Problem 51 - Calculus and its applications

Question

The magnitude \(R\) (measured on the Richter scale) of an earthquake of intensity I is defined as \(R=\log \frac{I}{I_{0}},\) where \(I_{0}\) is the minimum intensity (used for comparison). a) Find the rate of change \(d R / d I\). b) Interpret the meaning of \(d R / d I\).

Step-by-Step Solution

Answer

a) \(\frac{dR}{dI} = \frac{1}{I}\); b) It shows how magnitude changes with intensity, decreasing as intensity increases.

Step 1: Differentiate the definition of magnitude with respect to intensity

The given formula for the magnitude is \( R = \log \frac{I}{I_0} \). To find \( \frac{dR}{dI} \), we first express the magnitude formula in a more convenient form for differentiation: \( R = \log(I) - \log(I_0) \). Since \( I_0 \) is constant, its derivative with respect to \( I \) is zero, so we only need to differentiate \( \log(I) \) with respect to \( I \).

Step 2: Apply the derivative of logarithmic functions

The derivative of \( \log(I) \) with respect to \( I \) is \( \frac{1}{I} \). Thus, the derivative of \( R \) with respect to \( I \) is: \[ \frac{dR}{dI} = \frac{1}{I} \].

Step 3: Interpret the derivative

The derivative \( \frac{dR}{dI} = \frac{1}{I} \) represents the rate of change of earthquake magnitude, \( R \), with respect to a change in intensity, \( I \). This indicates that for a small change in intensity \( \Delta I \), the change in magnitude \( \Delta R \) is approximately \( \frac{\Delta I}{I} \), emphasizing that as intensity increases, the change in magnitude becomes smaller for the same change in \( I \).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a handy technique typically used to find derivatives of functions that are products, quotients, or have exponents. In the context of our exercise, it allows for differentiation of the earthquake magnitude formula, which involves a logarithm of the ratio of intensities.

To understand this better, imagine differentiating the formula for the Richter scale magnitude, given by \(R = \log \left(\frac{I}{I_0}\right)\). Applying the properties of logarithms, this becomes \(R = \log(I) - \log(I_0)\).

Since \(I_0\) is a constant, its derivative is zero, simplifying our task. We then only need to differentiate \(\log(I)\) with respect to \(I\). Through logarithmic differentiation, the process becomes clear and manageable:

• Recognize the formula structure.
• Apply logarithmic rules to simplify.
• Differentiate using known derivatives of log functions.

As a result of this method, we find that \(\frac{dR}{dI} = \frac{1}{I}\). This simple derivative shows how the magnitude reacts to intensity changes.

Rate of Change

The concept of the rate of change is crucial in understanding how one quantity varies in response to a change in another. In the case of earthquakes and the Richter scale, it helps to quantify how magnitude changes react to alterations in seismic intensity.

From the derived derivative \(\frac{dR}{dI} = \frac{1}{I}\), we observe a fascinating relationship. As intensity \(I\) increases, the rate of change of the magnitude, \(R\), becomes smaller. This is because the derivative suggests a reciprocal relationship:

• High intensity results in a smaller rate of change in magnitude.
• Low intensity yields a larger rate of change in magnitude.

This behavior reflects the logarithmic nature of the Richter scale where larger seismic intensities lead to less dramatic increases in reported magnitude.

Such an understanding is critical when analyzing earthquake data, as it implies that larger shifts in intensity may only translate to minor shifts in magnitude.

Earthquake Intensity

Earthquake intensity is a measure of the strength or energy released by an earthquake at a location. The Richter scale quantifies this by comparing the earthquake's intensity \(I\) to a standard minimum intensity \(I_0\).

The formula \(R = \log \left(\frac{I}{I_0}\right)\) effectively captures this comparison. Intensity \(I\) reflects the energy transmitted through seismic waves. The higher the intensity, the more powerful the earthquake.

This formula reveals a logarithmic relationship where each whole number increase in \(R\) represents a tenfold increase in measured amplitude and roughly 31.6 times the energy release. Consequently:

• Magnitude \(R\) indicates relative destruction potential.
• Intensities \(I\) provide tangible energy measures.

Understanding the link between intensity and magnitude offers important context when interpreting seismic activity reports and assessing potential impacts on affected areas.