What is Taylor Series
Technical Definition: A way to approximate (very) smooth functions (for which derivatives up to high orders exist and are continuous)
Simple Definition: Taylor series is the infinite sum of all the terms for a specific function which is a very close approximation to the real value of the function.
Mathematical definition:
f(x) = f(a) + (f'(a)(x-a))/1! + (f''(x)(x-a)2)/2! + (f'''(x)(x-a)3)/3! + ... +
Why do we have an infinite number of terms?:
The more terms we include in our approximated function, the better the approximation to the real value. For a graph this means that it will represent the actual graph function more.
Special Case (Maclaurin Series):
Maclaurin series is based of the Taylor Series, but we choose the function to be around origin (value = 0) rather than anywhere else.
Advantage of using Taylor/Maclaurin series
its allows for incredibly accurate approximations of a function (depending on the number of terms included)
Provide for integration and differentiation of functions to arrive at representations of other function
Disadvantage of using Taylor/Maclaurin series
some calculations become tedious or the series doesn’t converge quickly
many of the functions are limited to a certain domain given a specific range for convergence (some Taylor series are only valid for a small domain)
Related Maths A Level answers
All answers ▸Consider the functions f(x) = −x^3 + 2x^2 + 3x and g(x) = −x^3 + 3x^2 − x + 3. (a) Find df/dx (x) and hence show that f(x) has turning points at when x = 2 /3 ± √ 13/ 3 . [5] (b) Find the points where f(x) and g(x) intersect. [4]
When would you apply the product rule in differentiation and how do you do this?
A curve is described by f(x) = x^2 + 2x. A second curve is described by g(x) = x^2 -5x + 7. Find the point (s) where both curves intersect.
The quadratic equation 2x^2+ 6x+7 has roots a and b. Write down the value of a+b and the value of ab.