Problem 15 - Calculus

Question

expand the quantity about 0 in terms of the variable given. Give four nonzero terms. $$\sqrt{T+h} \text { in terms of } \frac{h}{T}$$

Step-by-Step Solution

Answer

The expansion in terms of \( \frac{h}{T} \) is: \( \sqrt{T} \left( 1 + \frac{1}{2} \frac{h}{T} - \frac{1}{8} \left(\frac{h}{T}\right)^2 + \frac{1}{16} \left(\frac{h}{T}\right)^3 \right) \).

Step 1: Identify the Function and the Expansion Point

We need to expand the function \( \sqrt{T+h} \) about \( h = 0 \). We will use the Taylor series expansion method, focusing on expressing this function in terms of \( \frac{h}{T} \).

Step 2: Set up the Taylor Series Expansion for the Function

The Taylor series for \( \sqrt{x} \) about \( x = a \) is given by \( \sqrt{a} + \frac{1}{2\sqrt{a}}(x-a) - \frac{1}{8a^{3/2}}(x-a)^2 + \cdots \). Here, we set \( a = T \) and \( x = T+h \).

Step 3: Substitute and Expand

Substitute \( a = T \) and \( x = T+h \) into the Taylor series formula: \[ \sqrt{T+h} = \sqrt{T} + \frac{1}{2\sqrt{T}}h - \frac{1}{8T^{3/2}}h^2 + \frac{1}{16T^{5/2}}h^3 + \cdots \].

Step 4: Express in Terms of \( \frac{h}{T} \)

To express the series in terms of \( \frac{h}{T} \), factor out appropriate powers of \( T \) from each term: \[ \sqrt{T+h} = \sqrt{T} \left( 1 + \frac{1}{2} \frac{h}{T} - \frac{1}{8} \left(\frac{h}{T}\right)^2 + \frac{1}{16} \left(\frac{h}{T}\right)^3 + \cdots \right) \].

Step 5: Collect the First Four Nonzero Terms

The first four nonzero terms of the expansion in terms of \( \frac{h}{T} \) are: \( \sqrt{T} \), \( \frac{1}{2} \frac{h}{T} \cdot \sqrt{T} \), \(- \frac{1}{8} \left(\frac{h}{T}\right)^2 \cdot \sqrt{T} \), and \( \frac{1}{16} \left(\frac{h}{T}\right)^3 \cdot \sqrt{T} \).

Key Concepts

Square Root Function

In mathematics, understanding the square root function is essential as it appears in various fields, ranging from geometry to algebra. The square root function is denoted as \( \sqrt{x} \), which represents a number that, when multiplied by itself, yields \( x \). This function has unique properties that make it both useful and challenging to work with. For instance, the square root function is defined only for non-negative numbers in real number systems, which means the input \( x \) must be zero or positive.

The derivative of the square root function \( \sqrt{x} \) is an important aspect because it indicates the rate of change of the function. This derivative can be found using basic calculus rules and is given by \( \frac{1}{2\sqrt{x}} \). Here's why it's important: understanding the behavior of the curve helps when performing expansions, such as using Taylor series.

Thus, to effectively expand a function like \( \sqrt{T+h} \), knowing these properties is crucial as it provides a foundation for further mathematical operations. These help break down more complex expressions into manageable parts.

Expansion about a Point

The expansion of a function about a particular point is a powerful technique used to approximate non-linear functions with a series of terms. Taylor series, one of the most popular methods, helps in expanding functions around a given point. The general form of a Taylor series expansion for a function \( f(x) \) about a point \( a \) is:

• \( f(a) \)
• \( f'(a)(x-a) \)
• \( \frac{f''(a)}{2!}(x-a)^2 \)
• \( \frac{f'''(a)}{3!}(x-a)^3 \)
• And so on...

By utilizing these components, it's possible to represent functions like \( \sqrt{T+h} \) in simpler terms based on increments from a known value.

When expanding the function \( \sqrt{T+h} \) about \( h = 0 \), you use \( T \) as the baseline or the constant part of the function, allowing you to see how small changes in \( h \) impact the function's value. The goal is to express the function with reduced complexity while maintaining accuracy for small \( h \) values.

Terms of a Series

When breaking down functions into series, the individual terms provide insights into how the function behaves around the expansion point. Each term of the Taylor series indicates a specific level of detail or accuracy in the approximation.

And so, when expanding \( \sqrt{T+h} \), each term corresponds to a degree of \( h \). For example:

• The first term \( \sqrt{T} \) represents the value of the function at the point of expansion's baseline.
• The second term, \( \frac{1}{2} \frac{h}{T} \cdot \sqrt{T} \), indicates how the function changes initially as \( h \) increases.
• Subsequent terms like \(- \frac{1}{8} \left(\frac{h}{T}\right)^2 \cdot \sqrt{T} \) and \( \frac{1}{16} \left(\frac{h}{T}\right)^3 \cdot \sqrt{T} \) provide higher-order corrections, adding more layers of precision.

These terms form an infinite series, but usually, only the first few are necessary to create a useful approximation.

Understanding which terms to retain and which to ignore is crucial. Often, in practical applications, higher-order terms (especially where \( h \) is small) have negligible effects and can be left out without sacrificing much accuracy. This highlight underscores the balance between computational efficiency and functional accuracy.