Problem 4 - Calculus for Biology and Medicine

Question

produce a table for \(t=0,1,2, \ldots, 5\) and graph the function \(N_{t}\) $$ N_{t}=(0.3)(0.9)^{t} $$

Step-by-Step Solution

Answer

Calculate each \(N_t\), organize in a table, and plot the points on a graph.

Step 1: Understand the Function

The function given is \(N_t = 0.3 \times 0.9^t\). This represents a sequence where the value at each point in time \(t\) is determined by multiplying 0.3 by 0.9 raised to the power of \(t\). This type of function is an exponential decay.

Step 2: Calculate Values for Each t

We need to calculate \(N_t\) for each value of \(t\) from 0 to 5.- For \(t = 0\): \(N_0 = 0.3 \times 0.9^0 = 0.3 \times 1 = 0.3\)- For \(t = 1\): \(N_1 = 0.3 \times 0.9^1 = 0.3 \times 0.9 = 0.27\)- For \(t = 2\): \(N_2 = 0.3 \times 0.9^2 = 0.3 \times 0.81 = 0.243\)- For \(t = 3\): \(N_3 = 0.3 \times 0.9^3 = 0.3 \times 0.729 = 0.2187\)- For \(t = 4\): \(N_4 = 0.3 \times 0.9^4 = 0.3 \times 0.6561 = 0.19683\)- For \(t = 5\): \(N_5 = 0.3 \times 0.9^5 = 0.3 \times 0.59049 = 0.177147\)

Step 3: Create the Table

Now we organize the results into a table:\[\begin{array}{|c|c|}\hlinet & N_t \\hline0 & 0.3 \1 & 0.27 \2 & 0.243 \3 & 0.2187 \4 & 0.19683 \5 & 0.177147 \\hline\end{array}\]

Step 4: Graph the Function

To graph this function, plot the points \((t, N_t)\) calculated in Step 2:- Plot the points (0, 0.3), (1, 0.27), (2, 0.243), (3, 0.2187), (4, 0.19683), (5, 0.177147) on a coordinate plane.- Connect these points with a smooth curve that shows the decay as \(t\) increases. This curve will decrease exponentially.

Key Concepts

Function Graphing

Graphing a function is a fundamental part of understanding its behavior visually. In this exercise, the function given is an exponential decay model represented by \(N_t = 0.3 \times 0.9^t\). The graph for this function displays how \(N_t\) values change as \(t\) increases.

• Start by calculating the values of \(N_t\) for different values of \(t\).
• The goal is to understand how the function decreases over time.
• Exponential decay functions characteristically decrease rapidly at first and then level off more slowly.

For this graph, you'll plot the points calculated from the function for the given values of \(t\). Connecting these points with a smooth curve rather than straight lines reflects the continuous nature of the function. The curve will trend downwards more gently as \(t\) increases, visually representing the concept of exponential decay.

Sequence Calculation

Calculating sequences involves finding the individual terms in a series, which is a fundamental aspect of exploring functions. In the given problem, you needed to calculate \(N_t = 0.3 \times 0.9^t\) for \(t = 0\) to \(5\).

• For \(t = 0\), \(N_0 = 0.3 \times 0.9^0 = 0.3\).
• For \(t = 1\), \(N_1 = 0.3 \times 0.9 = 0.27\).
• Similar calculations follow for \(t = 2\) to \(t = 5\).

In this sequence, each step involves multiplying the previous term by the decay factor, 0.9.
This consistent multiplication by a number less than one is the hallmark of sequence calculations involving exponential decay. This sequence serves as our stepping stone to predict further values or understand how rapidly values decrease.

Coordinate Plane Plotting

Plotting points on the coordinate plane gives a visual understanding of the function's behavior over time. It reveals trends and patterns in sequences that might not be immediately obvious with numerical data alone.

• Each point on the graph corresponds to a pair \((t, N_t)\).
• In this task, the points - \((0, 0.3)\),- \((1, 0.27)\), - \((2, 0.243)\), and so on, are plotted.

Position these points precisely on the coordinate plane. For accuracy, ensure the scales on your axes reflect the range of your values effectively.
Once these points are marked, they are connected with a curve that shows the rate of decay, emphasizing the exponential nature of the function. Plotting in this way aids in grasping complex behavior and long-term trends within the function.