For analysing a function #f# in the neighbourhood of a point #p#, it is useful to approximate the function on a small interval around #p# by a polynomial. For arbitrary small intervals around #p#, the constant function #f(p)# is the best approximation of #f# by means of a constant and the function #f(p)+x\cdot f'(p)# is the best linear approximation of #f#. More generally, we will study Taylor polynomials about #p# of order #n# for each #n#, which are best approximations among polynomials of degree at most #n# of #f# on small intervals around #p#.
Let #n# be a natural number and #f# an #n+1# times differentiable function on an open interval around a point #p#.
The Taylor polynomial of #f# about #p# of order #n# is the polynomial #P_n(x)# given by \[P_n(x)=f(p)+f'(p)(x-p)+\frac{1}{2!}f^{(2)}(p)(x-p)^2+\cdots+\frac{1}{n!}f^{(n)}(p)(x-p)^n\]
If #f# is infinitely differentiable, and the limit #\lim_{n\to\infty}P_n(x)# exists for every #x# in an interval around #p#, then the limit is called the Taylor series of #f# about #p#.
If #p=0#, then the Taylor series is also called the Maclaurin series.
The Maclaurin series of the sine function is given by
\[\sin(x) = x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots\]
Indeed, by induction with respect to #n# it can be shown that the #n#-th derivative #\sin^{(n)}(x) # is equal to # (-1)^{\lfloor n/2\rfloor}\sin(x)# if #n# is even and equal to # (-1)^{\lfloor n/2\rfloor}\cos(x)# if #n# is odd. Consequently, #\sin^{(n)}(0) = 0 # if #n# is even and #\sin^{(n)}(0) =(-1)^{\lfloor n/2\rfloor}# if #n# is odd, so the Maclaurin series of the sine function, which is the Taylor series about #0# of #\sin(x)#, is equal to
\[\begin{array}{rcl}\sin(x) &=&\displaystyle\sin(0)+\sin^{(1)}(0)\cdot x+\frac{1}{2!}\sin^{(2)}(0)\cdot x^2+\frac{1}{3!}\sin^{(3)}(0)\cdot x^3+\cdots\\ &=&\displaystyle0+1\cdot x-\frac{0}{2!} \cdot x^2-\frac{1}{3!}\cdot x^3+\frac{0}{4!} \cdot x^4+\cdots\\ &=&\displaystyle x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots\end{array}\]
Thus, the sine function around the origin can be approximated by the sum of the first few terms. The approximations become better as the order increases.
The Taylor polynomial of #f# about #p# of order #1# is the polynomial #P_1(x)# given by \[P_1(x)=f(p)+f'(p)(x-p)\] This is the function rule of the well-known tangent line to the graph of #f# at #p#, also known as the linear approximation of the function #f# around #p#.
The following table shows some well-known functions in #x# and their Maclaurin series.
| #\phantom{rrr}#function |
#\phantom{rrr}#Maclaurin series |
| #\phantom{rrr}x^n# #(n>0)# |
#\phantom{rrr}x^n# |
| #\phantom{rrr}\dfrac{1}{1-x}# |
#\phantom{rrr}\displaystyle\sum_{k=0}^\infty x^k# |
| #\phantom{rrr}\dfrac{1}{(1-x)^2}# |
#\phantom{rrr}\displaystyle\sum_{k=1}^\infty kx^{k-1}# |
| #\phantom{rrr}\sin(x)# |
#\phantom{rrr}\displaystyle\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!}x^{2k+1}# |
| #\phantom{rrr}\cos(x)# |
#\phantom{rrr}\displaystyle\sum_{k=0}^\infty \frac{(-1)^k}{(2k)!}x^{2k}# |
| #\phantom{rrr}\ln(1+x)# |
#\phantom{rrr}\displaystyle\sum_{k=1}^\infty\frac{(-1)^{k-1}}{k}x^k # |
| #\phantom{rrr}\e^x# |
#\phantom{rrr}\displaystyle\sum_{k=0}^\infty\frac{x^k}{k!} # |
Let #f# be an #n+1# times differentiable function on an open interval around #p# with continuous #(n+1)#-st derivative. With some knowledge of this #(n+1)#-st derivative of #f#, we can estimate how close the Taylor approximation comes to #f#.
More specifically, the Taylor polynomial #P_n(x)# around #p# satisfies, for all #x# in the interval, \[\left|f(x)-P_n(x)\right|\le\frac{1}{(n+1)!}\max_{|y-p|\le |x-p|}\left|f^{(n+1)}(y)(x-p)^{n+1}\right|\]
In particular, if #f^{(n+1)}# is bounded on the open interval, then the right hand side of the inequality gives an upper bound for the error term #\left|f(x)-P_n(x)\right|# of the approximation.
For example, if #f(x) = \sin(x)#, then all higher derivatives are #\pm\cos# or #\pm\sin# and so #\left|f^{(n+1)}(y)\right|# is at most #1#, so the error term of the approximation of #\sin(x)# by the Taylor polynomial of order #n# about the origin is at most #\dfrac{|x|^{n+1}}{(n+1)!}#.
The Taylor polynomial #P_n(x)# is an approximation of the function #f# around #p#. That is why we also use the term Taylor approximation instead of Taylor polynomial.
We speak of order #n# rather than degree #n# because the degree of #P_n(x)# need not be #n#. Of course, the degree of #P_n(x)# is at most #n#.
Below graphs can be found of some well-known functions and their first few Taylor approximations.
Determine the Taylor polynomial #P_{3}(x)# of order #3# of the function #f(x)={{1}\over{x^2+3}}# about the point #1#.
Present your answer in the form \[a +b\cdot\left(x-1\right) +c\cdot\left(x-1\right)^2 +d\cdot\left(x-1\right)^3 \] where #a ,b , c , d # are constants.
#P_{3}(x)=# # {{1}\over{4}}-{{1}\over{8}} \cdot \left(x-1\right)+0 \cdot \left(x-1\right)^2+{{1}\over{32}} \cdot \left(x-1\right)^3#
The Taylor polynomial of order #{3}# of a function #f# about a point #p# is given by
\[P_ { 3 } (x)= f (p) + f'(p)(x-p) +\frac{1}{2}f''(p)(x-p)^2 +\frac{1}{6}f'''(p)(x-p)^3) \]
First we calculate the required derivatives:
\[ \begin{array}{rcl}\displaystyle f(p)&=&\displaystyle {{1}\over{p^2+3}} \\ f'(p)&=&\displaystyle-{{2\cdot p}\over{\left(p^2+3\right)^2}}\\ f''(p)&=&\displaystyle{{8\cdot p^2}\over{\left(p^2+3\right)^3}}-{{2}\over{\left(p^2+3\right)^2}}\\ f'''(p)&=&\displaystyle{{24\cdot p}\over{\left(p^2+3\right)^3}}-{{48\cdot p^3}\over{\left(p^2+3\right)^4}}\end{array}\]
Next we substitute #p=1# in the derivatives:
\[\begin{array}{rcl}\displaystyle f \left( 1\right) &=&\displaystyle {{1}\over{4}} \\\displaystyle f'\left(1\right)&=&\displaystyle -{{1}\over{8}} \\ \displaystyle f''\left(1\right)&=&\displaystyle 0 \\ \displaystyle f'''\left(1\right)&=&\displaystyle {{3}\over{16}} \end{array} \]
Finally, we find the requested polynomial by substituting #p=1# and the values of the derivatives at this point in the formula for #P_{3}(x)#:
\[P_ { 3 } (x) = {{1}\over{4}}-{{1}\over{8}} \cdot \left(x-1\right)+0 \cdot \left(x-1\right)^2+{{1}\over{32}} \cdot \left(x-1\right)^3 \]
Taylor series
