Question

Alternative definitions of means Consider the function $$f(t)=\frac{\int_{a}^{b} x^{t+1} d x}{\int_{a}^{b} x^{t} d x}$$ Show that the following means can be defined in terms of \(f\) a. Arithmetic mean: \(f(0)=\frac{a+b}{2}\) b. Geometric mean: \(f\left(-\frac{3}{2}\right)=\sqrt{a b}\) c. Harmonic mean: \(f(-3)=\frac{2 a b}{a+b}\) d. Logarithmic mean: \(f(-1)=\frac{b-a}{\ln b-\ln a}\)

Short answer

Question: Show that the function \(f(t)\) defined by $$f(t) = \frac{\int_{a}^{b} x^{t+1} d x}{\int_{a}^{b} x^{t} d x}$$ can be used to define the following means of two numbers \(a\) and \(b\): 1. Arithmetic mean, \(f(0)\) 2. Geometric mean, \(f\left(-\frac{3}{2}\right)\) 3. Harmonic mean, \(f(-3)\) 4. Logarithmic mean, \(f(-1)\) Answer: By evaluating the function \(f(t)\) at the given values of \(t\), we found that: 1. \(f(0) = \frac{a + b}{2}\), which is the definition of the arithmetic mean. 2. \(f\left(-\frac{3}{2}\right) = \sqrt{ab}\), which is the definition of the geometric mean. 3. \(f(-3) = \frac{2ab}{a + b}\), which is the definition of the harmonic mean. 4. \(f(-1) = \frac{b - a}{\ln b - \ln a}\), which is the definition of the logarithmic mean. Therefore, the function \(f(t)\) can be used to define the specified means of two numbers \(a\) and \(b\) when evaluated at the given values of \(t\).

Step-by-step solution

Calculate the integrals for the function \(f(t)\)

To evaluate \(f(t)\) at different values of \(t\), we first need to find the general expressions for the integrals in the numerator and denominator: For the numerator, we have $$\int_{a}^{b} x^{t+1} d x$$ To find this integral, we can use the power rule, which states that $$\int x^p d x = \frac{x^{p + 1}}{p + 1} + C$$ where \(p\) is a constant and \(C\) is an arbitrary constant of integration. Using this rule, we get $$\int_{a}^{b} x^{t+1} d x = \frac{x^{t + 2}}{t + 2}\bigg|_{a}^{b} = \frac{b^{t+2}-a^{t+2}}{t+2}$$ For the denominator, we have $$\int_{a}^{b} x^{t} d x$$ Applying the power rule again, $$\int_{a}^{b} x^{t} d x = \frac{x^{t + 1}}{t + 1}\bigg|_{a}^{b} = \frac{b^{t+1}-a^{t+1}}{t+1}$$ Now, we can write \(f(t)\) as $$f(t) = \frac{\frac{b^{t+2}-a^{t+2}}{t+2}}{\frac{b^{t+1}-a^{t+1}}{t+1}}$$

Calculate \(f(t)\) for given values of \(t\) and compare with definitions of means

Now, we evaluate \(f(t)\) at the given values of \(t\) and compare the results with the provided definitions of different means: a. Arithmetic mean, \(f(0)\): $$f(0) = \frac{\frac{b^2-a^2}{2}}{\frac{b - a}{1}} = \frac{(b - a)(b + a)}{2(b - a)} = \frac{a + b}{2}$$ This is the definition of the arithmetic mean. b. Geometric mean, \(f\left(-\frac{3}{2}\right)\): $$f\left(-\frac32\right) = \frac{\frac{b^{\frac12} - a^{\frac12}}{\frac12}}{\frac{b^{-\frac12} - a^{-\frac12}}{-\frac12}} = \frac{b^{\frac12} - a^{\frac12}}{a^{-\frac12} - b^{-\frac12}} = \frac{\sqrt{b} - \sqrt{a}}{\frac{1}{\sqrt{a}} - \frac{1}{\sqrt{b}}} = \sqrt{ab}$$ This is the definition of the geometric mean. c. Harmonic mean, \(f(-3)\): $$f(-3) = \frac{\frac{b^{-1}-a^{-1}}{-2}}{\frac{b^{-2}-a^{-2}}{-3}} = \frac{\frac{1}{a} - \frac{1}{b}}{\frac{1}{a^2} - \frac{1}{b^2}} = \frac{\frac{b - a}{ab}}{\frac{b - a}{a^2b^2}} = \frac{2ab}{a + b}$$ This is the definition of the harmonic mean. d. Logarithmic mean, \(f(-1)\): $$f(-1) = \frac{\frac{\ln b - \ln a}{0}}{\frac{b^{-1}-a^{-1}}{-2}} = \frac{\ln \frac{b}{a}}{\frac{1}{a} - \frac{1}{b}} = \frac{\ln \frac{b}{a}}{\frac{b - a}{ab}} = \frac{b - a}{\ln b - \ln a}$$ This is the definition of the logarithmic mean. We have now shown that the specific values of \(t\) in the function \(f(t)\) correspond to the definitions of the different means as required.

Key concepts

Arithmetic Mean

The arithmetic mean, often simply called the average, is one of the most common and easily understood measures of central tendency. To compute the arithmetic mean of two numbers, you sum them up and divide by two. This is encapsulated in the formula: \( \text{Arithmetic Mean} = \frac{a+b}{2} \).

• Simplicity: The arithmetic mean is straightforward and provides a quick summary of a data set.
• Common Usage: It is used in various fields such as economics, statistics, and everyday decision-making.
• Understanding: If numbers are all the same, the arithmetic mean will equal to the numbers themselves, showing it as a reliable center measure.

Additionally, in calculus, the arithmetic mean emerges when evaluating certain integrals. In the given context, it's demonstrated by setting the function \( f(t) \) to \( f(0) \), producing the familiar result.

Geometric Mean

The geometric mean is critical when dealing with exponential growth or multiplicative processes, like in finance when calculating average growth rates. It is defined as the square root of the product of two numbers, presented as \( \text{Geometric Mean} = \sqrt{ab} \).

• Applicability: It is useful when comparing different items with vastly different ranges.
• Fair Representation: Unlike the arithmetic mean, it reduces the impact of extreme values.
• Proportional Relationships: Particularly useful in scenarios involving ratios and rates.

When the function \( f(t) \) is evaluated at \( t = -\frac{3}{2} \), it reflects the definition of the geometric mean, making it a valuable tool in certain integration analyses.

Harmonic Mean

The harmonic mean offers a different perspective, emphasizing smaller numbers in a data set. It's calculated as the reciprocal of the average of reciprocals: \( \text{Harmonic Mean} = \frac{2ab}{a+b} \).

• Usefulness: It is especially potent in situations like averaging speeds or rates.
• Emphasizing Small Values: Smaller numbers heavily influence the harmonic mean, more than the arithmetic mean.
• Downweighting Large Values: It is less sensitive to unusually large outliers.

In calculus, evaluating the function \( f(t) \) at \( t = -3 \) aligns with the concept of the harmonic mean, perfectly illustrating its significance in cases where averages of rates are considered.

Logarithmic Mean

The logarithmic mean is a sophisticated measure, less commonly used than others but vital in specific scientific calculations. It is defined as \( \text{Logarithmic Mean} = \frac{b-a}{\ln b - \ln a} \). This mean fills a niche when dealing with information theory and fluid dynamics.

• Specialized Usage: Often employed in contexts involving heat transfer or processes where logs naturally arise.
• Transformation Sensitivity: The logarithmic mean smoothly handles comparisons of different logarithmic scales.
• Fills Gaps: It provides a value more precise than arithmetic or geometric means in certain transformations.

Here, setting \( f(t) \) to \( f(-1) \) illustrates the practical application of calculus in defining the logarithmic mean, aligning integrals with real-life applications in temperature and chemical concentration fields.