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Taylor's theorem

Taylor's theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Taylor's theorem rather than just read about it. In short: In calculus, Taylor's theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree k {\textstyle k} , called the k {\textstyle k} -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order k {\textstyle k} of the Taylor series of the function.

Taylor's theorem — main illustration
Taylor's theorem — illustration

Key takeaways

  • Taylor's theorem belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Taylor's theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Taylor's theorem from memory before moving on to harder problems.

Reference excerpt

In calculus, Taylor's theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree k {\textstyle k} , called the k {\textstyle k} -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order k {\textstyle 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 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. It provided the mathematical basis for some landmark early computing machines: Charles Babbage's difference engine calculated sines, cosines, logarithms, and other transcendental functions by numerically integrating the first 7 terms of their Taylor series.

Motivation

If a real-valued function f ( x ) {\textstyle f(x)} is differentiable at the point x = a {\textstyle x=a} , then it has a linear approximation near this point. This means that there exists a function h1(x) such that

f ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + h 1 ( x ) ( x − a ) , lim x → a h 1 ( x ) = 0. {\displaystyle f(x)=f(a)+f'(a)(x-a)+h_{1}(x)(x-a),\quad \lim _{x\to a}h_{1}(x)=0.}

Here

P 1 ( x ) = f ( a ) + f ′ ( a ) ( x − a ) {\displaystyle P_{1}(x)=f(a)+f'(a)(x-a)}

is the linear approximation of f ( x ) {\textstyle f(x)} for x near the point a, whose graph y = P 1 ( x ) {\textstyle y=P_{1}(x)} is the tangent line to the graph y = f ( x ) {\textstyle y=f(x)} at x = a. The error in the approximation is:

R 1 ( x ) = f ( x ) − P 1 ( x ) = h 1 ( x ) ( x − a ) . {\displaystyle R_{1}(x)=f(x)-P_{1}(x)=h_{1}(x)(x-a).}

As x tends to a, this error goes to zero much faster than ( x − a ) {\displaystyle (x-a)} , making f ( x ) ≈ P 1 ( x ) {\displaystyle f(x)\approx P_{1}(x)} a useful approximation.

For a better approximation to f ( x ) {\textstyle f(x)} , we can fit a quadratic polynomial instead of a linear function:

P 2 ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) 2 ( x − a ) 2 . {\displaystyle P_{2}(x)=f(a)+f'(a)(x-a)+{\frac {f''(a)}{2}}(x-a)^{2}.}

Instead of just matching one derivative of f ( x ) {\textstyle f(x)} at x = a {\textstyle x=a} , this polynomial has the same first and second derivatives, as is evident upon differentiation. Taylor's theorem ensures that the quadratic approximation is, in a sufficiently small neighborhood of x = a {\textstyle x=a} , more accurate than the linear approximation. Specifically,

… excerpt ends here. Continue reading the full article.

Illustrations

Taylor's theorem: The exponential function y = e x {\textstyle y=e^{x}} (red) and the corresponding Taylor polynomial of degree four (dashed green) around the origin.
The exponential function y = e x {\textstyle y=e^{x}} (red) and the corresponding Taylor polynomial of degree four (dashed green) around the origin.
Taylor's theorem: Graph of f ( x ) = e x {\textstyle f(x)=e^{x}} (blue) with its linear approximation P 1 ( x ) = 1 + x {\textstyle P_{1}(x)=1+x} (red) at a = 0 {\textstyle a=0} .
Graph of f ( x ) = e x {\textstyle f(x)=e^{x}} (blue) with its linear approximation P 1 ( x ) = 1 + x {\textstyle P_{1}(x)=1+x} (red) at a = 0 {\textstyle a=0} .
Taylor's theorem: Graph of f ( x ) = e x {\textstyle f(x)=e^{x}} (blue) with its quadratic approximation P 2 ( x ) = 1 + x + x 2 2 {\displaystyle P_{2}(x)=1+x+{\dfrac {x^{2}}{2}}} (red) at a = 0 {\textstyle a=0} . Note the improvement in the approximation.
Graph of f ( x ) = e x {\textstyle f(x)=e^{x}} (blue) with its quadratic approximation P 2 ( x ) = 1 + x + x 2 2 {\displaystyle P_{2}(x)=1+x+{\dfrac {x^{2}}{2}}} (red) at a = 0 {\textstyle a=0} . Note the improvement in the approximation.
Taylor's theorem: Approximation of f ( x ) = 1 1 + x 2 {\textstyle f(x)={\dfrac {1}{1+x^{2}}}} (blue) by its Taylor polynomials P k {\textstyle P_{k}} of order k = 1 , … , 16 {\textstyle k=1,\ldots ,16} centered at x = 0 {\textstyle x=0} (red) and x = 1 {\textstyle x=1} (green). The approximations do not improve at all outside ( − 1 , 1 ) {\displaystyle (-1,1)} and ( 1 − 2 , 1 + 2 ) {\textstyle (1-{\sqrt {2}},1+{\sqrt {2}})} , respectively.
Approximation of f ( x ) = 1 1 + x 2 {\textstyle f(x)={\dfrac {1}{1+x^{2}}}} (blue) by its Taylor polynomials P k {\textstyle P_{k}} of order k = 1 , … , 16 {\textstyle k=1,\ldots ,16} centered at x = 0 {\textstyle x=0} (red) and x = 1 {\textstyle x=1} (green). The approximations do not improve at all outside ( − 1 , 1 ) {\displaystyle (-1,1)} and ( 1 − 2 , 1 + 2 ) {\textstyle (1-{\sqrt {2}},1+{\sqrt {2}})} , respectively.
Taylor's theorem: Approximation of e x {\textstyle e^{x}} (blue) by its Taylor polynomials P k {\displaystyle P_{k}} of order k = 1 , … , 7 {\textstyle k=1,\ldots ,7} centered at x = 0 {\textstyle x=0} (red).
Approximation of e x {\textstyle e^{x}} (blue) by its Taylor polynomials P k {\displaystyle P_{k}} of order k = 1 , … , 7 {\textstyle k=1,\ldots ,7} centered at x = 0 {\textstyle x=0} (red).

Worked examples

Example 1 — a first encounter with Taylor's theorem

Start with the simplest possible case. Write down what Taylor's theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Taylor's theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Taylor's theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Taylor's theorem

In research
Taylor's theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Taylor's theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Taylor's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Approximations, Theorems in calculus, Theorems in real analysis, so understanding it makes those chapters shorter.
In everyday life
Look for Taylor's theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Taylor's theorem in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Taylor's theorem means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Taylor's theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Taylor's theorem in simple terms?

In calculus, Taylor's theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree k {\textstyle k} , called the k {\textstyle k} -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the…

Why does Taylor's theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Taylor's theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Taylor's theorem.

Tags

  • Approximations
  • Theorems in calculus
  • Theorems in real analysis

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