Problem 65 - Thomas Calculus in SI Units

Question

When hyperbolic function keys are not available on a calculator, it is still possible to evaluate the inverse hyperbolic functions by expressing them as logarithms, as shown here. (We give one derivation in Section 8.4.) $$\begin{aligned}&\sinh ^{-1} x=\ln (x+\sqrt{x^{2}+1}), \quad-\infty < x<\infty\\\&\cosh ^{-1} x=\ln (x+\sqrt{x^{2}-1}), \quad x \geq 1\\\ &\tanh ^{-1} x=\frac{1}{2} \ln \frac{1+x}{1-x}, \quad|x| < 1\\\&\operatorname{sech}^{-1} x=\ln \left(\frac{1+\sqrt{1-x^{2}}}{x}\right), \quad0 < x \leq 1\\\&\operatorname{csch}^{-1} x=\ln \left(\frac{1}{x}+\frac{\sqrt{1+x^{2}}}{|x|}\right), \quad x \neq 0\\\&\operatorname{coth}^{-1} x=\frac{1}{2} \ln \frac{x+1}{x-1}, \quad |x|>1\end{aligned}$$ Use the formulas in the box here to express the numbers in terms of natural logarithms. $$\operatorname{sech}^{-1}(3 / 5)$$

Step-by-Step Solution

Answer

\( \operatorname{sech}^{-1}(3/5) = \ln(3) \).

Step 1: Identify the Given Expression

We are asked to evaluate \( \operatorname{sech}^{-1}(3 / 5) \) using its logarithmic form. From the provided formulas, we know that: \[ \operatorname{sech}^{-1} x = \ln \left(\frac{1+\sqrt{1-x^{2}}}{x}\right) \] where \( 0 < x \leq 1 \). Here, \( x = \frac{3}{5} \).

Step 2: Substitute the Value

Substitute \( x = \frac{3}{5} \) into the formula: \[ \operatorname{sech}^{-1} \left(\frac{3}{5}\right) = \ln \left(\frac{1+\sqrt{1-\left(\frac{3}{5}\right)^2}}{\frac{3}{5}}\right) \]

Step 3: Simplify Inside the Square Root

Calculate \( \left(\frac{3}{5}\right)^2 \): \( \left(\frac{3}{5}\right)^2 = \frac{9}{25} \).Subtract from 1: \( 1 - \frac{9}{25} = \frac{16}{25} \).Calculate the square root: \( \sqrt{\frac{16}{25}} = \frac{4}{5} \).

Step 4: Simplify the Expression

Substitute the square root value back: \[ \operatorname{sech}^{-1} \left(\frac{3}{5}\right) = \ln \left(\frac{1 + \frac{4}{5}}{\frac{3}{5}}\right) \]Simplify: \( 1 + \frac{4}{5} = \frac{9}{5} \).Divide: \( \frac{9/5}{3/5} = 3 \).

Step 5: Evaluate the Natural Logarithm

Calculate the natural logarithm of the simplified expression: \[ \operatorname{sech}^{-1} \left(\frac{3}{5}\right) = \ln(3) \]

Key Concepts

Hyperbolic Functions

Hyperbolic functions are mathematical functions that are analogous to trigonometric functions but based on hyperbolas instead of circles. These functions are significant in various fields such as calculus and complex analysis. Just as sine and cosine are the building blocks for trigonometric functions, the hyperbolic sine (\(\sinh x\)) and hyperbolic cosine (\(\cosh x\)) serve a similar role for hyperbolic functions. They describe the shape of a hyperbola, hence the name.

The basic hyperbolic functions include:

• Hyperbolic sine: \[sinh(x) = \frac{e^x - e^{-x}}{2}\]
• Hyperbolic cosine: \[cosh(x) = \frac{e^x + e^{-x}}{2}\]
• Hyperbolic tangent: \[tanh(x) = \frac{sinh(x)}{cosh(x)}\]

Each of these functions has a corresponding inverse, such as inverse hyperbolic sine (\(\sinh^{-1} x\)) which are often used in solving equations where it’s convenient to work with hyperbolic identities. The inverse hyperbolic functions can be expressed in terms of logarithmic functions, which allow us to calculate them even if our calculator lacks specific hyperbolic keys.

Natural Logarithms

Natural logarithms are a fundamental concept in mathematics, particularly useful for their properties in calculus and analysis. The natural logarithm, denoted as \(\ln(x)\), is the logarithm to the base of Euler's number, \(e\), which is approximately 2.71828.

One of the primary reasons natural logarithms are useful is their behavior in differentiation and integration. They are particularly simple when it comes to these operations, which makes them critical tools in calculus.

• Differentiation: If \(f(x) = \ln(x)\), then \(f'(x) = \frac{1}{x}\).
• Integration: The integral of \(\frac{1}{x}\) is \(\ln|x| + C\), where \(C\) is the constant of integration.

Natural logarithms are also directly connected to exponential functions: \[e^{\ln(x)} = x\]\, allowing for the transformation and manipulation of expressions, which is crucial for understanding how inverse hyperbolic functions can be represented using natural logs.

Calculus

Calculus is the branch of mathematics that deals with rates of change and the accumulation of quantities. It encompasses two main areas: differential calculus and integral calculus. Differential calculus focuses on the concept of a derivative, which represents the rate at which a quantity changes. Integral calculus, on the other hand, is concerned with the accumulation of quantities, such as areas under a curve.

In the context of inverse hyperbolic functions, calculus provides the framework for understanding how these functions behave and change. For instance, the derivative of inverse hyperbolic sine (\(\sinh^{-1}(x)\)) is \(\frac{1}{\sqrt{x^2 + 1}}\), derived using implicit differentiation and the chain rule. Similarly, calculus helps in deriving and understanding the integrals involving hyperbolic functions.

• For example, the integral of \(\sinh(x)\) is straightforward: \[\int sinh(x)\, dx = cosh(x) + C\]
• Conversely, integration involving inverse hyperbolic functions can be expressed using logarithms, a link to natural logarithms previously described.

Calculus also plays a vital role in functions transformation, optimization, and the solving of complex equations, making it indispensable for analyzing hyperbolic functions and their inverses.