Problem 433 - Calculus Volume 1

Question

For the following exercises, solve each problem. Prove the expression for \(\sinh ^{-1}(x)\). Multiply \(x=\sinh (y)=(1 / 2)\left(e^{y}-e^{-y}\right)\) by \(2 e^{y}\) and solve for \(y\). Does your expression match the textbook?

Step-by-Step Solution

Answer

Yes, the expression matches: \( \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \).

Step 1: Understanding the Problem

We need to solve the equation by rewriting it from a hyperbolic sine identity to solve for the inverse hyperbolic sine, denoted as \( \sinh^{-1}(x) \). The given equation is related to the hyperbolic sine function: \( x = \sinh(y) = \frac{1}{2} (e^y - e^{-y}) \). Our goal is to manipulate this equation to express \( y \) in terms of \( x \).

Step 2: Multiply by \(2e^y\)

Starting with the equation \( x = \frac{1}{2} (e^y - e^{-y}) \), we first multiply both sides by \( 2e^y \) to eliminate the fraction:\[ 2xe^y = (e^y)^2 - 1 \].Here, \((e^y)^2\) comes from \( e^y \times e^{-y} = 1 \), simplifying the right side.

Step 3: Reorganize into a Quadratic Equation

Rearrange the equation \( 2xe^y = (e^y)^2 - 1 \) to form a standard quadratic equation:\[ (e^y)^2 - 2xe^y - 1 = 0 \].Now, this is in the form of \( ax^2 + bx + c = 0 \), where \( e^y \) is your variable.

Step 4: Solve the Quadratic Equation

Use the quadratic formula to solve for \( e^y \). The quadratic formula is \( e^y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). For our equation:- \( a = 1 \)- \( b = -2x \)- \( c = -1 \)Plug these values into the quadratic formula:\[ e^y = \frac{-(-2x) \pm \sqrt{(-2x)^2 - 4\cdot1\cdot(-1)}}{2\cdot1} \].

Step 5: Simplify the Solution

Simplify the quadratic solution to get:\[ e^y = \frac{2x \pm \sqrt{4x^2 + 4}}{2} \].This simplifies further to:\[ e^y = x \pm \sqrt{x^2 + 1} \].Since \( e^y \) must be positive, we take the positive solution:\[ e^y = x + \sqrt{x^2 + 1} \].

Step 6: Solve for \(y\)

Taking the natural logarithm of both sides to solve for \( y \):\[ y = \ln(x + \sqrt{x^2 + 1}) \].Thus, the inverse hyperbolic sine function is expressed as:\[ \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \].

Step 7: Verify the Solution

Compare your final expression \( \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \) with references or the textbook formula for the inverse hyperbolic sine to confirm accuracy. Generally, this matches standard mathematical references.

Key Concepts

Understanding Hyperbolic Sine

The hyperbolic sine function, denoted as \( \sinh(y) \), is a unique function that is similar yet distinct from the trigonometric sine function. It is defined by the equation:

• \( \sinh(y) = \frac{1}{2}(e^y - e^{-y}) \)

This expression involves exponential functions \( e^y \) and \( e^{-y} \), which are key in understanding the behavior of hyperbolic functions.
Unlike trigonometric functions, hyperbolic functions are based on the symmetrical properties of exponential growth and decay.
Understanding \( \sinh(y) \) helps us in dealing with curves similar to hyperbolas, showcasing its importance in many mathematical and engineering concepts.

Quadratic Equation and its Application

In solving equations like \( x = \sinh(y) \), we often encounter quadratic equations. A quadratic equation is generally written as:

• \( ax^2 + bx + c = 0 \)

In our context, by manipulating the original hyperbolic sine equation \( x = \frac{1}{2} (e^y - e^{-y}) \), we can reorganize it to become a form of quadratic equation:

• \((e^y)^2 - 2xe^y - 1 = 0 \)

The quadratic equation provides a structured method to find unknown variables using the quadratic formula, allowing us to solve for \( e^y \) effectively.

Role of Exponential Functions

Exponential functions are crucial components of hyperbolic functions. An exponential function is expressed as \( e^x \), where \( e \) is a mathematical constant approximately equal to 2.71828.
These functions describe processes involving growth or decay in various scientific and engineering fields. In our problem, we see exponential functions such as \( e^y \) and \( e^{-y} \) within the expressions for hyperbolic functions.
This exponential behavior allows the manipulation and transformation necessary to form quadratic equations in the solution process. Consequently, understanding exponential functions aids in grasping more complex relationships and transformations in mathematics.

Importance of the Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the inverse function of the exponential function \( e^x \). It is used to solve for variables that are exponentiated in equations.
In mathematics, when we encounter expressions like \( e^y = x + \sqrt{x^2 + 1} \), applying the natural logarithm to both sides allows us to "undo" the exponential operation.
Thus, using \( \ln \), the solution for \( y \) becomes:

• \( y = \ln(x + \sqrt{x^2 + 1}) \)

This indicates \( \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \), showing the critical role of natural logarithms in deriving the inverse hyperbolic functions efficiently.