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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.8.41a
Chapter 10, Problem 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)

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1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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
Related Practice
Textbook Question

Intervals of Convergence

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

∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]

Textbook Question

Power Series

In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.

∑ (from n = 1 to ∞) xⁿ/nⁿ

Textbook Question

Determining Convergence of Sequences

Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.


aₙ = 1 + (0.9)ⁿ

1
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Textbook Question

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)

Textbook Question

The series

eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + x⁵/5! + ⋯

converges to eˣ for all x.

a. Find a series for (d/dx)eˣ. Do you get the series for eˣ? Explain your answer.

Textbook Question

Intervals of Convergence

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

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