Problem 48 - Essential Calculus Early Transcendentals

Question

A metal cable has radius \(r\) and is covered by insulation, so that the distance from the center of the cable to the exterior of the insulation is \(R .\) The velocity \(v\) of an electrical impulse in the cable is $$v=-c\left(\frac{r}{R}\right)^{2} \ln \left(\frac{r}{R}\right)$$ where \(c\) is a positive constant. Find the following limits and interpret your answers. $$\text {(a)}\lim _{R \rightarrow r^{+}} v \quad \text { (b) } \lim _{r \rightarrow 0^{+}} v$$

Step-by-Step Solution

Answer

(a) 0; (b) 0. As the insulation approaches no thickness or cable's radius goes to zero, the velocity goes to zero.

Step 1: Understand the Given Function

The velocity function is given by: \[ v = -c\left(\frac{r}{R}\right)^{2} \ln \left(\frac{r}{R}\right) \] where \( c \) is a positive constant. The function describes the electrical impulse speed across a metal cable with insulation properties determined by \( r \) and \( R \).

Step 2: Limit as R approaches r from the right

For \( \lim_{R \to r^+} v \), substitute \( \frac{r}{R} \) with \( 1 \). The expression becomes: \[ \lim_{R \to r^+} -c \left(1\right)^2 \ln\left(1\right) \]Since \( \ln(1) = 0 \), this results in \( 0 \).

Step 3: Limit as r approaches 0 from the right

For \( \lim_{r \to 0^+} v \), the expression inside the limit is \( -c\frac{r^2}{R^2} \ln\left(\frac{r}{R}\right) \). As \( r \to 0^+ \), \( \frac{r}{R} \to 0^+ \), thus we use L'Hospital's Rule to resolve the form: \( 0 \cdot (-\infty) \). We see that \( x \ln(x) \) as \( x \to 0^+ \) approaches \( 0 \). Therefore, the expression limits to 0.

Key Concepts

L'Hospital's Rule

L'Hospital's Rule is a powerful tool in calculus that helps us find limits of indeterminate forms. Indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) often occur in limit problems, and this rule provides a way to solve them. Whenever you encounter such a form, consider using L'Hospital's Rule, which allows you to differentiate the numerator and the denominator separately and then take the limit again.

In our exercise, we faced the limit \( \lim_{r \to 0^+} -c\frac{r^2}{R^2} \ln\left(\frac{r}{R}\right) \). When \( \frac{r}{R} \) approaches zero, the expression inside becomes a 0 multiplied by \(-\infty\), an indeterminate form. L'Hospital's Rule was used to tackle this, leading to simpler expressions that can be further evaluated.

In summary, L'Hospital's Rule can simplify what looks complex. Keep an eye out for indeterminate forms and apply this rule as needed. You'll often find a path to the solution becomes much clearer!

Logarithmic Functions

Logarithmic functions are essential in calculus as they describe slow-growth behaviors. The natural logarithm \( \ln(x) \) is a logarithmic function frequently used in mathematics and sciences because it has properties that make it convenient, especially in calculus.

In this exercise, the natural logarithm appears in the form \( \ln\left(\frac{r}{R}\right) \). One important property to remember is that \( \ln(1) = 0 \), which helps in evaluating limits like \( \lim_{R \to r^+} v \). Logarithmic functions have unique behaviors at certain points: as \( x \to 0^+ \), \( \ln(x) \to -\infty \). This knowledge can help assess limits when variables approach critical points.

Logarithms can transform multiplicative relationships into additive ones, which is an often used property in solving equations and understanding exponential growth or decay processes. Understanding logarithmic functions will provide you with a vital tool for tackling diverse problems across different fields.

Velocity of Electrical Impulse

The velocity of an electrical impulse in a cable describes how quickly an electrical signal travels through the medium. This velocity is crucial, especially in designing cables for telecommunications and power transmission where speed and efficiency are important.

In our example, the impulse velocity \( v \) is defined by the equation \(v = -c\left(\frac{r}{R}\right)^{2} \ln \left(\frac{r}{R}\right)\), featuring dependencies on both the radius \( r \) and the insulation \( R \) of the cable. The structure of this formula showcases how engineering design, through variables \( r \) and \( R \), directly influences the signal's velocity.

Analyzing limits such as \( \lim_{R \to r^+} v \) and \( \lim_{r \to 0^+} v \) gives insights into how changes in physical dimensions affect signal speed. Understanding the relationships within the formula can assist in optimizing cable properties to achieve desired performance metrics. So, having a strong grasp of these concepts is essential for fields involving signal transmission and electronic engineering.