Problem 29 - Calculus for Biology and Medicine

Question

\mathrm{\\{} I n ~ P r o b l e m s ~ , ~ g r a p h ~ t h e ~ f u n c t i o n s ~ \(f(x)=a^{x}, x \in[0, \infty)\), \mathrm{\\{} a n d ~ \(N_{t}=R^{t}, t \in \mathbf{N}\), together in one coordinate system for the indicated values of a and \(R\). $$ a=R=1 / 2 $$

Step-by-Step Solution

Answer

Both functions, \(f(x)\) and \(N_t\), are plotted as exponential decay with \(a = R = 1/2\). The graph shows a continuous curve for \(f(x)\) and discrete points for \(N_t\) along the same curve.

Step 1: Understand the Functions

We need to graph two functions: \(f(x) = a^x\) for \(x \in [0, \infty)\) and \(N_t = R^t\) for \(t \in \mathbb{N}\). In this exercise, we are given \(a = 1/2\) and \(R = 1/2\). This means both functions will have the form of \(\left( \frac{1}{2} \right)^x\) and \(\left( \frac{1}{2} \right)^t\) respectively.

Step 2: Plotting f(x) = (1/2)^x

Let's start with plotting \(f(x) = \left( \frac{1}{2} \right)^x\). This function is an exponential decay function, indicating that as \(x\) increases, \(f(x)\) approaches 0. Plot points for several values of \(x\) such as 0, 1, 2, etc., to observe the trend.

Step 3: Plotting Nt = (1/2)^t

Now plot \(N_t = \left( \frac{1}{2} \right)^t\) where \(t\) is an integer. This function is essentially the same as \(f(x)\) but defined only for natural numbers \(t\). Plot points for \(t = 0, 1, 2,...\), which will align with the integer points of \(f(x)\).

Step 4: Combine the Graphs

Graph the functions \(f(x) = \left( \frac{1}{2} \right)^x\) and \(N_t = \left( \frac{1}{2} \right)^t\) on the same coordinate axes. Both should reflect the exponential decay from 1 towards 0 as \(x\) or \(t\) increase. Notice that the graph of \(N_t\) consists of discrete points along the same curve as \(f(x)\).

Step 5: Analyze the Graphs

Both functions demonstrate an exponential decay because their bases are less than 1. As \(x\) and \(t\) increase, their values diminish sharply towards zero. The graph of \(f(x)\) is continuous, while \(N_t\) is plotted as discrete points on the same graph.

Key Concepts

Graphing Functions

Graphing functions can help visualize how they behave. By plotting a graph, you can see the relationship between the variables and how changes in one affect the other.

In the case of exponential functions like \(f(x) = \frac{1}{2}^x\) and \(N_t = \frac{1}{2}^t\), you will notice the characteristic "L" shape as the graph approaches zero.

• Both graphs start at 1, when \(x = 0\) and \(t = 0\).
• As \(x\) and \(t\) increase, the functions decrease towards zero.
• These graphs help to intuitively understand the concept of exponential decay.

Graphing provides an intuitive way to compare continuous and discrete functions. It shows how they behave similarly at integer points but differently over intervals.

Exponential Decay

Exponential decay describes a process where quantities reduce by a consistent factor over equal intervals. In our example, both functions share a base \(\frac{1}{2}\), indicating a decay rate of 50%.

This means every increase in \(x\) or \(t\) cuts the value in half.

• At \(x = 1\), the function becomes \(\left( \frac{1}{2} \right)^1 = \frac{1}{2}\).
• At \(x = 2\), it becomes \(\left( \frac{1}{2} \right)^2 = \frac{1}{4}\) and so on.

The decay continues as \(x\) or \(t\) increases, producing values that approach zero but never actually reach it. This illustrates how exponential decay is rapid at first, but then slows down, illustrating the concept of asymptotic behavior.

Discrete vs Continuous Graphs

Understanding the difference between discrete and continuous graphs is crucial for mastering functions like these. A continuous graph shows every possible value, connecting them smoothly.

For \(f(x) = \frac{1}{2}^x\), the function can take any real value in \([0, \infty)\), and the graph forms a smooth curve.

• This smoothness reflects how continuous functions model continuous processes.

Conversely, \(N_t = \frac{1}{2}^t\) is only defined at integer values of \(t\), displaying as separate points.

• This reflects how some processes, like population counts, happen in distinct steps.

Bringing both functions to the same graph highlights the differences, where \(N_t\) appears as a series of points following the curve of \(f(x)\). Discrete points lie exactly on the continuous function at integer values, showing their shared properties.