7: Transcendental Functions
- Page ID
- 212061
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)An algebraic number is any number that can be a solution to a polynomial equation with integer coefficients. Any rational number is algebraic: for example, \(\frac57\) is a solution of \(7x-5=0\). Any square root or cube root is an algebraic number: \(\sqrt{3}\) is a solution of \(x^2-3=0\) and \(-\sqrt[3]{5}\) is a solution of \(x^3+5 = 0\). In fact, any number that can be expressed as the sum, difference, product, quotient or rational power of integers is an algebraic number, including something as horrible-looking as:\[\sqrt[5]{\frac{\sqrt[4]{93}-\sqrt[3]{436+\sqrt{2}}}{17+\sqrt{37}}}\]Any real number that is not an algebraic number is called a transcendental number. We have already met (and used extensively) two very important transcendental numbers: \(\pi\) and \(e\).
It turns out that proving that these numbers are transcendental is rather difficult, although by the end of Chapter 8 we will have most of the tools necessary to prove that \(e\) is transcendental.
Functions defined using sums, differences, products, quotients or rational powers of rational coefficients and a real-valued variable \(x\) are called algebraic functions. Any non-algebraic functions of a real variable are called transcendental functions. Examples of transcendental functions with which you should be very familiar are \(\sin(x)\), \(\cos(x)\), \(\tan(x)\) and the other trigonometric functions, as well as \(e^x\). (The geometric definitions of the trigonometric functions involve \(\pi\), while the exponential function obviously involves \(e\).)
The inverse functions of these transcendental functions are also transcendental functions: \(\arcsin(x)\), \(\arccos(x)\), \(\arctan(x)\) and the other inverse trigonometric functions, as well as \(\ln(x)\).
Because many important transcendental functions are defined as inverse functions, this chapter begins with a review of inverse functions and a discussion of finding derivatives of inverse functions. It continues with a review of inverse trigonometric functions and a discussion of their derivatives (and the usefulness of these derivative patterns in finding certain antiderivatives), then ties up some (very important) loose ends related to the definitions and properties of \(e^x\) and \(\ln(x)\). The chapter concludes by introducing some new transcendental functions: the hyperbolic functions and their inverses, which play important roles in calculus and in applications.