Question

A study conducted at the University of New Mexico found that the mass \(m(t)\) (in grams) of a juvenile desert tortoise \(t\) days after a switch to a particular diet is described by the function \(m(t)=m_{0} e^{0.004 t},\) where \(m_{0}\) is the mass of the tortoise at the time of the diet switch. If \(m_{0}=64\) evaluate \(m^{\prime}(65)\) and interpret the meaning of this result. (Source: Physiological and Biochemical Zoology, 85,1,2012 )

Short answer

Answer: The desert tortoise is gaining weight at a rate of approximately 0.261 grams per day after 65 days on the new diet.

Step-by-step solution

Understand the given function

The mass function of the desert tortoise is given as \(m(t)=m_{0}e^{0.004t}\). Here, \(m(t)\) is the mass of the tortoise at time \(t\), \(m_{0}\) is the mass of the tortoise at the start of the diet, and \(0.004t\) is the power of the exponential function, indicated by e. In this case, we are given that \(m_{0}=64\).

Find the derivative of the mass function

To find the rate of change of mass with respect to time, we need to compute the derivative of the mass function \(m(t)\) with respect to time \(t\). To accomplish this, we must differentiate \(m(t) = 64e^{0.004t}\) with respect to \(t\). Recall that the derivative of \(e^{kt}\) is \(ke^{kt}\), where k is a constant. In this case, we can write: $$ m'(t) = 64(0.004)e^{0.004t} $$

Evaluate the derivative of the mass function at \(t=65\) days.

Now, we need to evaluate the derivative \(m'(t)\) at \(t=65\) days: $$ m'(65) = 64(0.004)e^{0.004(65)} $$ Calculate the expression: $$ m'(65) \approx 0.261 $$

Interpret the meaning of this result.

The value we found for \(m'(65)\) is approximately 0.261. In this context, \(m'(t)\) represents the rate of change of the mass of the desert tortoise with respect to time. Thus, when \(t = 65\) days, the mass of the tortoise is increasing at a rate of approximately 0.261 grams per day. This means that after 65 days on the new diet, the desert tortoise gains weight at a rate of roughly 0.261 grams per day.

Key concepts

Derivative of Exponential Functions

Understanding the derivative of exponential functions is a fundamental concept in calculus. Specifically, when dealing with the natural exponential function, which is of the form e^{kt}, where e is the base of the natural logarithm and k is a constant, the differentiation process is straightforward. The derivative of e^{kt} with respect to t is k e^{kt}. In our tortoise mass example, this process translates into finding the rate at which the mass is changing over time, with the function m(t) = m_0 e^{0.004t}.

Once we apply this rule, we see that the rate of change of mass, m'(t), is 64(0.004)e^{0.004t}. This expression illustrates how the mass varies with each passing day, capturing the essence of exponential growth. The derivative here is also an exponential function, revealing that the growth rate itself is not constant but increases as time progresses—an intrinsic property of exponential growth.

Rate of Change in Biology

In biology, understanding rates of change can provide insights into various growth processes. When biologists study phenomena such as population dynamics, cell growth, or, as in our case, the mass of a desert tortoise, they often encounter relationships that are best modeled by exponential functions. The rate of change in such scenarios is not uniform; instead, it's proportional to the current state.

For the juvenile desert tortoise, the derivative m'(t) describes how quickly the tortoise's mass is growing over time. In real terms, this can be influenced by factors like diet, metabolism, and environmental conditions. In the given study, the mass increase rate on day 65 is approximately 0.261 grams per day, showcasing how derivatives can apply to biological growth and decay, providing a snapshot at a moment in time of a continuously evolving process.

Interpreting Derivatives in Real-World Contexts

Derivatives serve as a powerful tool not just in theoretical calculus, but also in real-world application, offering a way to quantify how variables such as time, space, or in our example, mass, change instantaneously. In the context of the desert tortoise, m'(65) ≈ 0.261 gives us a tangible measure of growth: the tortoise is expected to gain about 0.261 grams every day after 65 days on a specific diet.

This concept is pivotal in fields ranging from physics to finance, where interpreting the derivative provides insights into speeds, rates of return, or optimization problems. With this understanding, students can grasp not only the mechanics behind taking derivatives but also their valuable interpretations which can guide decision-making, predictions, and a range of analyses in scientific and business endeavors.