Problem 4 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Write each of the following functions in the form \(f(t)=A e^{k t} .\) a. \(\quad f(t)=5 \cdot 10^{t}\) b. \(\quad f(t)=5 \cdot 10^{-t}\) c. \(\quad f(t)=7 \cdot 2^{t}\) d. \(\quad f(t)=5 \cdot 2^{-t}\) e. \(\quad f(t)=5\left(\frac{1}{2}\right)^{t}\) f. \(\quad f(t)=5\left(\frac{1}{2}\right)^{-t}\)

Step-by-Step Solution

Answer

(a) A = 5, k = \ln(10); (b) A = 5, k = -\ln(10); (c) A = 7, k = \ln(2); (d) A = 5, k = -\ln(2); (e) A = 5, k = -\ln(2); (f) A = 5, k = \ln(2).

Step 1: Express the Function in Exponential Form (a)

The given function is \( f(t) = 5 \cdot 10^t \). We can express the base 10 exponent as an exponential function using base \( e \):\[ 10^t = e^{t \ln(10)} \]Substitute this into the original function:\[ f(t) = 5 \cdot e^{t \ln(10)} \]

Step 2: Express the Function in Exponential Form (b)

The given function is \( f(t) = 5 \cdot 10^{-t} \). We can express the base 10 exponent as an exponential function using base \( e \):\[ 10^{-t} = e^{-t \ln(10)} \]Substitute this into the original function:\[ f(t) = 5 \cdot e^{-t \ln(10)} \]

Step 3: Express the Function in Exponential Form (c)

The given function is \( f(t) = 7 \cdot 2^t \). We can express the base 2 exponent as an exponential function using base \( e \):\[ 2^t = e^{t \ln(2)} \]Substitute this into the original function:\[ f(t) = 7 \cdot e^{t \ln(2)} \]

Step 4: Express the Function in Exponential Form (d)

The given function is \( f(t) = 5 \cdot 2^{-t} \). We can express the base 2 exponent as an exponential function using base \( e \):\[ 2^{-t} = e^{-t \ln(2)} \]Substitute this into the original function:\[ f(t) = 5 \cdot e^{-t \ln(2)} \]

Step 5: Express the Function in Exponential Form (e)

The given function is \( f(t) = 5 \left( \frac{1}{2} \right)^t \). Recognizing that \( \frac{1}{2} = 2^{-1} \), we rewrite the function:\[ \left( \frac{1}{2} \right)^t = 2^{-t} \]Now, express the base 2 exponent in terms of base \( e \):\[ 2^{-t} = e^{-t \ln(2)} \]Substitute this into the original function:\[ f(t) = 5 \cdot e^{-t \ln(2)} \]

Step 6: Express the Function in Exponential Form (f)

The given function is \( f(t) = 5 \left( \frac{1}{2} \right)^{-t} \). Recognizing that \( \left( \frac{1}{2} \right)^{-t} = 2^t \), express it in terms of the base \( e \):\[ 2^t = e^{t \ln(2)} \]Substitute this into the original function:\[ f(t) = 5 \cdot e^{t \ln(2)} \]

Key Concepts

Base Conversion

Base conversion is the process of changing a number from one base to another. In the context of exponential functions, it often involves converting expressions to base \(e\), commonly known as the natural exponential base. This is essential because many mathematical models in science and engineering utilize the base \(e\) due to its natural properties in calculus. To convert a base-\(b\) exponentiation to base \(e\), we use the formula \(b^t = e^{t \ln(b)}\). Here, \(\ln(b)\) is the natural logarithm of \(b\). So, when you encounter a function such as \(5 \cdot 10^t\), it can be expressed as \(5 \cdot e^{t \ln(10)}\). This conversion simplifies further calculations and assessments when dealing with exponential growth or decay models.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is the logarithm to the base \(e\), where \(e\) is approximately 2.71828. It is a critical concept in mathematics due to its relationship with exponential functions and calculus. When you take the natural logarithm of a number, you're essentially asking "To what power must \(e\) be raised to get \(x\)?" For example, in converting a base 2 exponent to base \(e\), we utilize \(\ln(2)\). If you have \(2^t\), and you want it in the form \(e\), you calculate \(e^{t \ln(2)}\). By using the natural logarithm, we can linearize exponential growth or decay, making it easier to manipulate algebraically and integrate into broader mathematical models.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions and equations to represent real-world phenomena. Exponential functions are frequently used in modeling natural processes because they can describe growth and decay. Examples include population growth, radioactive decay, and compound interest. The form \(f(t) = A e^{k t}\) is especially powerful because it reflects continuous growth (or decay) compounded over time. In this representation, \(A\) is the initial value or starting point, \(e\) is the base of natural logarithms that reflects continuous processes, and \(k\) is the rate of growth or decay. By transforming other bases into exponential models with base \(e\), mathematicians can more effectively integrate and differentiate them, enhancing the model's application and predictive power.

Function Transformation

Function transformation involves changing a function's formula to produce variations that can still model the same phenomena in different ways. Changing exponential functions between different bases is an example of this concept. This can involve converting a function from one base to another to simplify the process of analyzing the behavior of exponential relationships. In the exercise given, each function is transformed to the form \(f(t) = Ae^{kt}\), to simplify analysis and calculations. You also see this principle in transforming \(\left( \frac{1}{2} \right)^t\) to its equivalent form based on base \(e\), which is\( e^{-t \ln(2)}\). With transformations, it becomes easier to identify and work with properties such as shifts, stretches, and compressions, making it a robust tool in both theoretical and applied mathematics.