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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 10.8.43a
Chapter 10, Problem 10.8.43a

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)

Verified step by step guidance
1
Identify the point at which the Taylor polynomial is to be generated. Since the problem does not specify a value for \( a \), assume it is at \( a = 0 \) unless otherwise stated.
Recall the formula for the Taylor polynomial of order 1 (linearization) of a function \( f(x) \) at \( x = a \): \[ L(x) = f(a) + f'(a)(x - a) \]
Calculate \( f(a) \) by substituting \( x = a \) into the function \( f(x) = \frac{1}{\sqrt{1 - x^2}} \).
Find the first derivative \( f'(x) \) using the chain rule. For \( f(x) = (1 - x^2)^{-1/2} \), differentiate to get \( f'(x) = \frac{d}{dx} (1 - x^2)^{-1/2} \).
Evaluate \( f'(a) \) by substituting \( x = a \) into the derivative expression, then write the linearization \( L(x) = f(a) + f'(a)(x - a) \) explicitly.

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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 approximation useful for estimating function values close to the point.
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Taylor Polynomials

Derivatives of the Function f(x) = 1 / √(1 − x²)

To find Taylor polynomials, you need the function's derivatives at the point of approximation. For f(x) = 1/√(1−x²), calculating the first and second derivatives involves applying the chain and power rules carefully due to the square root and the composite function.
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Derivative of the Natural Exponential Function (e^x)
Related Practice
Textbook Question

Assume that bₙ is a sequence of positive numbers converging to 1/3. Determine if the following series converge or diverge.

a. ∑ (from n = 1 to ∞) [(bₙ₊₁ + bₙ) / n 4ⁿ]

1
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Textbook Question

Assume that the series ∑ aₙ(x − 2)ⁿ converges for x = −1 and diverges for x = 6. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.

a. Converges absolutely for x = 1

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) = ln(cos x)

Textbook Question

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 ]

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 ]