Ch. 10 - Infinite Sequences and Series

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 10 - Infinite Sequences and Series

Problema 10.8.41a

Capitolo 10, Problema 10.8.41a

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function f(x) at x = a is called the quadratic approximation of f at x = a. In Exercises 41–46, find the (a) linearization (Taylor polynomial of order 1)
f(x) = ln(cos x)

Guida verificata passo dopo passo

Identify the point at which the Taylor polynomial is to be generated, denoted as \(a\). This is the center of the approximation.

Recall that the linearization (Taylor polynomial of order 1) of a function \(f(x)\) at \(x = a\) is given by the formula: \[L(x) = f(a) + f'(a)(x - a)\]

Calculate the value of the function at \(x = a\): \[f(a) = \ln(\cos a)\]

Find the first derivative of the function \(f(x) = \ln(\cos x)\). Use the chain rule: \[f'(x) = \frac{d}{dx} \ln(\cos x) = \frac{1}{\cos x} \cdot (-\sin x) = -\tan x\]

Evaluate the first derivative at \(x = a\): \[f'(a) = -\tan a\] Then substitute \(f(a)\) and \(f'(a)\) into the linearization formula to write the linear approximation.

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Taylor Polynomial

A Taylor polynomial approximates a function near a point using derivatives at that point. The polynomial of order n uses derivatives up to the nth order to create a polynomial that closely matches the function's behavior near the chosen point.

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Taylor Polynomials

Linearization (Taylor Polynomial of Order 1)

Linearization is the first-order Taylor polynomial, which approximates a function near a point using the function's value and its first derivative at that point. It provides a linear function that is tangent to the original function at the chosen point.

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Taylor Polynomials

Derivatives of Composite Functions

To find the Taylor polynomial of f(x) = ln(cos x), you must compute derivatives involving the chain rule, since f is a composition of ln and cos. Understanding how to differentiate composite functions is essential for finding accurate polynomial approximations.

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Evaluate Composite Functions - Special Cases

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∑ (from n=1 to ∞) (1 / √(n + 1)) diverges

b. What should n be in order that the partial sum sₙ = ∑ (from i=1 to n) (1 / √(i + 1)) satisfies sₙ > 1000?

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Assume that bₙ is a sequence of positive numbers converging to 4/5. Determine if the following series converge or diverge.

b. ∑ (from n = 1 to ∞) (5/4)ⁿ (bₙ)

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Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function f(x) at x = a is called the quadratic approximation of f at x = a. In Exercises 41–46, find the (a) linearization (Taylor polynomial of order 1)

f(x) = 1 / √(1 − x²)

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Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence.

∑ (from n = 1 to ∞) [ (√(n + 1) − √n)(x − 3)ⁿ ]

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Intervals of Convergence

In Exercises 1–36, for what values of x does the series converge (b) absolutely?

∑ (from n = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]

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