Domanda del libro di testo
Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.
f(x) = sin x, 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 11.1.49
{Use of Tech} Estimating errors Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.
e⁰ᐧ²⁵, n=4
Guida verificata passo dopo passo1
Identify the function to approximate: here, it is \(f(x) = e^x\), and we want to approximate \(e^{0.25}\) using the 4th-order Taylor polynomial centered at 0.
Recall the Taylor polynomial of order \(n\) for \(f(x)\) centered at 0 is given by:
\[T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(0)}{k!} x^k,\] where \(f^{(k)}(0)\) is the \(k\)th derivative of \(f\) evaluated at 0.
The remainder (error) term for the Taylor polynomial approximation is given by the Lagrange form:
\[R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1}\] for some \(c\) between 0 and \(x\).
Since \(f(x) = e^x\), all derivatives are \(e^x\), which are positive and increasing. To find an upper bound on the error, evaluate the maximum of \(|f^{(n+1)}(c)|\) on the interval between 0 and 0.25, which is \(e^{0.25}\) because \(e^x\) is increasing.
Substitute \(n=4\), \(x=0.25\), and the maximum derivative value into the remainder formula to get the error bound:
\[|R_4(0.25)| \leq \frac{e^{0.25}}{5!} (0.25)^5.\] This expression gives a bound on the error when approximating \(e^{0.25}\) with the 4th-order Taylor polynomial centered at 0.
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Taylor Polynomial Approximation
A Taylor polynomial approximates a function near a point using a finite sum of its derivatives at that point. The nth-order Taylor polynomial centered at 0 (Maclaurin polynomial) uses derivatives up to order n to estimate the function's value close to zero.
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Taylor Polynomials
Remainder (Error) Term in Taylor Series
The remainder term quantifies the difference between the actual function value and its Taylor polynomial approximation. It provides an upper bound on the error, often expressed using the Lagrange form involving the (n+1)th derivative evaluated at some point in the interval.
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08:42Taylor Series
Bounding the Error for Exponential Functions
For functions like e^x, all derivatives are e^x, which are positive and increasing for positive x. To bound the error, evaluate the maximum value of the (n+1)th derivative on the interval from 0 to the point of approximation, ensuring the error estimate is valid.
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Pratica correlata
Domanda del libro di testo
Remainders Find the remainder in the Taylor series centered at the point a for the following functions. Then show that lim ₙ→∞ Rₙ(x)=0, for all x in the interval of convergence.
f(x) = e⁻ˣ, a = 0
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Domanda del libro di testo
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (2x)ᵏ/k!
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Domanda del libro di testo
How are the Taylor polynomials for a function f centered at a related to the Taylor series of the function f centered at a?
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Domanda del libro di testo
Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.
tan ⁻¹ (1/2)
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Domanda del libro di testo
{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.
√(1+x) ≈ 1 + x/2 on [−0.1,0.1]
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