Problem 31 - Calculus for Biology and Medicine

Question

In Problems 31-34, 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 \(R\). $$ a=R=2 $$

Step-by-Step Solution

Answer

Graph by plotting the continuous curve \( f(x) = 2^x \) and discrete points \( N_t = 2^t \) together.

Step 1: Understand the Functions

The problem presents two exponential functions: \( f(x) = a^x \) and \( N_t = R^t \). We are given that \( a = R = 2 \). So, the functions are \( f(x) = 2^x \) and \( N_t = 2^t \). The focus is on combining these functions in a single coordinate system.

Step 2: Determine the Domain and Range

The domain for \( f(x) = 2^x \) is \( x \in [0, \infty) \), meaning it includes all non-negative real numbers. The domain for \( N_t = 2^t \) is \( t \in \mathbb{N} \), which means it includes all positive natural numbers (1, 2, 3, ...). Therefore, both functions will have outputs that start from 1 and increase as inputs increase.

Step 3: Create a Table of Values

For both functions, create a table of values to identify key points. For \( f(x) \), choose several values like x = 0, 1, 2, 3 (resulting in 1, 2, 4, 8). For \( N_t \), choose natural numbers like t = 1, 2, 3, 4 (also resulting in 2, 4, 8, 16).

Step 4: Plot the Functions on the Same Graph

Using the table of values, plot \( f(x) = 2^x \) and \( N_t = 2^t \) on the same coordinate system. For \( f(x) \), plot the continuous curve (connecting points for all non-negative real x). For \( N_t \), plot distinct points at natural numbers (1, 2, 3, ...) corresponding to t.

Step 5: Analyze the Graph

Upon graphing, both curves should appear exponential and increasing as x/t increases. \( f(x) \) will be a smooth continuous curve, while \( N_t \) will be a series of discrete points aligned with the curve of \( f(x) \). Both reflect the exponential growth behavior with base 2.

Key Concepts

Graphing Exponential Functions

Graphing exponential functions might sound difficult at first, but it's quite straightforward once you understand the pattern. Exponential functions have the form \( f(x) = a^x \), where \( a \) is a constant, and \( x \) is the variable. For exponential growth as in \( f(x) = 2^x \), each step to the right on the graph increases the value by the base, \( a \), raised to the corresponding power of \( x \).

To plot these functions effectively:

• Select key points, such as when \( x = 0, 1, 2, 3 \), and compute \( f(x) \) for each.
• Note that \( f(x) \) for \( x = 0 \) is always 1, as any number to the power of zero is 1.
• Draw a smooth curve connecting the plotted points for the best visual representation.

This smooth curve will continue to rise to the right without bending downwards, reflecting continuous growth.

Exponential Growth

Exponential growth is a fascinating concept where quantities increase rapidly by a consistent percentage over time. In mathematical terms, exponential growth is modeled by functions like \( f(x) = a^x \), where \( a > 1 \). For functions like \( f(x) = 2^x \) and \( N_t = 2^t \), the growth is doubled at each subsequent value of \( x \) or \( t \).

Important characteristics of exponential growth:

• The growth rate becomes more pronounced over time, speeding up as values increase.
• Exponential functions never touch the x-axis, as their range is from a positive minimum (greater than zero) to infinity.

Graphs of exponential growth have a characteristic "J" shape, initially flat but increasingly steep.

Coordinate Systems

Understanding coordinate systems is key to successfully graphing functions. A coordinate system is a grid defined by horizontal (x-axis) and vertical (y-axis) lines. Each point on this grid is identified by a pair of numbers \((x, y)\), indicating its horizontal and vertical positions respectively.

When plotting exponential functions:

• Identify the domain and range, which for \( f(x) = 2^x \), is \( x \geq 0 \) and \( y > 0 \).
• For discrete functions like \( N_t = 2^t \), plot only at natural number intervals since \( t \in \mathbb{N} \).
• On the graph, \( f(x) \) should appear as a continuous curve, while \( N_t \) would have distinct dot plots.

Understanding and using coordinate systems correctly allows you to effectively visualize and compare both continuous and discrete representations of exponential functions.