Question

When observations begin at \(t=0,\) a cell culture has 1200 cells and continues to grow according to the function \(p(t)=1200 e^{t},\) where \(p\) is the number of cells and \(t\) is measured in days. a. Compute \(p^{\prime}(t) .\) What units are associated with the derivative and what does it measure? b. On the interval \([0,4],\) when is the growth rate \(p^{\prime}(t)\) the least? When is it the greatest?

Short answer

Answer: The least growth rate is 1200 cells/day² at \(t=0\), and the greatest growth rate is approximately 95326.9 cells/day² at \(t=4\).

Step-by-step solution

Compute the first derivative of p(t)

To find the first derivative of \(p(t) = 1200e^t\), derive the given function with respect to t. \[ p'(t) = 1200 \cdot \frac{d}{dt}(e^t) = 1200 \cdot e^t \]

Identify the units associated with the derivative and what it measures

The units associated with the original function \(p(t)\) are cells/day, as \(p\) represents the number of cells and \(t\) represents time in days. Thus, the units for the derivative \(p'(t)\) are cells/day² (cells per day per day). The derivative \(p'(t)\) measures the rate of change of the number of cells with respect to time. For part (b):

Determine the least and greatest growth rate on the interval [0, 4]

Since it's an exponential function, the least growth rate of \(p'(t)=1200e^t\) occurs at the lower bound of the interval, and the greatest growth rate occurs at the upper bound of the interval. The least growth rate on the interval [0, 4] can be computed by substituting \(t=0\) into the derivative function \(p'(t)\): \[ p'(0) = 1200 \cdot e^0 = 1200 \cdot 1 = 1200 \, \text{cells/day²} \] The greatest growth rate on the interval [0, 4] can be computed by substituting \(t=4\) into the derivative function \(p'(t)\): \[ p'(4) = 1200 \cdot e^4 \approx 95326.9 \, \text{cells/day²} \] Thus, the growth rate \(p'(t)\) is least at \(t=0\) and has the value 1200 cells/day², and it is greatest at \(t=4\) and has the value approximately 95326.9 cells/day².

Key concepts

Exponential Growth

Exponential growth describes a process where the quantity increases at a rate proportional to its current value. In biological contexts, like the cell culture example, this means that the more cells there are, the faster the cell population grows. The function used to model such growth, like the given function for cell culture growth, is of the form \( p(t) = P_0 e^{rt} \), where \( P_0 \) is the initial amount, \( r \) is the growth rate, and \( t \) is time.

The remarkable characteristic of exponential growth is that it leads to large numbers very quickly. In nature, resources often limit exponential growth, but in controlled environments, like a laboratory, cell cultures can exhibit this type of growth for a period. It's crucial in research and industry to understand and predict this growth to optimize conditions and harvest times.

Derivative of Exponential Functions

The derivative of an exponential function provides information about the rate of change of the process it models. For an exponential function \( p(t) = P_0 e^{rt} \), the derivative is \( p'(t) = P_0 r e^{rt} \), which maintains the same exponential form. In the cell culture example, once you calculate the derivative \( p'(t) \), you have a formula that tells you how fast the number of cells is increasing at any moment in time.

Since the base of the natural exponential function is Euler’s number \( e \), which is approximately equal to 2.71828, it has a unique property where the derivative of \( e^t \) is simply \( e^t \). This property makes calculating the rate of change for natural exponential growth quite straightforward.

Rate of Change

In calculus, the rate of change is synonymous with the derivative. It measures how quickly a quantity changes when something else, usually time, changes. For our cell culture, the rate of change of cells with respect to time is given by the derivative \( p'(t) \).

This is not just an abstract concept but has real-world implications. In biology, a high rate of change in a cell population might suggest a need for more nutrients or space to sustain growth. It could also indicate the effectiveness of a drug in a pharmaceutical test. The units given to rates of change, like cells per day squared \( \text{cells/day}^2 \), help us quantify, analyze and compare different rates in meaningful ways.

Identifying Intervals

Identifying intervals in the context of calculus usually refers to determining where on the graph a function is increasing or decreasing, has a positive or negative rate of change, or reaches a maximum or minimum. In the exercise, we focus on the interval \( [0, 4] \), representing days, to see how the growth rate of the cell culture changes over time.

Exponential functions, by their nature, are always increasing if their rate is positive, which is the case in our example. It implies that the smallest value for the rate of change will occur at the beginning of the interval, and the largest value at the end, as the function's slope steepens. This knowledge can be crucial, for example, in financially based decisions on investment returns or biologically based decisions on the necessary scaling-up of cell culture facilities.