Problem 30 - 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 \(R\). $$ a=R=1 / 3 $$

Step-by-Step Solution

Answer

Graph exponential decay for both functions with \( a = R = \frac{1}{3} \).

Step 1: Understand the functions

We have two functions to graph: - The first function is an exponential function given by \( f(x) = a^x \) where \( a = \frac{1}{3} \). The domain for \( x \) is \( [0, \infty) \).- The second function is \( N_t = R^t \) where \( R = \frac{1}{3} \) and \( t \in \mathbb{N} \). The domain for \( t \) consists of natural numbers, so \( t \geq 0 \).

Step 2: Prepare for Plotting

We need to graph these functions on the same coordinate system. The x-axis will represent both \( x \) and \( t \). For \( f(x) = (\frac{1}{3})^x \), we consider \( x \geq 0 \). For \( N_t = (\frac{1}{3})^t \), we plot only at integer values of \( t \geq 0 \).

Step 3: Graph the continuous function \( f(x) \)

Graph \( f(x) = (\frac{1}{3})^x \) by evaluating the function at several points: - At \( x = 0 \), \( f(0) = 1 \).- At \( x = 1 \), \( f(1) = \frac{1}{3} \).- At \( x = 2 \), \( f(2) = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \).Connect these points with a smooth curve that approaches zero as \( x \) increases.

Step 4: Plot the discrete function \( N_t \)

Plot \( N_t = (\frac{1}{3})^t \) at integer values of \( t \):- At \( t = 0 \), \( N_0 = 1 \).- At \( t = 1 \), \( N_1 = \frac{1}{3} \).- At \( t = 2 \), \( N_2 = \frac{1}{9} \).Add these points to the graph and indicate them with distinct markers to differentiate from the continuous curve.

Key Concepts

Graphing Functions

Graphing functions is like creating a visual story of a mathematical equation. It helps us see patterns and relationships. For exponential functions, like the ones given in our exercise, the key is to understand how quantities grow or decay. An exponential function has the form \( f(x) = a^x \), where \( a \) is a constant and \( x \) is a variable. Here, the constant \( a \) is \( \frac{1}{3} \), which tells us that the function is decreasing, because \( \frac{1}{3} \) is a fraction less than 1.

When graphing, it's important to choose several points to evaluate the function, plot those points, and then connect them. This gives us a curve. With \( f(x) = (\frac{1}{3})^x \), we plot values like \( x = 0, 1, 2, \) etc., and connect them smoothly.

This graph will show a curve starting from 1 at \( x = 0 \) and approaching but never quite reaching zero. Understanding this concept is crucial for interpreting the growth or decay exhibited by the function.

Continuous and Discrete Functions

Understanding continuous and discrete functions is vital when learning about graphs. They tell us about the nature of the data we are dealing with. A continuous function, like \( f(x) = (\frac{1}{3})^x \), is one that is smooth and unbroken over its domain. This means at any point within its range, the function is defined.

On the other hand, a discrete function has outputs at only specific points, rather than forming a continuous line. The function \( N_t = (\frac{1}{3})^t \) from our exercise is discrete, because it's only defined at natural number values for \( t \). It jumps from one point to the next rather than connecting smoothly.

So, while \( f(x) \) gives us a smooth, continuous curve, \( N_t \) results in individual points on the graph. Each point corresponds to an integer \( t \), representing counts, items, or discrete events.

Function Plotting

Function plotting is the process of taking an equation and turning it into a visual graph. This is useful for understanding and analyzing the behavior of functions. For exponential functions, like those given in the exercise, plotting involves several steps.

• Evaluate the function at several key points.
• Mark each point on a coordinate system—the horizontal axis (x-axis) for inputs and the vertical axis (y-axis) for outputs.
• Connect the plotted points for continuous functions and mark distinct points for discrete ones.

The function \( f(x) = (\frac{1}{3})^x \) is plotted by marking points like \( x=0,1,2 \). Connect these for a smooth decay curve approaching zero. For \( N_t = (\frac{1}{3})^t \), plot only integer \( t \) values, adding distinct markers without connecting them.

The essence of plotting functions is in showing how they behave visually, making it easier to comprehend complex relationships and changes.