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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 11 - Power SeriesProblema 11.1.65d
Capitolo 11, Problema 11.1.65d

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. Suppose f'' is continuous on an interval that contains a, where f has an inflection point at a. Then the second−order Taylor polynomial for f at a is linear.

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Recall that an inflection point at \(x = a\) means the concavity of the function changes at \(a\), which implies that the second derivative \(f''(a) = 0\) if \(f''\) is continuous around \(a\).
The second-order Taylor polynomial for \(f\) at \(a\) is given by: \[T_2(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2\]
Since \(f''(a) = 0\) at the inflection point, the quadratic term in the Taylor polynomial disappears, leaving: \[T_2(x) = f(a) + f'(a)(x - a)\]
This polynomial is linear in \((x - a)\) because it only contains terms up to the first power of \((x - a)\).
Therefore, the second-order Taylor polynomial at an inflection point where \(f''\) is continuous is indeed linear, confirming the statement.

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Concetti chiave

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Inflection Point

An inflection point of a function f is a point where the concavity changes, meaning f'' changes sign. At this point, the second derivative f''(a) is typically zero or undefined, indicating a transition between concave up and concave down behavior.
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Critical Points

Second-Order Taylor Polynomial

The second-order Taylor polynomial of a function f at a point a approximates f near a using terms up to the second derivative: P_2(x) = f(a) + f'(a)(x - a) + (f''(a)/2)(x - a)^2. Its degree depends on whether f''(a) is zero or not.
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Taylor Polynomials

Continuity of the Second Derivative

If f'' is continuous near a, then f''(a) exists and is finite. For an inflection point at a with continuous f'', f''(a) = 0, which affects the form of the Taylor polynomial by eliminating the quadratic term, making it linear.
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The Second Derivative Test: Finding Local Extrema
Pratica correlata
Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. If f has a Taylor series that converges only on (−2,2), then f(x²) has a Taylor series that also converges only on (−2,2).

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{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals

S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt

d. How many terms of the Maclaurin series are required to approximate S(0.05) with an error no greater than 10⁻⁴?

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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.

d. If p(x) is the Taylor series for f centered at 0, then p(x−1) is the Taylor series for f centered at 1.

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Taylor series and interval of convergence


c. Determine the interval of convergence of the series.


f(x)=2/(1−x)³, a=0

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Domanda del libro di testo

Matching functions with polynomials Match functions a–f with Taylor polynomials A–F (all centered at 0). Give reasons for your choices.


d. 1/(1 + 2x)


A. p₂(x)= 1 + 2x + 2x²

B. p₂(x) = 1 − 6x + 24x²

C. p₂(x) = 1 + x − x²/2

D. p₂(x) = 1 − 2x + 4x²

E. p₂(x) = 1 − x + (3/2)x²

F. p₂(x) = 1 − 2x + 2x²

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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.


If f(x)=∑ₖ₌₀∞ cₖ xᵏ=0, for all x on an interval (−a, a), then cₖ = 0, for all k.

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