Problem 32 - Calculus for Biology and Medicine

Question

In Problems graph the functions \(f(x)=a^{x}, x \in[0, \infty)\), and \(N_{t}=R^{t}, t \in \mathbf{N}\), together in one coordinate system for the indicated values of a and \(R\). $$ a=R=3 $$

Step-by-Step Solution

Answer

Plot both functions, \(f(x) = 3^x\) and \(N_t = 3^t\), on the same graph. The curve for \(f(x)\) is continuous, while \(N_t\) consists of discrete dots at natural numbers.

Step 1: Identify the Functions

We have two functions to graph: 1. The exponential function \( f(x) = a^x = 3^x \) for \( x \geq 0 \).2. The discrete exponential function \( N_t = R^t = 3^t \) for \( t \in \mathbb{N} \).

Step 2: Determine the Domain and Range

For \( f(x) = 3^x \), the domain is \([0, \, \infty)\) and the range is \((0, \, \infty)\), as it is an exponential function that grows without bound as \(x\) increases. For \( N_t = 3^t \), since \(t\) is a natural number, the function is only defined at integer points starting from 0, and the same range \((0, \, \infty)\) applies.

Step 3: Plot Points for f(x) = 3^x

Calculate and plot points for \( f(x) = 3^x \). For example, - When \( x = 0 \), \( f(x) = 3^0 = 1 \). - When \( x = 1 \), \( f(x) = 3^1 = 3 \). - When \( x = 2 \), \( f(x) = 3^2 = 9 \), and so on. Connect these points to form the curve.

Step 4: Plot Points for N_t = 3^t

Because \( N_t \) is only defined at natural numbers, plot the following points:- When \( t = 0 \), \( N_t = 3^0 = 1 \).- When \( t = 1 \), \( N_t = 3^1 = 3 \).- When \( t = 2 \), \( N_t = 3^2 = 9 \), and so on.These points should form a sequence of dots on the graph.

Step 5: Graphing Together

On a coordinate system, plot both the continuous curve for \( f(x) = 3^x \) and the discrete points for \( N_t = 3^t \). The continuous curve will smoothly pass through (0,1), (1,3), (2,9), etc., while the discrete graph will have distinct points at the same coordinates: (0,1), (1,3), (2,9), etc.

Key Concepts

Continuous vs Discrete Functions

Understanding the difference between continuous and discrete functions is crucial when working with exponential functions like those in the problem. A continuous function is smooth and connected, meaning you can draw it without lifting your pencil. For example, the function \(f(x) = 3^x\) is continuous because you can plug in any real number for \(x\), forming a smooth curve across its domain, \([0, \infty)\). The graph of this function shows an exponential rise as \(x\) increases.

In contrast, a discrete function consists of separate, distinct points. The function \(N_t = 3^t\), where \(t\) is a natural number, is discrete. Since it's defined only at integer values, you will plot individual points like (0,1), (1,3), (2,9) instead of a continuous curve.

To summarize:

• Continuous Functions: Smooth, uninterrupted graphs (e.g., \(f(x) = 3^x\)).
• Discrete Functions: Consists of distinct points (e.g., \(N_t = 3^t\)).

Understanding these differences helps in visualizing and analyzing the behavior of different types of exponential functions.

Domain and Range

The domain and range of a function are foundational concepts that describe the inputs and outputs of a function, respectively. For the continuous exponential function \(f(x) = 3^x\), the domain is all non-negative real numbers, \([0, \infty)\). This means \(x\) can take any value from 0 to infinity.

The range of \(f(x) = 3^x\) is \((0, \infty)\). As the function increases infinitely, the output values are positive and can grow without bound. In simpler terms, \(f(x)\) will never touch or go below zero.

For the discrete function \(N_t = 3^t\), the domain is the set of natural numbers, \(\mathbb{N}\). This reflects that only integer inputs are valid. However, the range remains the same as that of \(f(x)\), which is \((0, \infty)\).

• Domain of \(f(x) = 3^x\): \([0, \infty)\)
• Range of \(f(x) = 3^x\): \((0, \infty)\)
• Domain of \(N_t = 3^t\): Natural numbers (\(\mathbb{N}\))
• Range of \(N_t = 3^t\): \((0, \infty)\)

Grasping these concepts is vital for understanding what values an exponential function can take and what output it can produce.

Graphing Mathematical Functions

Graphing functions gives insight into their behavior and can enhance understanding of their growth and characteristics. For the continuous function \(f(x) = 3^x\), graphing involves plotting the smooth curve that reflects exponential growth. The points might start at (0,1), (1,3), (2,9), etc., connecting smoothly into a curve that rises sharply as \(x\) increases.

In comparison, graphing the discrete function \(N_t = 3^t\) involves plotting distinct points, like dots, since \(t\) can only be a natural number. These points align vertically over the graph of \(f(x) = 3^x\) at (0,1), (1,3), (2,9), and so on, but they do not connect to form a curve.

When graphing both functions together:

• The continuous graph of \(f(x) = 3^x\) provides a base, a smooth rising curve, symbolizing unlimited mathematical input possibilities.
• The graph of \(N_t = 3^t\) appears as discrete, separate dots representing specific, countable inputs.

By graphing these functions together, students can visually differentiate the continuous nature of \(f(x) = 3^x\) and the discrete nature of \(N_t = 3^t\), which enhances their understanding of exponential functions.