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) = e⁻ˣ, a = 0
76
views
Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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 11.1.23
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.
Find the Taylor polynomial p₃ centered at a=e for f(x)=ln x.
Guida verificata passo dopo passo1
Recall that the Taylor polynomial of degree 3 for a function \( f(x) \) centered at \( a \) is given by:
\[ p_3(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 \]
This polynomial approximates \( f(x) \) near \( x = a \).
Identify the function and center: here, \( f(x) = \ln x \) and \( a = e \). We need to compute \( f(a) \), \( f'(a) \), \( f''(a) \), and \( f'''(a) \).
Calculate the derivatives of \( f(x) = \ln x \):
- First derivative: \( f'(x) = \frac{1}{x} \)
- Second derivative: \( f''(x) = -\frac{1}{x^2} \)
- Third derivative: \( f'''(x) = \frac{2}{x^3} \)
Evaluate each derivative at \( x = a = e \):
- \( f(e) = \ln e \)
- \( f'(e) = \frac{1}{e} \)
- \( f''(e) = -\frac{1}{e^2} \)
- \( f'''(e) = \frac{2}{e^3} \)
Substitute these values into the Taylor polynomial formula:
\[ p_3(x) = f(e) + f'(e)(x - e) + \frac{f''(e)}{2}(x - e)^2 + \frac{f'''(e)}{6}(x - e)^3 \]
This gives the cubic approximation of \( \ln x \) near \( x = e \).
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
4mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Taylor Polynomials
Taylor polynomials approximate a function near a point using derivatives at that point. The nth-degree Taylor polynomial uses derivatives up to order n to create a polynomial that closely matches the function's behavior near the center. For example, the third-degree Taylor polynomial p₃ at a point a approximates f(x) using f(a), f'(a), f''(a), and f'''(a).
Video consigliato:
07:00
Taylor Polynomials
Linear and Quadratic Approximations
Linear approximation uses the first-degree Taylor polynomial to estimate function values near a point, relying on the function's value and first derivative. Quadratic approximation extends this by including the second derivative, providing a more accurate estimate with a second-degree polynomial. These approximations simplify complex functions for easier calculation.
Video consigliato:
Percorso guidato
07:17Linearization
Derivatives of Logarithmic Functions
Understanding the derivatives of f(x) = ln(x) is essential for constructing Taylor polynomials. The first derivative is 1/x, the second is -1/x², and the third is 2/x³. Evaluating these at the center a = e allows building the polynomial terms needed for accurate approximations near that point.
Video consigliato:
05:18Derivative of the Natural Logarithmic Function
Pratica correlata
Domanda del libro di testo
{Use of Tech} Remainders Let
f(x) = ∑ₖ₌₀∞ xᵏ = 1/(1−x) and Sₙ(x) = ∑ₖ₌₀ⁿ⁻¹ xᵏ
The remainder in truncating the power series after n terms is Rₙ = f(x) − Sₙ(x), which depends on x.
a. Show that Rₙ(x) = xⁿ /(1−x).
b. Graph the remainder function on the interval |x| < 1, for n=1, 2, and 3 . Discuss and interpret the graph. Where on the interval is |Rₙ(x)| largest? Smallest?
c. For fixed n, minimize |Rₙ(x)| with respect to x. Does the result agree with the observations in part (b)?
d. Let N(x) be the number of terms required to reduce |Rₙ(x)| to less than 10⁻⁶. Graph the function N(x) on the interval |x|<1.
45
views
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 (ln (1 + t))/t dt
88
views
Domanda del libro di testo
Combining power series Use the geometric series
f(x) = 1/(1-x) = ∑ₖ₌₀∞ xᵏ, for |x| < 1,
to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series.
f(x³) = 1/(1 − x³)
59
views
Domanda del libro di testo
Suppose you use a second-order Taylor polynomial centered at 0 to approximate a function f. What matching conditions are satisfied by the polynomial?
76
views
Domanda del libro di testo
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.∫₀⁰ᐧ²⁵ e⁻ˣ² dx
106
views