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Find a Taylor series for f centered at 2 given that f⁽ᵏ⁾(2)=1, for all nonnegative integers k.
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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 11.2.39
Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
ƒ(x) = tan⁻¹(4x), a = 0
Guida verificata passo dopo passo1
Recall that the Taylor series of a function \( f(x) \) centered at \( a \) is given by \( f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x - a)^n \). Here, \( a = 0 \), so the series is centered at zero (Maclaurin series).
Identify the function: \( f(x) = \tan^{-1}(4x) \). We can use the known Maclaurin series for \( \tan^{-1}(x) = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1} \) for \( |x| < 1 \).
Substitute \( 4x \) in place of \( x \) in the series to get \( \tan^{-1}(4x) = \sum_{n=0}^\infty (-1)^n \frac{(4x)^{2n+1}}{2n+1} = \sum_{n=0}^\infty (-1)^n \frac{4^{2n+1} x^{2n+1}}{2n+1} \).
Write out the first three nonzero terms by plugging in \( n = 0, 1, 2 \):
- For \( n=0 \): \( \frac{4^1 x^1}{1} = 4x \)
- For \( n=1 \): \( - \frac{4^3 x^3}{3} = - \frac{64 x^3}{3} \)
- For \( n=2 \): \( \frac{4^5 x^5}{5} = \frac{1024 x^5}{5} \).
Finally, write the Taylor series in summation notation as \( \sum_{n=0}^\infty (-1)^n \frac{4^{2n+1}}{2n+1} x^{2n+1} \).
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Taylor Series Expansion
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. Each term involves the nth derivative evaluated at the center point, multiplied by (x - a)^n and divided by n!. This allows approximation of functions near the center point.
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Taylor Series
Derivatives of Inverse Trigonometric Functions
Understanding how to differentiate inverse trigonometric functions like arctan(x) is essential. For example, the derivative of arctan(x) is 1/(1 + x^2). When the function is arctan(4x), the chain rule applies, multiplying by the derivative of 4x, which is 4.
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06:35Derivatives of Other Inverse Trigonometric Functions
Summation Notation for Series
Summation notation concisely expresses infinite series using the sigma symbol (∑). It includes an index of summation, limits, and a general term formula. Writing the Taylor series in summation form captures all terms compactly and highlights the pattern in coefficients and powers.
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Percorso guidato
06:00Geometric Series
Pratica correlata
Domanda del libro di testo
Use of Tech Linear and quadratic approximation
a. Find the linear approximating polynomial for the following functions centered at the given point a.
b. Find the quadratic approximating polynomial for the following functions centered at a.
c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.
f(x) = 8x^(3/2), a=1; approximate 8 ⋅ 1.1^(3/2)
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{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.
ln 1.04, n=3
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Domanda del libro di testo
{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.
∫₀⁰ᐧ² sin x² dx
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Domanda del libro di testo
Derivative trick Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Then f⁽ᵏ⁾(a)=k! multiplied by the coefficient of (x−a)ᵏ. Use this idea to evaluate f⁽³⁾(0) and f⁽⁴⁾(0) for the following functions. Use known series and do not evaluate derivatives.
f(x) = eᶜᵒˢ ˣ
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Domanda del libro di testo
{Use of Tech} Approximating sin x Let f(x)=sin x, and let pₙ and qₙ be nth−order Taylor polynomials for f centered at 0 and π, respectively.
a. Find p₅ and q₅
b. Graph f, p₅, and q₅ on the interval [−π, 2π]. On what interval is p₅ a better approximation to f than q₅? On what interval is q₅ a better approximation to f than p₅?
c. Complete the following table showing the errors in the approximations given by p₅ and q₅ at selected points.
d. At which points in the table is p₅ a better approximation to f than q₅? At which points do p₅ and q₅ give equal approximations to f? Explain your observations.
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