Textbook Question
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x)=3ˣ, a=0
Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 11, Problem 11.1.76b
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
b. Estimate f(0.2) and give a bound on the error in the approximation.
f(x) = sin ⁻¹ x ≈ x
Verified step by step guidance1
Identify the function and the approximation given: the function is \(f(x) = \sin^{-1}(x)\), and the approximation near zero is \(f(x) \approx x\).
To estimate \(f(0.2)\) using the approximation, substitute \(x = 0.2\) into the approximation: \(f(0.2) \approx 0.2\).
To find a bound on the error, recall that the error in a linear approximation can be estimated using the remainder term from Taylor's theorem. For \(\sin^{-1}(x)\) expanded at 0, the next term after \(x\) involves \(x^3\).
The second derivative of \(f(x) = \sin^{-1}(x)\) is \(f''(x) = \frac{x}{(1 - x^2)^{3/2}}\). Use this to find a bound on the error by evaluating or bounding \(|f''(c)|\) for some \(c\) between 0 and 0.2.
Use the Lagrange form of the remainder: the error \(R_2\) satisfies \(|R_2| \leq \frac{M}{3!} |x|^3\), where \(M\) is the maximum value of \(|f''(c)|\) on the interval \([0, 0.2]\). Calculate or estimate \(M\) and then compute the error bound.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Small Angle Approximations
Small angle approximations simplify functions like sin⁻¹(x) near zero by approximating them with simpler expressions, such as sin⁻¹(x) ≈ x. This is valid because the function and its approximation have the same value and slope at zero, making calculations easier for small x values.
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Determining Error and Relative Error
Error Bound Using Taylor's Remainder Theorem
Taylor's Remainder Theorem provides a way to estimate the error when approximating a function by a polynomial. It bounds the difference between the actual function value and the approximation using the next derivative term, ensuring the approximation's accuracy is quantifiable.
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Inverse Sine Function Properties
The inverse sine function, sin⁻¹(x), is continuous and differentiable near zero, with its derivative at zero equal to 1. Understanding these properties helps justify the linear approximation and calculate error bounds based on higher derivatives.
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Related Practice
Textbook Question
{Use of Tech} Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is
J₀(x) = ∑ₖ₌₀∞ (−1)ᵏ/(2²ᵏ(k!)²) x²ᵏ
b. Find the radius and interval of convergence of the power series for J₀.
Textbook Question
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = ln (x − 2), a = 3
Textbook Question
Probability: sudden−death playoff Teams A and B go into suddendeath overtime after playing to a tie. The teams alternate possession of the ball, and the first team to score wins. Assume each team has a 1/6 chance of scoring when it has the ball, and Team A has the ball first.
b. The expected number of rounds (possessions by either team) required for the overtime to end is (1/6) ∑ₖ₌₁∞ k(5/6)ᵏ⁻¹. Evaluate this series.
Textbook Question
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = e²ˣ, a = 0
Textbook Question
{Use of Tech} Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is
J₀(x) = ∑ₖ₌₀∞ (−1)ᵏ/(2²ᵏ(k!)²) x²ᵏ
a. Write out the first four terms of J₀.