Domanda del libro di testo
Taylor series
b. Write the power series using summation notation.
f(x) = 1/x, a = 1
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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 11.1.74b
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
b. Estimate f(0.2) and give a bound on the error in the approximation.
f(x) = ln (1 + x) ≈ x − x²/2
Guida verificata passo dopo passo1
Identify the function and its approximation: The function given is \(f(x) = \ln(1 + x)\), and the approximation near zero is \(f(x) \approx x - \frac{x^2}{2}\).
Calculate the approximate value at \(x = 0.2\) by substituting into the approximation formula: \(f(0.2) \approx 0.2 - \frac{(0.2)^2}{2}\).
Understand that the error bound can be estimated using the remainder term from the Taylor series expansion of \(\ln(1+x)\) around \(x=0\).
Recall that the Taylor series for \(\ln(1+x)\) is \(x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots\), so the next term after the approximation is \(\frac{x^3}{3}\), which helps in bounding the error.
Use the Lagrange form of the remainder to bound the error: the error \(R_2(x)\) satisfies \(|R_2(x)| \leq \max_{c \in [0, x]} \left| \frac{f^{(3)}(c)}{3!} x^3 \right|\), where \(f^{(3)}(x)\) is the third derivative of \(\ln(1+x)\).
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Taylor Series Approximation
A Taylor series approximates a function near a point by a polynomial whose terms involve derivatives of the function at that point. For small values of x near zero, ln(1 + x) can be approximated by the first few terms of its Taylor series, such as x − x²/2, simplifying calculations.
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Taylor Series
Error Bound in Taylor Approximations
The error bound quantifies the maximum difference between the actual function value and its Taylor polynomial approximation. It is often estimated using the remainder term, which depends on higher-order derivatives evaluated within the interval, ensuring the approximation's accuracy is understood.
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08:42Taylor Series
Properties of the Natural Logarithm Function
The natural logarithm function ln(1 + x) is defined for x > -1 and is smooth and differentiable near zero. Understanding its behavior and derivatives helps in constructing accurate approximations and estimating errors when x is close to zero.
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Percorso guidato
06:21Properties of Functions
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S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
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Domanda del libro di testo
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
b. Estimate f(0.2) and give a bound on the error in the approximation.
f(x) = sin x ≈ x
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Probability: sudden−death playoff Teams A and B go into suddendeath overtime after playing to a tie. The teams alternate possession of the ball, and the first team to score wins. Assume each team has a 1/6 chance of scoring when it has the ball, and Team A has the ball first.
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Sine integral function The function Si(x) = ∫₀ˣ f(t) dt, where f(t) = {(sin t)/t if t ≠ 0, 1 if t = 0, is called the sine integral function.
b. Integrate the series to find a Taylor series for Si.
120
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Domanda del libro di testo
Taylor series
b. Write the power series using summation notation.
f(x) = ln x, a = 3
110
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