Problem 44 - Calculus Early Transcendentals

Question

A quantity grows exponentially according to \(y(t)=y_{0} e^{i t} .\) What is the relationship between \(m, n,\) and \(p\) such that \(y(p)=\sqrt{y(m) y(n)} ?\)

Step-by-Step Solution

Answer

Question: Find the relationship between \(m\), \(n\), and \(p\) such that \(y(p) = \sqrt{y(m) y(n)}\), where \(y(t) = y_0 e^{it}\) is an exponential growth formula. Answer: The relationship between \(m\), \(n\), and \(p\) is given by \(p = \frac{m + n}{2}\).

Step 1: Write down exponential growth formulas for \(y(m)\), \(y(n)\), and \(y(p)\)

We have the exponential growth formula: \(y(t) = y_0 e^{it}\). Plugging in \(m\), \(n\), and \(p\) for \(t\), we can express \(y(m)\), \(y(n)\), and \(y(p)\) as follows: $$ y(m) = y_0 e^{im}, \quad y(n) = y_0 e^{in},\quad y(p) = y_0 e^{ip} $$

Step 2: Substitute \(y(m)\) and \(y(n)\) in the given equation

We are given \(y(p) = \sqrt{y(m) y(n)}\). Let's substitute the expressions for \(y(m)\) and \(y(n)\) from Step 1: $$ y_0 e^{ip} = \sqrt{y_0 e^{im} \cdot y_0 e^{in}} $$

Step 3: Simplify the equation

Now, let's simplify the equation by using the product-to-exponent rule: \(e^{a} \cdot e^{b} = e^{a + b}\) and square-root property: \(\sqrt{ab}=\sqrt{a}\sqrt{b}\) to obtain: $$ y_0 e^{ip} = \sqrt{y_0^2 e^{i(m + n)}} $$

Step 4: Further simplify and relate the exponentials

The equation may be further simplified by taking the square root of the product of the constant \(y_0^2\) and the exponential term \(e^{i(m + n)}\) separately, resulting in: $$ y_0 e^{ip} = y_0 e^{i\frac{(m + n)}{2}} $$

Step 5: Determine the relationship

Since the exponential terms must be equal, we can conclude that their exponents must also be equal, so we derive the required relationship: $$ ip = i\frac{(m + n)}{2} $$ Divide both sides by \(i\), we get: $$ p = \frac{m + n}{2} $$ Hence, the relationship between \(m\), \(n\), and \(p\) is \(p = \frac{m + n}{2}\).

Key Concepts

Exponential Functions

Exponential functions represent a pattern of growth or decay that increases at a consistent rate. In calculus and other fields of mathematics, these functions are described by the equation of the form \( y(t) = y_0 e^{kt} \), where \( y(t) \) describes the value of the growing quantity at time \( t \), \( y_0 \) represents the initial value, \( e \) is the base of natural logarithms, and \( k \) is a constant that determines the rate of growth or decay.

Understanding exponential functions is vital since they model a wide variety of real-world phenomena such as population growth, radioactive decay, and interest calculation. In the context of the problem in question, we see that not only is the base of the exponent an important number (\( e \)), but the exponent itself is a complex number, which is a segue to another important mathematical concept.

Complex Numbers

Complex numbers expand the idea of the 'number' beyond the traditional concept of the real number line. A complex number is written in the form \( a + bi \), where \( a \) is the real part, \( b \) is the imaginary part, and \( i \) is the imaginary unit with the property that \( i^2 = -1 \). In the original problem, the exponential function uses a complex exponent \( it \), suggesting that our growth pattern oscillates in the complex plane.

One particularly interesting property of complex numbers in the context of exponential growth is Euler's formula, which relates complex exponentials to trigonometric functions. This deep connection is pivotal in fields ranging from quantum physics to electrical engineering. The problem presented requires a clear understanding of complex numbers to solve the exponential equations correctly.

Properties of Exponents

Properties of exponents simplify the manipulation and solving of equations involving exponential terms. To understand and solve exponential growth questions, particularly with complex numbers, a grasp of these properties is essential. Key properties include the product rule \( e^a \cdot e^b = e^{a+b} \), the power rule \( (e^a)^b = e^{ab} \), and the quotient rule \( e^a / e^b = e^{a-b} \).

In the solution provided, we see the use of such properties to simplify the solution to reach the relationship between \( m \), \( n \), and \( p \). The product-to-exponent rule helps in combining the exponents when two exponential terms with the same base are multiplied. This is critical in finding the simplistic form required to solve for the relationship between the variables in the equation.