{f'(a)}{1!}(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f^{(3)}(a)}{3!}(x - a)^3 + \cdots\]
Substitute the values found and simplify, including only the nonzero terms.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor Series Definition
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. Each term involves the nth derivative evaluated at the center point, multiplied by (x - a)^n and divided by n!. This series approximates the function near the point a.
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Taylor Series
Derivatives of sin(x)
To find the Taylor series of sin(x), you need to compute its derivatives at the center point. The derivatives of sin(x) cycle every four steps: sin(x), cos(x), -sin(x), -cos(x), then back to sin(x). Evaluating these at x = π/2 helps determine the coefficients of the series.
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04:56Derivative of the Natural Exponential Function (e^x)
Centering the Series at a = π/2
Centering the Taylor series at a = π/2 means expanding the function around this point, so terms involve powers of (x - π/2). This shifts the approximation focus to values of x near π/2, requiring evaluation of derivatives specifically at π/2 to find accurate coefficients.
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Related Practice
Textbook Question
Matching functions with polynomials Match functions a–f with Taylor polynomials A–F (all centered at 0). Give reasons for your choices.
a. √(1 + 2x)
A. p₂(x)= 1 + 2x + 2x²
B. p₂(x) = 1 − 6x + 24x²
C. p₂(x) = 1 + x − x²/2
D. p₂(x) = 1 − 2x + 4x²
E. p₂(x) = 1 − x + (3/2)x²
F. p₂(x) = 1 − 2x + 2x²
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The function f(x) = √x has a Taylor series centered at 0.
Textbook Question
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
a. Estimate f(0.1) and give a bound on the error in the approximation.
f(x) = eˣ ≈ 1 + x
Textbook Question
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = cosh 3x, a = 0
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. Only even powers of x appear in the Taylor polynomials for f(x)=e⁻²ˣ centered at 0.
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