Domanda del libro di testo
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = cosh 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 11.1.76b
{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 ⁻¹ x ≈ x
Guida verificata passo dopo passo1
Identify the function and the approximation given: the function is \(f(x) = \sin^{-1}(x)\), and the approximation near zero is \(f(x) \approx x\).
To estimate \(f(0.2)\) using the approximation, substitute \(x = 0.2\) into the approximation: \(f(0.2) \approx 0.2\).
To find a bound on the error, recall that the error in a linear approximation can be estimated using the remainder term from Taylor's theorem. For \(\sin^{-1}(x)\) expanded at 0, the next term after \(x\) involves \(x^3\).
The second derivative of \(f(x) = \sin^{-1}(x)\) is \(f''(x) = \frac{x}{(1 - x^2)^{3/2}}\). Use this to find a bound on the error by evaluating or bounding \(|f''(c)|\) for some \(c\) between 0 and 0.2.
Use the Lagrange form of the remainder: the error \(R_2\) satisfies \(|R_2| \leq \frac{M}{3!} |x|^3\), where \(M\) is the maximum value of \(|f''(c)|\) on the interval \([0, 0.2]\). Calculate or estimate \(M\) and then compute the error bound.
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Small Angle Approximations
Small angle approximations simplify functions like sin⁻¹(x) near zero by approximating them with simpler expressions, such as sin⁻¹(x) ≈ x. This is valid because the function and its approximation have the same value and slope at zero, making calculations easier for small x values.
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Percorso guidato
04:57
Determining Error and Relative Error
Error Bound Using Taylor's Remainder Theorem
Taylor's Remainder Theorem provides a way to estimate the error when approximating a function by a polynomial. It bounds the difference between the actual function value and the approximation using the next derivative term, ensuring the approximation's accuracy is quantifiable.
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08:42Taylor Series
Inverse Sine Function Properties
The inverse sine function, sin⁻¹(x), is continuous and differentiable near zero, with its derivative at zero equal to 1. Understanding these properties helps justify the linear approximation and calculate error bounds based on higher derivatives.
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4:03Inverse Sine
Pratica correlata
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) = tan x ≈ x
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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.
b. Write the power series using summation notation.
f(x) = 1/x², a=1
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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.
b. The function f(x) = csc x has a Taylor series centered at π/2.
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Domanda del libro di testo
Taylor series
b. Write the power series using summation notation.
f(x)=sin x, a = π/2
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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.
b. Let f(x)=x⁵−1 The Taylor polynomial for f of order 10 centered at 0 is f itself.
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