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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 11 - Power SeriesProblema 11.1.5
Capitolo 11, Problema 11.1.5

Suppose f(0)=1, f'(0)=0, f''(0)=2, and f⁽³⁾(0)=6. Find the third-order Taylor polynomial for f centered at 0 and use it to approximate f(0.2).

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Recall the formula for the third-order Taylor polynomial of a function \( f \) centered at \( a = 0 \): \[ P_3(x) = f(0) + f'(0) x + \frac{f''(0)}{2!} x^2 + \frac{f^{(3)}(0)}{3!} x^3 \]
Substitute the given values into the polynomial: \[ f(0) = 1, \quad f'(0) = 0, \quad f''(0) = 2, \quad f^{(3)}(0) = 6 \] So, \[ P_3(x) = 1 + 0 \cdot x + \frac{2}{2} x^2 + \frac{6}{6} x^3 \]
Simplify the coefficients in the polynomial: \[ P_3(x) = 1 + x^2 + x^3 \]
To approximate \( f(0.2) \), substitute \( x = 0.2 \) into the polynomial: \[ P_3(0.2) = 1 + (0.2)^2 + (0.2)^3 \]
Calculate the powers of \( 0.2 \) (without final summation) to prepare for approximation: \[ (0.2)^2 = 0.04, \quad (0.2)^3 = 0.008 \] Then, sum these terms with 1 to get the approximate value of \( f(0.2) \).

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Taylor Polynomial

A Taylor polynomial approximates a function near a specific point using derivatives at that point. The nth-order Taylor polynomial includes terms up to the nth derivative, providing a polynomial that closely matches the function's behavior near the center.
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Derivatives and Their Role in Taylor Series

Derivatives at the center point determine the coefficients of the Taylor polynomial. Each term involves the function's nth derivative evaluated at the center, divided by n! and multiplied by (x - center)^n, capturing the function's local rate of change.
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Polynomial Approximation and Evaluation

Once the Taylor polynomial is constructed, it can be used to approximate function values near the center by substituting the desired x-value. This provides an estimate that becomes more accurate with higher-order polynomials and points closer to the center.
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L'Hôpital's Rule by Taylor series Suppose f and g have Taylor series about the point a.

a. If f(a) = g(a) = 0 and g′(a) ≠ 0, evaluate lim ₓ→ₐ f(x)/g(x) by expanding f and g in their Taylor series. Show that the result is consistent withl’Hôpital’s Rule.

b. If f(a) = g(a) =f′(a) = g′(a) = 0 and g′′(a) ≠ 0, evaluate lim ₓ→ₐ f(x)/g(x) by expanding f and g in their Taylor series. Show that the result is consistent with two applications of 1'Hôpital's Rule.

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b. Identify the function represented by the differentiated series.

c. Give the interval of convergence of the power series for the derivative.


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