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The Taylor series is the power series:
\qquad \qquad f(x) = f(a) + \displaystyle \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...
The following applet can be used to graph the Taylor polynomial of degree n, Tn(x), and/or the remainder function, Rn(x), for the given function y=f(x), centered at x=a. Simply enter the function f(x) and the values a and 0 ≤ n ≤ 50. The applet can also be used to approximate the value of f(b) with Tn(b), for b near a. The values a and b can be changed by simply typing a new value, such as "1.2345", "pi/2", "sqrt(5)+cos(3)", etc. You may also change this value by using the up/down arrow keys, clicking on the graph of y=f(x) or dragging the corresponding point left or right. To move the center of the graph, simply drag any point to a new location. To label the x-axis in radians (i.e. multiples of pi), click on the graph and press "control-r". To switch back, simply press "control-r" again.

This page uses an applet from David Little (Lecturer in the Mathematics Department, Penn State University, USA) and is used with his permission.

Glossary

degree

On a graph: the number of edges incident to a vertex.
In geometry: a measure of angle

function

A rule that connects one value in one set with one and only one value in another set.

graph

A diagram showing a relationship between two variables.
The diagram shows a vertical y axis and a horizontal x axis.

polynomial

an expression which has whole-numbered powers of the variable, x, multiplied by coefficients, and added together.
Examples: y = 2x, y = 1+x², y = 3-2x+7x³

power series

a series whose terms form a power sequence, often obtained from Maclaurin or Taylor series

remainder

The number remaining after one polynomial is divided into another an exact number of times.

series

the sum of terms in a sequence

taylor series

a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point

Full Glossary List

This question appears in the following syllabi:

SyllabusModuleSectionTopicExam Year
AP Calculus BC (USA)5Maclaurin and Taylor SeriesTaylor series-
AQA A2 Further Maths 2017Pure MathsExtraTaylor Series-
AQA AS/A2 Further Maths 2017Pure MathsExtraTaylor Series-
Edexcel A-Level (UK - Pre-2017)FP2Maclaurin and Taylor SeriesTaylor series-
Edexcel A2 Further Maths 2017Further Pure 1Methods in CalculusTaylor Series-
Edexcel AS/A2 Further Maths 2017Further Pure 1Methods in CalculusTaylor Series-
I.B. Higher Level9Maclaurin and Taylor SeriesTaylor series-
Methods (UK)M10Maclaurin and Taylor SeriesTaylor series-
OCR A2 Further Maths 2017Pure CoreExtraTaylor Series-
OCR MEI AS Further Maths 2017Numerical MethodsExtraTaylor Series-
Scottish Advanced HighersM3Maclaurin and Taylor SeriesTaylor series-
Scottish (Highers + Advanced)AM3Maclaurin and Taylor SeriesTaylor series-
Universal (all site questions)MMaclaurin and Taylor SeriesTaylor series-
WJEC A-Level (Wales)FP3Maclaurin and Taylor SeriesTaylor series-