Problem 57 - Calculus with Concepts in Calculus

Question

According to Weiss's Law of Excitation of Tissue, the strength \(S\) of an electric current is related to the time \(t\) the current takes to excite tissue by the formula $$ S(t)=\frac{a}{t}+b \quad \text { for } \quad t>0 $$ where \(a\) and \(b\) are constants. Then the limit \(\lim _{t \rightarrow \infty} S(t)\) is the threshold strength of current below which the tissue will never he excited. Find \(\lim _{t \rightarrow \infty} S(t)\).

Step-by-Step Solution

Answer

The limit is \( b \), which is the threshold strength of the current.

Step 1: Understand the Formula

According to the problem, the strength of an electric current as a function of time is given by the formula \( S(t) = \frac{a}{t} + b \), where \( a \) and \( b \) are constants.

Step 2: Identify the Limit

We are asked to find the limit as \( t \to \infty \) of \( S(t) \), which is expressed as \( \lim_{t \to \infty} S(t) \).

Step 3: Apply the Limit to each Term

The function \( S(t) = \frac{a}{t} + b \) has two components. The limit \( \lim_{t \to \infty} \frac{a}{t} \) needs to be evaluated first. As \( t \to \infty \), \( \frac{a}{t} \to 0 \), because \( a \) is a constant and \( t \) becomes very large, making the fraction approach 0.

Step 4: Simplifying with Constants

The other part of the equation is the constant \( b \). Since the limit of a constant is the constant itself, \( \lim_{t \to \infty} b = b \).

Step 5: Calculate the Final Limit

Using the limits from the previous steps, combine them: \( \lim_{t \to \infty} S(t) = \lim_{t \to \infty} \left( \frac{a}{t} + b \right) = 0 + b = b \).

Step 6: Interpret the Result

Therefore, the threshold strength of the current below which the tissue will never be excited is \( b \).

Key Concepts

Limit of a Function

The limit of a function is a core idea in calculus, describing the behavior of a function as it approaches a specified input value. In simpler terms, it tells us what value the function gets closer to as the input gets closer to a particular point. This point could be a finite value or infinity in some cases.

When we say "find the limit as \( t \to \infty \) of \( S(t) \)," we're interested in the value that the function \( S(t) = \frac{a}{t} + b \) approaches as \( t \) becomes very large. Understanding limits is crucial because it helps us determine how functions behave at boundary values, allowing us to gather meaningful insights even when such values aren't explicitly part of the function's domain.

• Limits are foundational to the understanding of continuity, derivatives, and integrals.
• They allow analysis of a function's behavior at infinity or at points of discontinuity.
• In this exercise, recognizing that \( \lim_{t \to \infty} \frac{a}{t} = 0 \) is key to solving the problem.

Weiss's Law of Excitation

Weiss's Law of Excitation refers to a mathematical model that describes the relationship between the strength of an electric current and the time needed to excite tissue. According to this law, the current strength \( S \) decreases with an increase in time \( t \), described by the formula \( S(t) = \frac{a}{t} + b \). This relation explains how less current is required over longer durations to excite tissue effectively.

The key takeaway here is the term \( \frac{a}{t} \), which reduces as \( t \) increases, making the limit approach the constant \( b \). Weiss's Law highlights critical characteristics in biological tissues' response to electrical stimuli, which are essential in understanding and designing biomedical devices such as pacemakers and nerve stimulators.

• The law explains the diminishing amplitude needed over time for excitation.
• Understanding this helps in determining the minimum constant current, or threshold \( b \), necessary for the tissue to be effectively excited.
• It has practical applications across various medical fields, annotating bio-electric signal interactions.

Asymptotic Behavior

Asymptotic behavior in mathematics refers to how a function behaves as its input grows towards a certain point or infinity. It's like understanding the long-term growth or decay trend of the function. In simpler words, it helps us grasp how the function stabilizes or changes as the input becomes extremely large or small.

For the function \( S(t) = \frac{a}{t} + b \), as \( t \to \infty \), the term \( \frac{a}{t} \) becomes negligible, causing \( S(t) \) to approach the constant value \( b \). This reflects the asymptotic behavior, implying that the function eventually stabilizes at \( b \). This concept is crucial because it tells us the threshold strength of the current that will reliably excite tissue.

• Asymptotic analysis provides insights into stability or bounding values of functions.
• In this context, it shows the long-run payoff of \( S(t) \) as being predetermined by \( b \).
• It helps in predicting behavior in extreme cases, which is particularly useful in scientific modeling and engineering applications.