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Chapter 5: Problem 62

Which hyperbolic functions take on only positive values? Which hyperbolic functions are increasing on their domains?

Short Answer

Expert verified
The hyperbolic functions that take on only positive values are cosh(x) and sech(x). The hyperbolic functions that are increasing on their domains are sinh(x), cosh(x), and tanh(x).

Step by step solution

01

Analyze Hyperbolic Functions

Identify the hyperbolic functions to be analyzed, which includes: sinh(x), cosh(x), tanh(x), coth(x), sech(x), and csch(x). These functions are defined with relations to e (Euler's number).
02

Determine Functions With Only Positive Values

Looking at the hyperbolic functions, it's clear that cosh(x) and sech(x) only return positive values. For cosh(x), this is because it's defined as \( (e^x + e^{-x})/2 \), the sum of two positive terms divided by two. And for sech(x), which is 1/cosh(x), anything divided by a positive number is always positive.
03

Analyze Increasing Functions

The functions sinh(x), cosh(x), and tanh(x) are always increasing. The derivative of sinh(x) is cosh(x), which is always positive, hence sinh(x) is increasing. The derivative of cosh(x) is sinh(x), which is positive for x > 0 and negative for x < 0, hence cosh(x) is increasing. The derivative of tanh(x) is sech^2(x), which is always positive, hence tanh(x) is increasing

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Most popular questions from this chapter

In Exercises 87鈥�90, solve the differential equation. $$ \frac{d y}{d x}=\frac{1}{(x-1) \sqrt{-4 x^{2}+8 x-1}} $$

Find the derivative of the function. \(y=x \arcsin x+\sqrt{1-x^{2}}\)

Prove each differentiation formula. (a) \(\frac{d}{d x}[\arctan u]=\frac{u^{\prime}}{1+u^{2}}\) (b) \(\frac{d}{d x}[\operatorname{arccot} u]=\frac{-u^{\prime}}{1+u^{2}}\) (c) \(\frac{d}{d x}[\operatorname{arcsec} u]=\frac{u^{\prime}}{|u| \sqrt{u^{2}-1}}\) (d) \(\frac{d}{d x}[\operatorname{arccsc} u]=\frac{-u^{\prime}}{|u| \sqrt{u^{2}-1}}\)

(a) Use a graphing utility to evaluate arcsin (arcsin 0.5) and \(\arcsin (\arcsin 1) .\) (b) Let \(f(x)=\arcsin (\arcsin x)\) Find the values of \(x\) in the interval \(-1 \leq x \leq 1\) such that \(f(x)\) is a real number.

A Function and Its Derivative Is there a function \(f\) such that \(f(x)=f^{\prime}(x)\) ? If so, identify it.
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