Approximate the specified function value as indicated and...

Approximate the specified function value as indicated and check your work by comparing your answer to the function value produced directly by your calculating utility.
Approximate sinh 0.5 to three decimal-place accuracy using the Maclaurin series for $\sinh x .$

Key Concepts

Maclaurin Series

The Maclaurin series is a special case of the Taylor series expansion centered at zero. It expresses a function as an infinite sum of its derivatives at zero, each multiplied by a corresponding power of the variable divided by the factorial of the term's degree. This series is a powerful tool for approximating functions, especially when evaluating them at points where direct computation might be complicated.

Hyperbolic Sine Function

The hyperbolic sine function, denoted as sinh(x), is defined as (e^x - e^(-x))/2. It is one of the key hyperbolic functions and has properties analogous to the trigonometric sine function. Its Maclaurin series expansion involves only the odd-powered terms of x, making it particularly amenable to series approximation methods.

Series Approximation

Series approximation involves truncating an infinite series after a finite number of terms to estimate the value of a function. When using the Maclaurin series for functions like sinh(x), the approximation improves as more terms are included. Determining the number of terms needed to achieve a specific accuracy, such as three-decimal-place accuracy, is a common part of numerical analysis.

Numerical Verification

Numerical verification refers to the process of validating the approximation obtained through the series expansion by comparing it to the value computed using a direct calculation method, such as a calculator or a computer algebra system. This step is essential to ensure that the truncation error in the series does not exceed the desired level of accuracy.

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