Problem 6 - Thomas Calculus

Question

Rewrite the expressions in Exercises \(5-10\) in terms of exponentials and simplify the results as much as you can. $$ \sinh (2 \ln x) $$

Step-by-Step Solution

Answer

\( \sinh(2 \ln x) = \frac{x^4 - 1}{2x^2} \)

Step 1: Recall Hyperbolic Sine Definition

The hyperbolic sine function is defined as \( \sinh(u) = \frac{e^u - e^{-u}}{2} \). Our task is to simplify \( \sinh(2 \ln x) \) using exponentials. The given argument \( 2 \ln x \) will be used in place of \( u \).

Step 2: Substitute and Expand Argument

Substitute \( u = 2 \ln x \) into the definition: \( \sinh(2 \ln x) = \frac{e^{2 \ln x} - e^{-2 \ln x}}{2} \).

Step 3: Use Exponential and Logarithmic Identity

Recall the identity \( e^{\ln a} = a \). Applying it, we have \( e^{2 \ln x} = (e^{\ln x})^2 = x^2 \) and \( e^{-2 \ln x} = (e^{-\ln x})^2 = \left(\frac{1}{x}\right)^2 = \frac{1}{x^2} \).

Step 4: Substitute Back and Simplify

Substitute these results back into the expression: \( \sinh(2 \ln x) = \frac{x^2 - \frac{1}{x^2}}{2} \). Combine the terms in the numerator over a common denominator to get \( \frac{x^4 - 1}{2x^2} \). Therefore, the simplified expression is \( \frac{x^4 - 1}{2x^2} \).

Key Concepts

Exponentials

Exponentials are a fundamental mathematical concept used extensively in various areas of mathematics, science, and engineering. The main idea is based on the exponent, which represents how many times a number is multiplied by itself. For example, in the expression \( e^x \), \( e \) is known as Euler’s number (approximately 2.718), and \( x \) is the exponent.

• Exponential functions grow very rapidly and are used to model growth processes, such as population growth or radioactive decay.
• They can express large amounts of growth or decay succinctly, thanks to the nature of their increase or decrease rates.

Exponentials are also valuable when working with transformations, as they convert multiplicative processes (like compound interest) into additive ones when logged. This is crucial for simplifying equations and understanding relationships in fields like calculus.

Logarithmic Identity

Logarithmic identities are powerful tools that help simplify expressions by transforming them—making complex multiplicative relationships linear. The basic idea behind a logarithm is that it represents the power to which a base must be raised to produce a given number. For instance, the natural logarithm \( \,\ln(x) \) is with base \( e \).

• A common identity is \( e^{\ln a} = a \), demonstrating how exponentials can be reversed using logarithms.
• This identity highlights the inverse relationship between natural logarithms and exponentials, allowing simplifications like converting \( e^{2\ln x} \) to \( x^2 \).

When using logarithmic identities, remember that they help connect the exponential growth or decay processes back to a simple number multiplication framework. This makes calculations more manageable and insights clearer when dealing with exponential equations.

Hyperbolic Sine

The hyperbolic sine function, denoted as \( \sinh(u) \), is a specific type of hyperbolic function analogous to the trigonometric sine function. It is defined using exponential functions:\[\sinh(u) = \frac{e^u - e^{-u}}{2}\]

• The hyperbolic sine function is used in various areas, such as hyperbolic geometry and in describing the shape of hanging cables, known as catenaries.
• It follows the identity, allowing transformations of arguments and simplifying expressions using underlying exponential principles.

In the example \( \sinh(2\ln x) \), the function helps transform the logarithmic argument into an expression solely in terms of \( x \). The use of exponentials here facilitates simplifying the expression further into a form that is easier to interpret mathematically, such as \( \frac{x^4 - 1}{2x^2} \). Understanding \( \sinh(u) \) thus provides insight into more complex behaviors using simpler exponential components.