Problem 3 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Find values of \(C\) and \(k\) so that \(P(t)=C e^{k t}\) matches the data. a. \(P(0)=5 \quad P(2)=10\) b. \(P(0)=10 \quad P(2)=5\) c. \(P(0)=2 \quad P(5)=10\) d. \(P(0)=10 \quad P(5)=10\) e. \(P(0)=5 \quad P(2)=2\) f. \(P(0)=8 \quad P(10)=6\) g. \(P(1)=5 \quad P(2)=10\) h. \(P(2)=10 \quad P(10)=20\)

Step-by-Step Solution

Answer

C and k are found using P(0) to get C and another condition to solve for k.

Step 1: Understanding Initial Condition P(0)

The initial condition for each part indicates that when \(t = 0\), \(P(0) = C\). This gives us the value of \(C\).

Step 2: Using Second Condition to Find k

For each part, use the equation \(P(t) = Ce^{kt}\) with the second condition given to find \(k\). Substitute \(C\) from Step 1 and solve for \(k\).

Key Concepts

Initial Value Problems

Initial value problems are a central concept in mathematical modeling involving differential equations. You typically start with a differential equation that describes a process or phenomenon, then use known values at specific points to find a solution that matches the scenario. In the context of exponential growth or decay, the initial condition gives you the starting point or the value of your function at time zero. This is crucial as it helps define the constant factor in your equation, often represented by the variable \( C \). For instance, in the problem presented, when \( t = 0 \), the function \( P(t) = C e^{kt} \) simplifies to \( P(0) = C \). This means that the initial value of the function directly equals the constant \( C \), establishing the baseline from which growth or decay is measured.

Exponential Functions

Exponential functions form the backbone of models describing processes that grow or shrink at rates proportional to their current size. The general form of an exponential function can be written as \( f(t) = C e^{kt} \), where:

• \( C \) is a constant representing the initial amount or size.
• \( e \) is the base of the natural logarithm, approximately 2.718.
• \( k \) is the growth (if positive) or decay (if negative) rate constant.

The exponential function is unique in that it scales with time in a multiplicative fashion. If you understand exponential functions, you can model scenarios involving continuous growth (like populations) or decay (like radioactive materials). The task now is to determine the values of \( C \) and \( k \) such that the function fits specific data points, highlighting the modeling power of exponential equations in representing real-world situations.

Parameter Estimation

Parameter estimation is the process of using data to determine the values of parameters within a mathematical model. For an exponential function like \( P(t) = C e^{kt} \), this means finding the values of \( C \) and \( k \) that best fit the given data points. This can be achieved through algebraic manipulation and solving.To estimate the parameters, you start with the data points provided:

• The initial condition usually gives you \( C \)
• The subsequent data point helps you solve for \( k \)

For example, once you know \( C \) from \( P(0) \) and have another point like \( P(t_1) = y \), plug these into the formula:

• \( y = C e^{kt_1} \)

After rearranging, find \( k \) by solving \( k = \frac{1}{t_1} \ln\left(\frac{y}{C}\right) \). This estimation process is crucial as it ensures your model accurately describes the observed behavior.