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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 11 - Power SeriesProblem 11.1.37
Chapter 11, Problem 11.1.37

{Use of Tech} Approximations with Taylor polynomials


a. Approximate the given quantities using Taylor polynomials with n = 3.


b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.


√1.06

Verified step by step guidance
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Identify the function to approximate: here, we want to approximate \( f(x) = \sqrt{x} \) near a point where the function and its derivatives are easy to compute. A good choice is \( a = 1 \) because \( \sqrt{1} = 1 \).
Write the Taylor polynomial of degree 3 for \( f(x) \) centered at \( a = 1 \). The general formula is: \[ T_3(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 \] Calculate the first, second, and third derivatives of \( f(x) = \sqrt{x} = x^{1/2} \).
Evaluate each derivative at \( x = 1 \) to find \( f(1), f'(1), f''(1), \) and \( f'''(1) \). Substitute these values into the Taylor polynomial formula.
Substitute \( x = 1.06 \) into the Taylor polynomial \( T_3(x) \) to approximate \( \sqrt{1.06} \). This gives the approximate value using the third-degree Taylor polynomial.
To find the absolute error, calculate the exact value of \( \sqrt{1.06} \) using a calculator, then subtract the Taylor polynomial approximation from this exact value. The absolute error is: \[ \text{Absolute Error} = |\sqrt{1.06} - T_3(1.06)| \]

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Key Concepts

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

Taylor Polynomials

Taylor polynomials approximate functions near a point by using derivatives at that point. For n=3, the polynomial includes terms up to the cubic degree, providing a close estimate of the function's value near the expansion point.
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Taylor Polynomials

Error Estimation in Approximations

The absolute error measures the difference between the exact value and the approximation. Calculating this helps assess the accuracy of the Taylor polynomial approximation.
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Determining Error and Relative Error

Function Expansion Point and Domain

Choosing the expansion point (often near the value to approximate) is crucial for accuracy. Understanding the domain and behavior of the function, like √x near x=1, ensures the Taylor polynomial converges well.
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Finding the Domain and Range of a Graph
Related Practice
Textbook Question

Use the Taylor series for cos x centered at 0 to verify that lim ₓ→ₐ (1− cos x)/x = 0.

Textbook Question

{Use of Tech} Approximations with Taylor polynomials


a. Approximate the given quantities using Taylor polynomials with n = 3.


b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.


e⁰ᐧ¹²

Textbook Question

Suppose a power series converges if |x−3|<4 and diverges if |x−3| ≥ 4. Determine the radius and interval of convergence.

Textbook Question

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.


∑ₖ₌₀∞ (x/3)ᵏ

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Textbook Question

Evaluating an infinite series Let f(x) = (eˣ − 1)/x, for x ≠ 0, and f(0)=1. Use the Taylor series for f centered at 0 to evaluate f(1) and to find the value of ∑ₖ₌₀∞ 1/(k+1)!

Textbook Question

{Use of Tech} Approximating sin x Let f(x)=sin x, and let pₙ and qₙ be nth−order Taylor polynomials for f centered at 0 and π, respectively.

a. Find p₅ and q₅

b. Graph f, p₅, and q₅ on the interval [−π, 2π]. On what interval is p₅ a better approximation to f than q₅? On what interval is q₅ a better approximation to f than p₅?

c. Complete the following table showing the errors in the approximations given by p₅ and q₅ at selected points.

d. At which points in the table is p₅ a better approximation to f than q₅? At which points do p₅ and q₅ give equal approximations to f? Explain your observations.

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