Problem 45 - Calculus Early Transcendentals

Question

The same exponential growth function can be written in the forms \(y(t)=y_{0} e^{k t}, y(t)=y_{0}(1+r)^{t}\) and \(y(t)=y_{0} 2^{t / T_{2}} .\) Write \(k\) as a function \(r, r\) as a function of \(T_{2}\) and \(T_{2}\) as a function of \(k\)

Step-by-Step Solution

Answer

Question: Express the relationships between the parameters k, r, and T_2 in exponential growth functions. Answer: 1. k = (ln(1+r))/t 2. r = e^(ln(2)/T_2) - 1 3. T_2 = ln(2)/k

Step 1: Equate the exponential growth functions

Since all three expressions denote the same function, we can equate the expressions to one another and solve for the relationships between the parameters. First, let's equate the first two: $$y_0e^{kt} = y_0(1+r)^t$$

Step 2: Solve for k in terms of r

To get \(k\), first divide by \(y_0\) and then apply the natural logarithm on both sides to simplify: $$e^{kt} = (1+r)^t $$ $$kt = ln(1+r)^t$$ $$k = \frac{ln(1+r)}{t}$$ Now we have \(k\) as a function of \(r\).

Step 3: Equate the second and the third functions

Now, let's equate the second exponential function with the third one: $$y_0(1+r)^t = y_0 2^{t / T_2}$$

Step 4: Solve for r in terms of T_2

Again, divide by \(y_0\) and then apply the natural logarithm on both sides: $$(1+r)^t = 2^{t / T_2}$$ $$t ln(1+r) = \frac{t}{T_2} ln(2)$$ Now, isolate \(r\): $$r = e^{\frac{ln(2)}{T_2}} - 1$$ So, we have \(r\) as a function of \(T_2\).

Step 5: Solve for T_2 in terms of k

Use the previously found expression of \(k\) and \(r\) to express \(T_2\) in terms of \(k\): $$k = \frac{ln(1+r)}{t} \Rightarrow r = e^{kt} - 1$$ Now, replace r with the expression in terms of \(T_2\): $$e^{kt} - 1 = e^{\frac{ln(2)}{T_2}} - 1$$ Now, solve for \(T_2\): $$kt = \frac{ln(2)}{T_2}$$ $$T_2 = \frac{ln(2)}{k}$$ In conclusion, the relationships between the parameters are: 1. \(k = \frac{ln(1+r)}{t}\) 2. \(r = e^{\frac{ln(2)}{T_2}} - 1\) 3. \(T_2 = \frac{ln(2)}{k}\)

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions used to model phenomena exhibiting multiplicative change over time. They are crucial in fields like biology, finance, and physics. Exponential functions can be written in several forms, such as \( y(t) = y_0 e^{kt} \), where \( y_0 \) is the initial quantity, \( k \) is the growth rate, and \( t \) is time.
These functions exhibit the property of exponential growth. This means the rate of growth is proportional to the current value, leading to faster changes as the quantity increases. Another common form is \( y(t) = y_0 (1 + r)^t \), where \( r \) signifies the growth rate per time unit. As \( t \) increases, the quantity grows much faster than it would with linear growth.
Exponential growth is not just a theoretical construct but a practical tool used to model population growth, financial investments, and even certain chemical reactions. Understanding the versatility of exponential functions allows you to approach diverse problems with confidence.

Logarithms

Logarithms are the inverses of exponential functions and play a vital role in solving equations involving exponential terms. If you have an exponential function \( y = b^x \), a logarithm helps you find \( x \) when \( y \) and \( b \) are known. Logarithms can be particularly useful when equating exponential functions to find unknown rates or times.
In the natural logarithm, denoted as \( \ln \), the base \( e \) is used. This is the mathematical constant approximately equal to 2.718. Converting between exponential and logarithmic forms can simplify the process of solving for various growth parameters. For example, if you need to solve for \( k \) in \( e^{kt} = (1+r)^t \), taking the natural logarithm of both sides simplifies the equation to \( kt = \ln(1+r)^t \).
Using logarithms in mathematical modeling can demystify complex exponential processes, turning challenging non-linear scenarios into more manageable linear ones. Understanding and applying logarithms ensures you can reverse exponential processes, facilitating deeper analyses of growth phenomena.

Mathematical Modeling

Mathematical modeling involves creating mathematical representations of real-world situations to predict or explain complex phenomena. This powerful tool employs equations and functions to simulate and understand the implications of real-world actions or processes.
In contexts requiring exponential models, models are constructed to understand how a system grows or shrinks over time. For example, when modeling population growth or radioactive decay, exponential functions represent the change. To create effective models, it's crucial to derive relationships like those between \( k \), \( r \), and \( T_2 \), as shown in the exercise.
Successful modeling balances complexity with simplicity. It requires understanding underlying principles and interpreting parameters accurately. By articulating the relationship between \( k = \frac{\ln(1+r)}{t} \), \( r = e^{\frac{\ln(2)}{T_2}} - 1 \), and \( T_2 = \frac{\ln(2)}{k} \), you can adjust models to fit different scenarios. This makes your solutions efficient and tailored to specific cases. Learning to model mathematically equips you to solve diverse problems with a structured and analytical approach.