Problem 73 - Calculus for Biology and Medicine

Question

Suppose that the length of a certain organism at age \(x\) is given by \(L(x)\), which satisfies the differential equation $$ \frac{d L}{d x}=e^{-0.1 x}, \quad x \geq 0 $$ Find \(L(x)\) if the limiting length \(L_{\infty}\) is given by $$ L_{\infty}=\lim _{x \rightarrow \infty} L(x)=25 $$ How big is the organism at age \(x=0\) ?

Step-by-Step Solution

Answer

The organism is 15 units long at age 0.

Step 1: Set Up the Integral

To find the function \(L(x)\), we need to integrate the differential equation. The equation given is \( \frac{dL}{dx} = e^{-0.1x} \). Integrate this expression with respect to \(x\) to find \(L(x)\).

Step 2: Perform the Integration

Integrate the function \(\frac{dL}{dx} = e^{-0.1x}\). The integral of \(e^{-0.1x}\) is \(-10e^{-0.1x} + C\), where \(C\) is a constant of integration. Thus, the expression for \(L(x)\) becomes:\[ L(x) = -10e^{-0.1x} + C \]

Step 3: Apply the Limiting Condition

We know that \(L_{\infty} = 25\), meaning as \(x\) approaches infinity, \(L(x)\) approaches 25. Substituting in and solving:\[ \lim_{x \to \infty} (-10e^{-0.1x} + C) = 25 \]Since \(\lim_{x \to \infty} e^{-0.1x} = 0\), it simplifies to:\[ C = 25 \]

Step 4: Express the Final Function

Now substitute \(C\) back into \(L(x)\):\[ L(x) = -10e^{-0.1x} + 25 \]This is the function describing the organism's length at any age \(x\).

Step 5: Calculate the Length at Age 0

We need to find \(L(0)\). Substitute \(x = 0\) into the equation:\[ L(0) = -10e^{-0.1 \cdot 0} + 25 = -10 \cdot 1 + 25 = 15 \]

Key Concepts

Integration

Integration is a fundamental concept in calculus, which essentially involves finding the function that describes the accumulation of quantities. In the context of the given exercise, integration helps us find a function for the length of an organism, knowing the rate of change of its length with respect to time.
To derive the function from the differential equation given, \( \frac{dL}{dx} = e^{-0.1x} \), we integrate the right-hand side with respect to \(x\) and obtain:

• The integral of \( e^{-0.1x} \) is calculated using basic integration rules. The antiderivative is \(-10e^{-0.1x} + C\), where \(C\) is the constant of integration. This is a typical pattern when dealing with exponential functions.
• Integration shows us how the small changes, described by the derivative, can be accumulated into a complete picture of the function \(L(x)\), which models the organism's length over time.

In the exercise, this integration step is essential to move from a differential equation, which gives rates, to a yield function \(L(x)\), illustrating the total length over time.

Limiting Condition

A limiting condition in mathematics and physical sciences refers to the behavior of a function as the input approaches a certain value, typically infinity or a boundary value. This concept plays a vital role in solving the problem of finding \(L(x)\) when given its limiting length.
The limiting length, \(L_\infty = 25\), reflects the organism's expected length as time goes toward infinity. In the equation \( \lim_{x \to \infty} ( -10e^{-0.1x} + C) = 25 \):

• As \(x\) approaches infinity, the term \(e^{-0.1x}\) approaches zero because exponential decay happens, which simplifies the expression to \(C\).
• Consequently, the constant \(C\) is determined to be 25, ensuring that the formula accounts for the ultimate size of the organism.

This condition helps us solve for \(C\), thereby refining the function \(L(x)\) to include all relevant constants, ensuring accurate predictions of length over time.

Exponential Functions

Exponential functions, characterized by the presence of variables in exponents, describe growth and decay processes, like population growth, radioactive decay, and in this case, the length of an organism. They are pivotal in modeling situations where changes happen proportionally to the current value.
In the expression \( e^{-0.1x} \):

• The negative exponent signifies decay, meaning as \(x\) increases, \( e^{-0.1x} \) decreases rapidly, approaching zero as \(x\) approaches infinity.
• This reflects how certain natural processes, like the growth of this organism, slow down over time and approach a limiting value, rather than continuing indefinitely.

Understanding how exponential functions operate allows us to predict the length behavior of the organism over time confidently. It provides insight into the natural limiting condition of the organism's growth curve and accurately reflects real-world biological processes.