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 11.1.75a
{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
Verified step by step guidance1
Identify the function and the approximation given: the function is \(f(x) = e^{x}\) and the approximation near zero is \(f(x) \approx 1 + x\).
Calculate the approximate value of \(f(0.1)\) using the linear approximation: substitute \(x = 0.1\) into \(1 + x\) to get the estimate.
Recall that the error in using the linear approximation can be bounded using the remainder term from Taylor's theorem. For \(e^{x}\), the remainder after the linear term is given by \(R_2 = \frac{e^{c} x^{2}}{2}\) for some \(c\) between 0 and \(x\).
To find an error bound, note that \(e^{c}\) is increasing, so the maximum value of \(e^{c}\) on the interval \([0, 0.1]\) is \(e^{0.1}\). Use this to bound the error: \(|R_2| \leq \frac{e^{0.1} (0.1)^{2}}{2}\).
Summarize the result: the approximate value of \(f(0.1)\) is \(1 + 0.1\), and the error in this approximation is at most \(\frac{e^{0.1} (0.1)^{2}}{2}\). This gives a clear estimate and a guaranteed bound on the error.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor Series Approximation
The Taylor series expresses a function as an infinite sum of terms calculated from its derivatives at a single point. Near zero, eˣ can be approximated by 1 + x, which is the first-order Taylor polynomial. This approximation simplifies calculations for small x values by using only the first two terms.
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Error Bound in Taylor Approximations
The error bound estimates how far the approximation is from the true function value. For the first-order Taylor approximation of eˣ, the remainder term involves the second derivative and can be bounded using the Lagrange form of the remainder. This helps quantify the maximum possible error when approximating f(0.1).
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Exponential Function Properties
The exponential function eˣ is smooth and infinitely differentiable, with all derivatives equal to eˣ. This property ensures the Taylor series converges for all real x and allows easy computation of derivatives needed for approximations and error bounds.
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Related Practice
Textbook Question
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
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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
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
{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
a. Compute S′(x) and C′(x).
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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