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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 11.1.27
Chapter 11, Problem 11.1.27

{Use of Tech} Graphing Taylor polynomials


a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n=1 and n=2.


b. Graph the Taylor polynomials and the function.


f(x)=sin x, a=π/4

Verified step by step guidance
1
Identify the function and the center point: here, the function is \(f(x) = \sin x\) and the center is \(a = \frac{\pi}{4}\).
Calculate the derivatives of \(f(x)\) up to the second order because we need the Taylor polynomials for \(n=1\) and \(n=2\). The derivatives are: \(f'(x) = \cos x\), \(f''(x) = -\sin x\).
Evaluate the function and its derivatives at the center point \(a = \frac{\pi}{4}\): compute \(f\left(\frac{\pi}{4}\right)\), \(f'\left(\frac{\pi}{4}\right)\), and \(f''\left(\frac{\pi}{4}\right)\).
Write the Taylor polynomials using the formula for the \(n\)th-order Taylor polynomial centered at \(a\): \[T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x - a)^k\] Specifically, for \(n=1\) and \(n=2\), write out \(T_1(x)\) and \(T_2(x)\) using the values from the previous step.
To graph the function and the Taylor polynomials, plot \(f(x) = \sin x\) along with \(T_1(x)\) and \(T_2(x)\) on the same coordinate axes, focusing on the interval around \(a = \frac{\pi}{4}\) to observe how well the polynomials approximate the function near the center.

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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 a function near a point a using a finite sum of derivatives at that point. The nth-order Taylor polynomial includes terms up to the nth derivative, providing increasingly accurate approximations as n increases. For example, the first-order polynomial is a linear approximation, while the second-order includes curvature.
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Taylor Polynomials

Derivatives of Trigonometric Functions

Understanding the derivatives of sine and cosine functions is essential for constructing Taylor polynomials of sin(x). The derivatives cycle every four steps: sin(x), cos(x), -sin(x), -cos(x), then repeat. Evaluating these at the center point a allows calculation of polynomial coefficients.
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Derivatives of Other Inverse Trigonometric Functions

Graphing and Comparing Functions and Approximations

Graphing the original function alongside its Taylor polynomials helps visualize how well the polynomials approximate the function near the center point. This comparison illustrates the accuracy and limitations of the approximation, especially as the order n changes.
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Graph of Sine and Cosine Function
Related Practice
Textbook Question

Use of Tech Linear and quadratic approximation


a. Find the linear approximating polynomial for the following functions centered at the given point a.


b. Find the quadratic approximating polynomial for the following functions centered at a.


c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.


Find the Taylor polynomials p₁, p₂, and p₃ centered at a=1 for f(x)=x³.

Textbook Question

Representing functions by power series Identify the functions represented by the following power series.

∑ₖ₌₁∞ (x²ᵏ)/k

Textbook Question

How are the Taylor polynomials for a function f centered at a related to the Taylor series of the function f centered at a?

Textbook Question

{Use of Tech} Remainders Let 


f(x) = ∑ₖ₌₀∞ xᵏ = 1/(1−x) and Sₙ(x) = ∑ₖ₌₀ⁿ⁻¹ xᵏ


The remainder in truncating the power series after n terms is Rₙ = f(x) − Sₙ(x), which depends on x.


a. Show that Rₙ(x) = xⁿ /(1−x).

b. Graph the remainder function on the interval |x| < 1, for n=1, 2, and 3 . Discuss and interpret the graph. Where on the interval is |Rₙ(x)| largest? Smallest?

c. For fixed n, minimize |Rₙ(x)| with respect to x. Does the result agree with the observations in part (b)?

d. Let N(x) be the number of terms required to reduce |Rₙ(x)| to less than 10⁻⁶. Graph the function N(x) on the interval |x|<1.

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Textbook Question

{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.

∫₀⁰ᐧ² (ln (1 + t))/t dt

Textbook Question

Combining power series Use the geometric series


f(x) = 1/(1-x) = ∑ₖ₌₀∞ xᵏ, for |x| < 1,


to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series.


g(x) = x³/(1 − x)

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