Problem 57 - Calculus for Biology and Medicine

Question

Assume that a population size at time \(t\) is \(N(t)\) and that \(N(t)=2^{t}, t \geq 0\) (a) Find the population size for \(t=0,1,2,3\), and \(4 .\) (b) Graph \(N(t)\) for \(t \geq 0\).

Step-by-Step Solution

Answer

(a) \(N(0) = 1\), \(N(1) = 2\), \(N(2) = 4\), \(N(3) = 8\), \(N(4) = 16\); (b) Graph illustrates an exponential increase in population size.

Step 1: Substitute Values of t

For each given time value, substitute it into the equation \(N(t) = 2^t\) to find the population size.- When \(t = 0\): \(N(0) = 2^0 = 1\).- When \(t = 1\): \(N(1) = 2^1 = 2\).- When \(t = 2\): \(N(2) = 2^2 = 4\).- When \(t = 3\): \(N(3) = 2^3 = 8\).- When \(t = 4\): \(N(4) = 2^4 = 16\).

Step 2: Plot the Points

Now plot the calculated points on a graph.- Plot the point \((0, 1)\) which represents \(N(0)\).- Plot the point \((1, 2)\) which represents \(N(1)\).- Plot the point \((2, 4)\) which represents \(N(2)\).- Plot the point \((3, 8)\) which represents \(N(3)\).- Plot the point \((4, 16)\) which represents \(N(4)\).

Step 3: Draw the Graph

Connect the plotted points with a smooth curve to illustrate how the population changes over time.- Notice that the curve is rising exponentially, since \(N(t)\) is doubling as \(t\) increases by 1.- Label the curve with \(N(t) = 2^t\) and ensure axes are correctly labeled: \(t\) (time) on the x-axis and \(N(t)\) (population) on the y-axis. This will help communicate the exponential nature of the function.

Key Concepts

Population Growth

Population growth often follows an exponential pattern, meaning it can increase rapidly over a period of time. Exponential growth can occur when each member of the population contributes to the production of another member, as is the case with bacteria multiplying or people spreading a viral video. In our model, the population size at time \( t \) is given by the expression \( N(t) = 2^t \). This represents a situation where the population doubles with each passing time unit.

Let's break it down with an example:

• At \( t = 0 \), we start with \( N(0) = 2^0 = 1 \). This is our initial population size.
• At \( t = 1 \), the population size becomes \( N(1) = 2^1 = 2 \), doubling the previous size.
• Continuing on, you’ll see that this pattern persists, resulting in a rapid increase as \( t \) grows.

This function is a classic example of exponential growth seen frequently in the natural and social sciences.

Graphing Functions

Graphing functions allows us to visualize the relationship between variables. In our scenario, we are plotting population size \( N(t) \) against time \( t \). This gives us a clear picture of how the population increases over time.

To graph the function \( N(t) = 2^t \):

• Begin by plotting specific points calculated from substituting \( t \) values into the function. From our example, these points are: \((0, 1), (1, 2), (2, 4), (3, 8), (4, 16)\).
• Connect these points with a smooth curve. The pattern you’ll see is characterized by a steep upward trajectory as each value of \( t \) doubles your \( N(t) \).

Ensure your axes are labeled, with \( t \) on the x-axis and \( N(t) \) on the y-axis, to effectively communicate this exponential trend.

Substituting Values

Substituting values is a simple mathematical technique used to calculate unknowns by replacing variables with specific numbers. In the context of the exponential function \( N(t) = 2^t \), substitution provides a method to ascertain specific population sizes for given points in time.

Here's how it works:

• Start with the general function \( N(t) \). For each time \( t \) you want to explore, replace \( t \) with the desired number. For example, substituting \( t = 2 \) gives \( N(2) = 2^2 = 4 \).
• Repeat this substitution process for several \( t \) values to observe how \( N(t) \) changes. Our example uses \( t = 0, 1, 2, 3, \) and \( 4 \), generating populations of \( 1, 2, 4, 8, \) and \( 16 \) respectively.

This step-by-step substitution not only helps find the population size for specific times but also aids in understanding the function's growth pattern.