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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 11.1.83
Chapter 11, Problem 11.1.83

Tangent line is p₁ Let f be differentiable at x=a


a. Find the equation of the line tangent to the curve y=f(x) at (a, f(a)).


b. Verify that the Taylor polynomial p_1 centered at a describes the tangent line found in part (a).

Verified step by step guidance
1
Recall that the equation of the tangent line to the curve \(y = f(x)\) at the point \((a, f(a))\) can be found using the point-slope form of a line. The slope of this tangent line is given by the derivative of \(f\) at \(x = a\), which is \(f'(a)\).
Write the equation of the tangent line using the point-slope form: \[y - f(a) = f'(a)(x - a)\] This expresses the line passing through \((a, f(a))\) with slope \(f'(a)\).
Recall the first-degree Taylor polynomial (linear approximation) of \(f\) centered at \(a\) is given by: \[p_1(x) = f(a) + f'(a)(x - a)\] This polynomial approximates \(f(x)\) near \(x = a\).
Compare the equation of the tangent line from part (a) with the Taylor polynomial \(p_1(x)\). Notice that both have the same form, representing a line with slope \(f'(a)\) passing through \((a, f(a))\).
Conclude that the Taylor polynomial \(p_1(x)\) centered at \(a\) indeed describes the tangent line to the curve \(y = f(x)\) at the point \((a, f(a))\), verifying the relationship between the two.

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Key Concepts

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

Derivative and Differentiability

The derivative of a function at a point measures the instantaneous rate of change or slope of the function's graph at that point. Differentiability at x = a means the derivative f'(a) exists, allowing us to find the slope of the tangent line to the curve y = f(x) at (a, f(a)).
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Equation of the Tangent Line

The tangent line to the curve y = f(x) at x = a is a straight line that touches the curve at (a, f(a)) with slope f'(a). Its equation is given by y = f(a) + f'(a)(x - a), representing the best linear approximation of the function near x = a.
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Taylor Polynomial of Degree One (Linear Approximation)

The first-degree Taylor polynomial p₁ centered at a approximates the function near x = a using p₁(x) = f(a) + f'(a)(x - a). This polynomial matches the value and slope of f at a, thus describing the tangent line and providing a linear approximation of the function.
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Related Practice
Textbook Question

Evaluating an infinite series Write the Maclaurin series for f(x) = ln (1+x) and find the interval of convergence. Evaluate f(−1/2) to find the value of ∑ₖ₌₁∞ 1/(k 2ᵏ)

Textbook Question

Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series


(1 + x)⁻² = 1 − 2x + 3x² − 4x³ + ⋯, for −1 < x < 1.


1/(3 + 4x)²

Textbook Question

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


∑ₖ₌₁∞ ((−1)ᵏ⁺¹(x−1)ᵏ)/k

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Textbook Question

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.

tan ⁻¹ (1/2)

Textbook Question

Power series for derivatives


a. Differentiate the Taylor series centered at 0 for the following functions.

b. Identify the function represented by the differentiated series.

c. Give the interval of convergence of the power series for the derivative.


f(x) = ln (1 + x)

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Textbook Question

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.∫₀⁰ᐧ²⁵ e⁻ˣ² dx

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