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, \( \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \).

Step 1: Understand the Definition

The hyperbolic sine function is defined as \( \sinh(y) = \frac{1}{2}(e^y - e^{-y}) \). We need to prove the expression for \( \sinh^{-1}(x) \). This means we will solve the equation \( x = \sinh(y) \) for \( y \).

Step 2: Set Up the Equation

We start with the equation \( x = \frac{1}{2}(e^y - e^{-y}) \). Our task is to isolate \( y \). Multiplying both sides of the equation by 2 to simplify, we have \( 2x = e^y - e^{-y} \).

Step 3: Multiply Through by \( e^y \)

Multiply both sides of the equation \( 2x = e^y - e^{-y} \) by \( e^y \) to eliminate the negative exponent: \( 2x \cdot e^y = e^{2y} - 1 \).

Step 4: Rearrange to Solve for \( e^{2y} \)

Rearrange the equation \( e^{2y} - 2x \cdot e^y - 1 = 0 \) and treat it as a quadratic in terms of \( e^y \).

Step 5: Solve the Quadratic Equation

Using the quadratic formula, \( e^y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), in this case, \( a = 1 \), \( b = -2x \), and \( c = -1 \). Substitute these values: \( e^y = \frac{2x \pm \sqrt{(2x)^2 + 4}}{2} \).

Step 6: Select the Correct Root

Select the positive root because \( e^y \) is always positive: \( e^y = x + \sqrt{x^2 + 1} \).

Step 7: Solve for \( y \)

Take the natural logarithm of both sides to solve for \( y \): \( y = \ln(x + \sqrt{x^2 + 1}) \).

Step 8: Verify the Expression

The expression \( y = \ln(x + \sqrt{x^2 + 1}) \) matches the standard expression for the inverse hyperbolic sine function, \( \sinh^{-1}(x) \).

Key Concepts

Hyperbolic Sine Function

The hyperbolic sine function, denoted as \( \sinh(y) \), is an important concept in mathematics, similar to the regular sine function, but it utilizes exponential functions. This function is defined by the equation \( \sinh(y) = \frac{1}{2}(e^y - e^{-y}) \). Here, \( e^y \) and \( e^{-y} \) are exponential functions, which are crucial in forming the hyperbolic functions that are widely used in various applications, such as engineering and hyperbolic geometry.

If we want to find the inverse of the hyperbolic sine function, or \( \sinh^{-1}(x) \), it means we're looking to find a value \( y \) given \( x \) such that \( x = \sinh(y) \). Solving for \( y \) involves manipulating the original equation and using properties of exponential functions to derive a formula that expresses \( y \) in terms of \( x \). The result can be used to easily compute the inverse hyperbolic sine value for any real number \( x \).

Understanding \( \sinh(y) \) and its inverse is crucial for solving problems where hyperbolic functions appear, providing insight into various complex calculations.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically in the form \( ax^2 + bx + c = 0 \). In our context, once we derived the expression \( e^{2y} - 2x \cdot e^y - 1 = 0 \), it became necessary to identify it as a quadratic equation in terms of \( e^y \).

The solution of quadratic equations allows for isolating terms or finding specific values for variables. Here, it involves the quadratic formula:

• \( e^y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

For the equation \( e^{2y} - 2x \cdot e^y - 1 = 0 \):

• \( a = 1 \)
• \( b = -2x \)
• \( c = -1 \)

Substituting these values into the formula helps solve for \( e^y \), which is a critical step in finding the inverse hyperbolic sine function.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the inverse operation of the exponential function with the base \( e \), where \( e \approx 2.71828 \). In the context of our exercise, after solving the quadratic equation for \( e^y \) and obtaining values, selecting the positive root is necessary. We arrive at the formulation:

• \( e^y = x + \sqrt{x^2 + 1} \)

Once we isolate \( e^y \), the natural logarithm is applied to move us from \( e^y \) to \( y \).

• \( y = \ln(x + \sqrt{x^2 + 1}) \)

This relationship simplifies the solution by providing a direct formula to compute the value of \( y \), which is significant for determining \( \sinh^{-1}(x) \).

Leveraging the natural logarithm is crucial when dealing with exponential equations, making it possible to transition between exponential and linear forms.

Exponential Functions

Exponential functions are expressed as \( f(x) = a^x \), with \( a \) being a positive constant. In particular, the base \( e \) is the most common due to its natural properties, leading to functions like \( e^x \) and \( e^{-x} \), which are featured in hyperbolic functions.

In our exercise, exponential functions are used to represent \( \sinh(y) \) in its standard form \( \frac{1}{2}(e^y - e^{-y}) \). Essential steps include manipulating expressions involving \( e^y \) to solve quadratic equations.

Exponential functions have the following characteristics:

• They grow or decay at rates proportional to their current value.
• \( e^x \) grows as \( x \) increases.
• \( e^{-x} \) decreases as \( x \) increases.

By understanding and using exponential functions, we effectively solve equations across various mathematical contexts, including the application involved in hyperbolic functions.