Problem 4 - Calculus for Biology and Medicine

Question

In Problems , produce a table for \(t=0,1,2, \ldots, 5\) and graph the function \(N_{t}\). $$ N_{t}=0.2(0.8)^{t} $$

Step-by-Step Solution

Answer

Calculate \( N_t \) for \( t=0,1,2,...,5 \), then graph the decay curve.

Step 1: Understanding the Function

The function given is \( N_t = 0.2(0.8)^t \). Here, \( N_t \) represents the value at time \( t \), \( 0.2 \) is the initial value or coefficient, and \( 0.8 \) is the base of the exponent that represents a common ratio in a geometric sequence.

Step 2: Calculate Values for Each t

We will evaluate \( N_t \) for each value of \( t \) from 0 to 5. - For \( t = 0 \): \ \( N_0 = 0.2(0.8)^0 = 0.2 \times 1 = 0.2 \)- For \( t = 1 \): \ \( N_1 = 0.2(0.8)^1 = 0.2 \times 0.8 = 0.16 \)- For \( t = 2 \): \ \( N_2 = 0.2(0.8)^2 = 0.2 \times 0.64 = 0.128 \)- For \( t = 3 \): \ \( N_3 = 0.2(0.8)^3 = 0.2 \times 0.512 = 0.1024 \)- For \( t = 4 \): \ \( N_4 = 0.2(0.8)^4 = 0.2 \times 0.4096 = 0.08192 \)- For \( t = 5 \): \ \( N_5 = 0.2(0.8)^5 = 0.2 \times 0.32768 = 0.065536 \)

Step 3: Construct the Table

Use the calculated values to construct the table:\[\begin{array}{|c|c|}\hline\text{t} & N_t \\hline0 & 0.2 \1 & 0.16 \2 & 0.128 \3 & 0.1024 \4 & 0.08192 \5 & 0.065536 \\hline\end{array}\]

Step 4: Graph the Function

Graph the values calculated on a coordinate plane with \( t \) on the x-axis and \( N_t \) on the y-axis.- Plot the points: (0, 0.2), (1, 0.16), (2, 0.128), (3, 0.1024), (4, 0.08192), (5, 0.065536).- Connect the points smoothly to display the decay in the function as \( t \) increases, noting the exponential decrease.

Key Concepts

Exponential Decay

Exponential decay refers to a decrease that happens rapidly at first, then slows down over time. In this exercise, we have the function \( N_t = 0.2(0.8)^t \), which is a perfect example of exponential decay. The number 0.8, known as the decay factor, is less than 1, which indicates that as \( t \) increases, \( N_t \) decreases. Each time \( t \) increases by 1, the previous value is multiplied by 0.8. Unlike linear decay, where values decrease by a constant amount, exponential decay decreases by a fixed percentage each time. This results in a curve on a graph that starts steep and curves gently as it moves right.

Function Graphing

When graphing functions, especially those involving exponential decay, it's essential to pay attention to the shape and behavior of the graph. For \( N_t = 0.2(0.8)^t \), with \( t \) on the x-axis and \( N_t \) on the y-axis, you start by plotting points calculated for different \( t \)-values. Here, with values such as (0, 0.2), (1, 0.16), you'll notice the points form a smooth curve that shows a consistent decrease.

• Start by marking each point accurately on graph paper or using graphing software.
• Ensure there's a clear representation of how the decrease happens, showcasing the curvature of exponential decay.
• Connect your dots not with straight lines, but with a curve that gently bends downward as \( t \) increases.

This visualization can help in understanding how quickly or slowly the function decays over the chosen interval of time.

Mathematical Tables

Creating a table provides a clear view of calculations and results for different values. It acts as a reference for both creating graphs and understanding how values change step-by-step. In the original exercise, a table for \( t=0,1,2 \ldots 5 \) was used to find values of \( N_t \).

• For \( t = 0 \), \( N_0 = 0.2 \)
• For \( t = 1 \), \( N_1 = 0.16 \)
• For \( t = 2 \), \( N_2 = 0.128 \)
• Continue this process for other \( t \) values, calculating each using the formula \( N_t = 0.2(0.8)^t \)

Tables help in ensuring all calculations are organized. They make it easier to cross-check results and provide a clear set of data points for graphing the exponential function effectively.