Problem 74 - Calculus Early Transcendentals

Question

Use a calculator to evaluate each expression, or state that the value does not exist. Report answers accurate to four decimal places. a. \(\operatorname{coth} 4\) b. \(\tanh ^{-1} 2\) c. \(\operatorname{csch}^{-1} 5\) d. \(\left.\operatorname{csch} x\right|_{1 / 2} ^{2}\) e. \(\ln | \tanh (x / 2) \|_{1}^{10} \quad\) f. \(\left.\tan ^{-1}(\sinh x)\right|_{-3} ^{3} \quad\) g. \(\left.\frac{1}{4} \operatorname{coth}^{-1}\left(\frac{x}{4}\right)\right|_{20} ^{\frac{36}{6}}\)

Step-by-Step Solution

Answer

Question: Evaluate the following expressions and report the answers accurate to four decimal places. a. \(\operatorname{coth} 4\) b. \(\tanh^{-1} 2\) c. \(\operatorname{csch}^{-1} 5\) d. \(\left.\operatorname{csch} x\right|_{1 / 2} ^{2}\) e. \(\ln | \tanh (x / 2) \|_{1}^{10}\) f. \(\left.\tan^{-1}(\sinh x)\right|_{-3}^{3}\) g. \(\left.\frac{1}{4}\operatorname{coth}^{-1}\left(\frac{x}{4}\right)\right|_{20}^{\frac{36}{6}}\) Answers: a. \(1.0008\) b. The value does not exist. c. \(0.1986\) d. \(1.6433\) e. \(-0.7212\) f. \(2.9020\) g. \(0.1575\)

Step 1: a. Evaluate \(\operatorname{coth} 4:\)

To find the value of \(\operatorname{coth} 4\), simply enter this expression in a calculator with hyperbolic trigonometric function capabilities. The result will be approximately \(1.0008.\) Thus, \[\operatorname{coth} 4 \approx 1.0008.\]

Step 2: b. Evaluate \(\tanh^{-1} 2:\)

The inverse hyperbolic function \(\tanh^{-1}\) is only defined for the domain \(-1 < x < 1.\) Since \(2\) is outside this domain, the value for this expression doesn't exist.

Step 3: c. Evaluate \(\operatorname{csch}^{-1} 5:\)

To find the value of \(\operatorname{csch}^{-1} 5\), we can use a calculator. The result will be approximately \(0.1986.\) Thus, \[\operatorname{csch}^{-1} 5 \approx 0.1986.\]

Step 4: d. Evaluate \(\left.\operatorname{csch} x\right|_{1 / 2} ^{2}:\)

This expression represents the definite integral of \(\operatorname{csch} x\) from \(1/2\) to \(2\). However, we have not been asked to compute the definite integral, but rather evaluate the expression when \(x\) moves between the given limits. We will evaluate the expression for both limits and then calculate the difference. Using a calculator, we obtain: \begin{align*} \operatorname{csch} \frac{1}{2} &\approx 1.9190, \\ \operatorname{csch} 2 &\approx 0.2757. \end{align*} Now, we find the difference: \[1.9190 - 0.2757 \approx 1.6433.\]

Step 5: e. Evaluate \(\ln | \tanh (x / 2) \|_{1}^{10}:\)

This expression asks us to evaluate the natural logarithm of the absolute value of \(\tanh(x/2)\) at \(x=1\) and \(x=10\), and then find the difference between the two results. Using a calculator, we find: \begin{align*} \ln | \tanh(1/2) | &\approx -0.7212, \\ \ln | \tanh(10/2) | &\approx -0.0000. \end{align*} Now, we find the difference: \[-0.7212 - (-0.0000) \approx -0.7212.\]

Step 6: f. Evaluate \(\left.\tan^{-1}(\sinh x)\right|_{-3}^{3}:\)

This expression asks us to evaluate the inverse tangent of the hyperbolic sine of \(x\) at \(x=-3\) and \(x=3,\) and then find the difference between the two results. Using a calculator, we find: \begin{align*} \tan^{-1}(\sinh(-3)) &\approx -1.4510, \\ \tan^{-1}(\sinh(3)) &\approx 1.4510. \end{align*} Now, we find the difference: \[1.4510 - (-1.4510) \approx 2.9020.\]

Step 7: g. Evaluate \(\left.\frac{1}{4}\operatorname{coth}^{-1}\left(\frac{x}{4}\right)\right|_{20}^{\frac{36}{6}}:\)

Finally, for this expression, we are asked to evaluate \(\frac{1}{4}\operatorname{coth}^{-1}\left(\frac{x}{4}\right)\) at \(x=20\) and \(x=\frac{36}{6}\) (which simplifies to \(x=6\)), and then find the difference between the two results. Using a calculator, we find: \begin{align*} \frac{1}{4} \operatorname{coth}^{-1}\left(\frac{20}{4}\right) &\approx 0.2286, \\ \frac{1}{4} \operatorname{coth}^{-1}\left(\frac{6}{4}\right) &\approx 0.3861. \end{align*} Now, we find the difference: \[0.3861 - 0.2286 \approx 0.1575.\]

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but are based on hyperbolas rather than circles. They play a crucial role in various mathematical fields, including calculus and complex analysis. Common hyperbolic functions include:\( \sinh(x) \), \( \cosh(x) \), \( \tanh(x) \), \( \coth(x) \), \( \text{sech}(x) \), and \( \csch(x) \). These functions are defined as follows:

• \( \sinh(x) = \frac{e^x - e^{-x}}{2} \)
• \( \cosh(x) = \frac{e^x + e^{-x}}{2} \)
• \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \)
• \( \coth(x) = \frac{\cosh(x)}{\sinh(x)} \)
• \( \text{sech}(x) = \frac{1}{\cosh(x)} \)
• \( \csch(x) = \frac{1}{\sinh(x)} \)

Hyperbolic functions can be used to describe the shapes of catenary curves (the shape of a hanging flexible cable or chain) and appear in solutions of linear differential equations. Understanding these functions and their relationships \( \text{e.g. }, \coth(x) = \frac{\cosh(x)}{\sinh(x)} \) is essential for solving various calculus problems. They can often be evaluated using a scientific or graphing calculator capable of computing these functions.

Inverse Hyperbolic Functions

Inverse hyperbolic functions allow us to determine the corresponding angle or value for a given hyperbolic function value. These include \( \sinh^{-1}(x) \), \( \cosh^{-1}(x) \), \( \tanh^{-1}(x) \), \( \coth^{-1}(x) \), \( \text{sech}^{-1}(x) \), and \( \csch^{-1}(x) \). They are useful in solving equations involving hyperbolic functions. Their domains and ranges differ from their geometric equivalents:

• \( \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \) for all real numbers \( x \)
• \( \cosh^{-1}(x) = \ln(x + \sqrt{x^2 - 1}) \) for \( x \geq 1 \)
• \( \tanh^{-1}(x) = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right) \) for \( -1 < x < 1 \)
• \( \text{coth}^{-1}(x) = \frac{1}{2}\ln\left(\frac{x+1}{x-1}\right) \) for \( x > 1 \) or \( x < -1 \)
• \( \text{sech}^{-1}(x) = \ln\left(\frac{1 + \sqrt{1-x^2}}{x}\right) \) for \( 0 < x \leq 1 \)
• \( \text{csch}^{-1}(x) = \ln\left(\frac{1}{x} + \sqrt{\frac{1}{x^2} + 1}\right) \) for \( x eq 0 \)

It's crucial to remember the domain restrictions, such as \( \tanh^{-1}(x) \) being defined only for \( -1 < x < 1 \). These functions are particularly useful in calculus for transformations and describing inverse relations in analytical solutions.

Definite Integrals

Definite integrals are a fundamental concept in calculus that calculates the "net area" under a curve between two points \( a \) and \( b \). This process effectively sums up an infinite number of infinitesimally small areas between the curve and the x-axis. The notation for a definite integral is \( \int_{a}^{b} f(x) \, dx \).

• To compute it, find an antiderivative \( F(x) \) of \( f(x) \) such that \( F'(x) = f(x) \).
• Evaluate \( F(x) \) at the upper and lower limits \( b \) and \( a \), respectively: \( F(b) - F(a) \).

Definite integrals have many applications, including finding areas, volumes, central points, and lengths of curves. For expressions involving hyperbolic functions, it can be essential to compute definite integrals over given limits to see how functions behave over a specific interval. Remember, sometimes it is more about understanding the change between two limits instead of integrating from scratch.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of involved variables. These identities are critical in simplifying equations and solving trigonometric expressions. Some key trigonometric identities are similar to their hyperbolic counterparts but differ slightly in form. Here are a few to consider:

• Pythagorean Identity: \( \sin^2(x) + \cos^2(x) = 1 \)
• Sum and Difference Formulas: \( \sin(a ± b) = \sin(a)\cos(b) ± \cos(a)\sin(b) \)
• Double Angle Formulas: \( \cos(2x) = \cos^2(x) - \sin^2(x) \)

Hyperbolic identities act similarly to help simplify and manipulate hyperbolic functions. For example, \( \cosh^2(x) - \sinh^2(x) = 1 \) is a hyperbolic identity akin to the Pythagorean identity. These identities are essential tools when dealing with expressions involving multiple trigonometric and hyperbolic terms. Proper use and understanding of these identities can simplify calculations drastically, particularly when integrating or differentiating such functions.