This page is a sub-page of our page on Calculus of One Real Variable.
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Related KMR-pages:
• Taylor Expansion in Several Real Variables
• Big-Ordo and Little-ordo notation
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Other related sources of information:
• Taylor series, Steven Strogatz on YouTube
• Taylor’s theorem
• James Gregory
• Colin Maclaurin
• Brook Taylor
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The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.
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Anchors into the text below:
• Differentials of higher order
• Taylor’s theorem
• Taylor’s theorem in one real variable
• Taylor series
• Some common Taylor series of functions from R to R
• Taylor polynomials of increasing degree for the function y = exp(x)
• Taylor polynomials (of increasing degree) for the function y = cos(x)
• Taylor polynomials (of degree 1 and 2) for the function y = cos(x)
• Taylor polynomials (of degree 1, 2 and 3) for the function y = cos(x)
• Taylor polynomials (of degree 1, 2, 3 and 4) for the function y = cos(x)
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Differentials of higher order
Let \, X \, and \, Y \, be two metric, linear spaces
and let \, f : X \rightarrow Y \, be an \, n \, times differentiable function.
Then \, L(X, Y) \, is a linear space if we define
\, (A + B) x \, \stackrel {\mathrm{def}}{=} \, Ax + Bx and \, (\alpha A) x \, \stackrel {\mathrm{def}}{=} \, \alpha A x .
Moreover, \, L(X, Y) \, is a metric space if we define
\, || A || \, \stackrel {\mathrm{def}}{=} \, \underset{|x|=1}{\mathrm{sup}} | A x | \, and \, d(A, B) \, \stackrel {\mathrm{def}}{=} \, || A - B || .
It follows that the function \, X \xrightarrow[]{f'} L(X, Y) \, is a mapping between two metric, linear spaces. Now, \, f' \, is differentiable at the point \, x \in X \, if there exists a linear function \, (f')'(x) \in L(X, L(X,Y)) \, such that
\, f'(x + h) = f'(x) + (f')'(x) h + o(h) .
Theorem: \, L(X, L(X,Y)) \, \simeq \, L_2(X,Y) \, where
\, L_2(X,Y) = \{ \mathrm{bilinear \, maps} : X × X \rightarrow Y \} .
That \, g : X × X \rightarrow Y \, is bilinear means that
\, g({\lambda}_1 \, a_1 + {\lambda}_2 \, a_2 \, , b) = {\lambda}_1 \, g(a_1, b) + {\lambda}_2 \, g(a_2, b) \, and
\, g(a \, , {\lambda}_1 \, b_1 + {\lambda}_2 \, b_2) = {\lambda}_1 \, g(a, b_1) + {\lambda}_2 \, g(a, b_2) .
Hence we have \, f''(x) \in L_2(X,Y) .
Theorem: If \, x \mapsto f''(x) \, is continuous, then we have
\;\;\;\;\;\;\;\;\;\;\;\;\; f''(x)(h_1, h_2) = f''(x)(h_2, h_1) .
Repeating the argument,
it follows that the function \, X \xrightarrow[]{f''} L(X, L(X, Y)) \, is a mapping between two metric, linear spaces. Now, \, f'' \, is differentiable at the point \, x \in X \, if there exists a linear function \, (f'')'(x) \in L(X, L(X, L(X,Y))) \, such that
\, f''(x + h) = f''(x) + (f'')'(x) h + o(h) .
And so on … for as many times (assumed to be \, n \, above) that the original function \, f \, is differentiable.
NOTE: It is important to remember that in these differentiations the point \, x \, is always held constant, whereas \, h \, provides the variability (in terms of small variations around the fixed point \, x . In general, the functions \, f, f', f'', f''', \cdots, f^{(n)} \, are non-linear, whereas the functions \, f(x), f'(x), f''(x), f'''(x), \cdots \, , f^{(n)}(x) \, are respectively constant, linear, bilinear, trilinear, … , n -linear.
Taylor’s theorem
/////// Quoting Wikipedia (on Taylor’s theorem):
In calculus, Taylor’s theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the kth-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order k of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor’s theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial.
Taylor’s theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, although an earlier version of the result was already mentioned in 1671 by James Gregory.
Taylor’s theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis. It gives simple arithmetic formulas to accurately compute values of many transcendental functions such as the exponential function and trigonometric functions. It is the starting point of the study of analytic functions, and is fundamental in various areas of mathematics, as well as in numerical analysis and mathematical physics. Taylor’s theorem also generalizes to multivariate and vector valued functions.
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Taylor’s theorem in one real variable
Taylor series:
\, f(x+h) = e^d f|_x \,
\, f(x+h) \, = \, \sum\limits_{\alpha = 0}^{\infty} \, \frac{1}{\alpha !} {f}^{( \alpha)}(x) h^{\alpha} \, = \, \sum\limits_{\alpha = 0}^{\infty} \, \frac{1}{\alpha !} h^{\alpha} {(\frac{d}{dx})}^{\alpha}f(x) .
\, f(x+h) \, = \, \sum\limits_{\alpha = 0}^{\infty} \, \frac{1}{\alpha !} h^{\alpha} {\partial}^{\alpha} f(x) \, .
NOTE: \, {\partial}^{ \, \alpha} {\partial}^{ \, \beta} f \, = \, {\partial}^{ \, \alpha + \beta} f .
One variable:
\, x \,
\, h \,
\, \partial \stackrel {\mathrm{def}}{=} \frac{d}{dx} \,
\, d \stackrel {\mathrm{def}}{=} h \cdot \partial = h \frac{d}{dx} \,
\, f(x+h) = e^{h \frac{d}{dx}} f(x) = \sum\limits_{\nu = 0}^{\infty} \frac{1}{\nu !} (h \frac{d}{dx})^{\nu} f|_x \,
Several variables:
\, x \stackrel {\mathrm{def}}{=} (x_1, x_2, ... \, , x_n) \,
\, h \stackrel {\mathrm{def}}{=} (h_1, h_2, ... \, , h_n) \,
\, \partial \stackrel {\mathrm{def}}{=} ({\partial}_1, {\partial}_2, ... \, , {\partial}_n) \,
\, d \stackrel {\mathrm{def}}{=} h \cdot \partial = h_1 {\partial}_1 + h_2 {\partial}_2 + ... + h_n {\partial}_n \,
\, f(x+h) = e^{(h_1 {\partial}_1 + ... + \, h_n {\partial}_n)} f(x) = \sum\limits_{\nu = 0}^{\infty} \frac{1}{\nu !} (h_1 {\partial}_1 + ... + h_n {\partial}_n)^{\nu} f|_x \,
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Some common Taylor series of functions from \, \mathbb{R} \, to \, \mathbb{R} \, :
\, e^x \, \equiv \, \sum\limits_{n = 0}^{\infty} \, \frac{x^n}{n \, !} \, \equiv \, 1 + x + \frac{1}{2!} x^2 + \frac{1}{3!} x^3 + \frac{1}{4!} x^4 + \frac{1}{5!} x^5 + \cdots \,
\, \sin x \, \equiv \, \sum\limits_{n = 0}^{\infty} \, \frac {(-1)^n}{(2n+1)!} x^{2n+1} \, \equiv \, x - \frac{1}{3!} x^3 + \frac{1}{5!} x^5 - \frac{1}{7!} x^7 + \cdots \,
\, \cos x \, \equiv \, \sum\limits_{n = 0}^{\infty} \, \frac {(-1)^n}{2n!} x^{2n} \, \equiv \, 1 - \frac{1}{2!} x^2 + \frac{1}{4!} x^4 - \frac{1}{6!} x^6 + \cdots \,
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NOTE:
\, e^{ix} \, \equiv \, \sum\limits_{n = 0}^{\infty} \, \frac{{(ix)}^n}{n \, !} \, \equiv \, 1 + ix - \frac{1}{2!} x^2 - i \frac{1}{3!} x^3 + \frac{1}{4!} x^4 + i \frac{1}{5!} x^5 + \cdots \, \equiv \, \;\;\;\;\;\;\, \equiv \, \cos x + i \sin x \,
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\, \tan x \, \equiv \, x + \frac{1}{3} x^3 + O( x^5 ) \,
\, \ln (1 + x) \, \equiv \, \sum\limits_{n = 0}^{\infty} \, \frac {(-1)^{n-1}}{n} x^n \, \equiv \, x - \frac{1}{2} x^2 + \frac{1}{3} x^3 - \frac{1}{4} x^4 + \cdots \,
\, (1 + x)^{\alpha} \, \equiv \, 1 + \frac{\alpha}{1} x + \frac{\alpha(\alpha -1)}{2!} x^2 + \frac{\alpha(\alpha -1)(\alpha -2)}{3!} x^3 + \cdots + \,
\, + \frac{\alpha (\alpha -1)(\alpha -2) \cdots (\alpha -(n-1)) } {n!} x^n + O( x^{n+1} ) \, = \,
\, = \, \sum\limits_{k = 0}^{n} \binom{\alpha}{k} x^k + O( x^{n+1} ) \,
If we set \, \alpha = \frac{1}{2} \, we get:
\, \sqrt{1 + x} \, \equiv \, 1 + \frac{1}{2} x - \frac{1}{8} x^2 + O( x^3 ) \,
where we have made explicit the first three terms of the Taylor series.
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Taylor polynomials (of increasing degree) for the function \, y = e^x \, :
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Taylor polynomials (of increasing degree) for the function \, y = \cos(x) \, :
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Taylor polynomials (of degree 1 and 2) for the function \, y = \cos(x) \, :
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Taylor polynomials (of degree 1, 2 and 3) for the function \, y = \cos(x) \, :
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Taylor polynomials (of degree 1, 2, 3 and 4) for the function \, y = \cos(x) \, :
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