Problem 2 - Calculus for Biology and Medicine

Question

In Problems , produce a table for \(t=0,1,2, \ldots, 5\) and graph the function \(N_{t}\). $$ N_{t}=6 \cdot 2^{t} $$

Step-by-Step Solution

Answer

The table for \( t=0,1,2, \ldots, 5 \) is: [(0,6), (1,12), (2,24), (3,48), (4,96), (5,192)]. Graph these points to show exponential growth.

Step 1: Understanding the Function

The function given is \( N_t = 6 \cdot 2^t \). This is an exponential function where \( N_t \) represents the value at time \( t \). The base of the exponent is 2, which means the function doubles as \( t \) increases by 1. The initial value (when \( t = 0 \)) is 6.

Step 2: Calculating Values for the Table

Let's calculate the function values for each \( t \) from 0 to 5:- \( t=0 \): \( N_0 = 6 \cdot 2^0 = 6 \cdot 1 = 6 \) - \( t=1 \): \( N_1 = 6 \cdot 2^1 = 6 \cdot 2 = 12 \) - \( t=2 \): \( N_2 = 6 \cdot 2^2 = 6 \cdot 4 = 24 \) - \( t=3 \): \( N_3 = 6 \cdot 2^3 = 6 \cdot 8 = 48 \) - \( t=4 \): \( N_4 = 6 \cdot 2^4 = 6 \cdot 16 = 96 \) - \( t=5 \): \( N_5 = 6 \cdot 2^5 = 6 \cdot 32 = 192 \)

Step 3: Constructing the Table

Create a table with the values calculated in Step 2:| \( t \) | \( N_t \) ||---|---|| 0 | 6 || 1 | 12 || 2 | 24 || 3 | 48 || 4 | 96 || 5 | 192 |This table shows the exponential growth of \( N_t \) as \( t \) increases.

Step 4: Graphing the Function

To graph the function, plot each \( (t, N_t) \) pair on a coordinate plane. The x-axis will represent \( t \) and the y-axis will represent \( N_t \). Plot the points (0,6), (1,12), (2,24), (3,48), (4,96), and (5,192). Connect these points with a smooth curve. Since this is an exponential function, the graph should continuously increase and should resemble a steadily upward-curving line.

Key Concepts

Exponential Growth

Exponential growth refers to the rapid increase in quantity over a certain time period. In mathematics, this is typically expressed with exponential functions. In the function given by the exercise, \( N_t = 6 \cdot 2^t \), the quantity \( N_t \) grows exponentially as time \( t \) increases.

• The number 6 is the initial value, meaning it's the starting point when \( t = 0 \).
• The base of the exponential part, 2, implies that the amount doubles as \( t \) increases by 1.

This characteristic doubling results in a very swift increase in value. For example, when \( t = 0 \), \( N_t = 6 \). At \( t = 5 \), the value has jumped all the way to 192. Understanding this rapid rate of increase is key to grasping the concept of exponential growth.

Mathematical Graphing

Graphing an exponential function like \( N_t = 6 \cdot 2^t \) helps visualize the change over time. To begin graphing this function:

• Place \( t \) on the x-axis (horizontal) and \( N_t \) on the y-axis (vertical).
• Identify and plot critical points such as (0,6), (1,12), (2,24), (3,48), (4,96), and (5,192).
• Connect these points.

The plotted line is not straight but curves upwards, showcasing how the function's value rises exponentially. Graphs like these are essential in identifying patterns and understanding how quantities interact and change over time in real-world situations like population growth or investments.

Function Table Creation

Creating a function table is a fundamental skill in understanding how a function behaves. When you construct a table for the exponential function \( N_t = 6 \cdot 2^t \), you track how the function's value changes over a range. For each time point \( t \):

• Input \( t \) into the function.
• Calculate \( N_t \) by multiplying 6 by 2 raised to the power of \( t \).
• Record both \( t \) and \( N_t \) in the table.

This process reveals a clear pattern, making it easier to predict future values. In our table, the sequence of numbers (6, 12, 24, 48, 96, 192) doesn’t just appear random; it follows a pattern dictated by exponential growth. By practicing this method, you gain insight into structured ways to approach complex mathematical problems.