Domanda del libro di testo
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)=3ˣ, a=0
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Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 11, Problema 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
Guida verificata passo dopo passo1
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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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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08:42
Taylor Series
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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08:42Taylor Series
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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Percorso guidato
06:21Properties of Functions
Pratica correlata
Domanda del libro di testo
{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.
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Domanda del libro di testo
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) = tan ⁻¹ (x/2), a = 0
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Domanda del libro di testo
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.
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Domanda del libro di testo
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) = e⁻ˣ, a=0
59
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Domanda del libro di testo
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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