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Wize University Calculus 1 Textbook > Applications of Differentiation

Taylor and MacLaurin and Polynomials

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Taylor Polynomials

While linear approximations (Linearization) can give us great approximation, we can get even better approximations by using more derivatives!

The nth Degree Taylor Polynomial

The polynomial of degree n which approximates f(x)f\left(x\right) aroundx=x0x=x_0 is the nth degree Taylor Polynomial for f(x)f\left(x\right) at x0x_0, and is given by
Tn=f(x0)+f′(x0)1!(x−x0)+f′′(x0)2!(x−x0)2+...+f(n)(x0)n!(x−x0)n\boxed{T_n=f\left(x_0\right)+\frac{f'\left(x_0\right)}{1!}\left(x-x_0\right)+\frac{f''\left(x_0\right)}{2!}\left(x-x_0\right)^2+...+\frac{f^{\left(n\right)}\left(x_0\right)}{n!}\left(x-x_0\right)^n}
The Taylor Polynomial about x0=0x_0=0 is called the Maclaurin Polynomial.

Note: Sometimes the letter aa will be used instead of x0x_0.

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Example: Taylor Polynomials

Find the third degree Taylor polynomial of f(x)=ln⁡(1−2x) at x=0.f(x)=\ln(1−2x)\ \text{at}\ x=0.

f(0)=ln⁡(1−2(0))=0f′(x)=11−2x(−2)=−2(1−2x)−1  ⟹  f′(0)=−2(1−2(0))−1=−2f′′(x)=2(1−2x)−2(−2)  ⟹  f′′(0)=−4(1−2(0))−2=−4f′′′(x)=8(1−2x)−3(−2)  ⟹  f′′′(0)=−16((1−2(0)))−3=−16  ⟹  T3=0+−21!(x−0)1+−42!(x−0)2+−163!(x−0)3=−2x−2x2−83x3\begin{array}{l} \displaystyle f\left(0\right)=\ln\left(1-2\left(0\right)\right)=\boxed0 \\ \\ \displaystyle f'\left(x\right)=\frac{1}{1-2x}\left(-2\right)=-2\left(1-2x\right)^{-1} \implies f'\left(0\right)=-2\left(1-2\left(0\right)\right)^{-1}= \boxed{-2} \\ \\ f''\left(x\right)=2\left(1-2x\right)^{-2}\left(-2\right) \implies f''\left(0\right)=-4\left(1-2\left(0\right)\right)^{-2}=\boxed{-4} \\ \\ f'''\left(x\right)=8\left(1-2x\right)^{-3}\left(-2\right) \implies f'''\left(0\right)=-16\left(\left(1-2\left(0\right)\right)\right)^{-3}=\boxed{-16} \\ \\ \\ \implies \displaystyle T_3=\boxed0+\frac{\boxed{-2}}{1!}\left(x-0\right)^1+\frac{\boxed{-4}}{2!}\left(x-0\right)^2+\frac{\boxed{-16}}{3!}\left(x-0\right)^3 \\ \\ \displaystyle =-2x-2x^2-\frac{8}{3}x^3 \end{array}
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Example: Maclaurin Polynomials

Find the 4th degree Maclaurin Polynomial for f(x)=cos⁡3xf(x)=\cos{3x}.


Remember a Maclaurin Polynomial is a Taylor Polynomial with x0=0x_0=0.

f(0)=cos⁡(0)=1f′(x)=−3sin⁡(3x)  ⟹  f′(0)=−3sin⁡(0)=0f′′(x)=−9cos⁡(3x)  ⟹  f′′(0)=−9cos⁡(0)=−9f′′′(x)=27sin⁡(3x)  ⟹  f′′′(0)=27sin⁡(0)=0f(4)(x)=81cos⁡(3x)  ⟹  f(4)(0)=81cos⁡(0)=81  ⟹  M4(x)=1+0(x−0)11!+−9(x−0)22!+0(x−3)33!+81(x−0)44!=1−9x22!+81x44!\begin{array}{l} f(0)=\cos (0)=\boxed1 \\ \\ f'(x)=-3\sin(3x) \implies f'(0)=-3\sin(0)=\boxed0 \\ \\ f''(x)=-9 \cos(3x) \implies f''(0)= -9\cos(0)=\boxed{-9} \\ \\ f'''(x)=27\sin(3x) \implies f'''(0)=27\sin(0)=\boxed0 \\ \\ f^{(4)}(x)=81\cos(3x) \implies f^{(4)}(0)=81\cos(0)=\boxed{81} \\ \\ \\ \displaystyle \implies M_4(x)=\boxed1+\frac{\boxed0 (x-0)^1}{1!} +\frac{\boxed{-9}(x-0)^2}{2!}+\frac{\boxed0(x-3)^3}{3!}+\frac{\boxed{81}(x-0)^4}{4!} \\ \\ =1 - \dfrac{9x^2}{2!}+\dfrac{81x^4}{4!} \end{array}



Find the 2nd degree Taylor polynomial of f(x)=e2xf\left(x\right)=e^{2x}centered at a=ln⁡3a=\ln\sqrt{3}.

Extra Practice
Taylor and MacLaurin and Polynomials
Find the 2nd degree Taylor polynomial of f(x)=e2xf\left(x\right)=e^{2x}centered at a=ln⁡3a=\ln\sqrt{3}.
(i.e. find the first 3 terms (i.e.n=0,1,2n=0,1,2) of the Taylor polynomial of f(x)f\left(x\right) with a=ln⁡3a=\ln\sqrt{3})
Taylor and MacLaurin and Polynomials
Find the 2nd degree Taylor polynomial of f(x)=e2xf\left(x\right)=e^{2x}centered at a=ln⁡3a=\ln\sqrt{3}.
(i.e. find the first 3 terms (i.e.n=0,1,2n=0,1,2) of the Taylor polynomial of f(x)f\left(x\right) with a=ln⁡3a=\ln\sqrt{3})
Taylor and MacLaurin and Polynomials
Find the 2nd degree Taylor polynomial of f(x)=e2xf\left(x\right)=e^{2x}centered at a=ln⁡3a=\ln\sqrt{3}.
(i.e. find the first 3 terms (i.e.n=0,1,2n=0,1,2) of the Taylor polynomial of f(x)f\left(x\right) with a=ln⁡3a=\ln\sqrt{3})
Determine the second order Taylor approximation of 3+x2\sqrt{3+x^2} about x=1x=1.
(a) 2+12x2+316x42+\frac12x^2+\frac{3}{16}x^4 (b) 2+12(x−1)+316(x−1)22+\frac{1}{2}(x-1)+\frac{3}{16}(x-1)^2 (c) 4+14(x−1)+38(x−1)24+\frac{1}{4}(x-1)+\frac{3}{8}(x-1)^2
Find the 2nd degree Taylor polynomial of f(x)=e2xf\left(x\right)=e^{2x}centered at a=ln⁡3a=\ln\sqrt{3}.
(a) 3+(x−ln⁡3)+(x−ln⁡3)22!3+\left(x-\ln\sqrt{3}\right)+\frac{\left(x-\ln\sqrt{3}\right)^2}{2!} (b) 3+6(x−ln⁡3)+6(x−ln⁡3)22!3+6\left(x-\ln\sqrt{3}\right)+\frac{6\left(x-\ln\sqrt{3}\right)^2}{2!}
(c) 3+6(x−ln⁡3)+6(x−ln⁡3)23+6\left(x-\ln\sqrt{3}\right)+6\left(x-\ln\sqrt{3}\right)^2 (d) (x−ln⁡3)+(x−ln⁡3)22!+(x−ln⁡3)33!\left(x-\ln\sqrt{3}\right)+\frac{\left(x-\ln\sqrt{3}\right)^2}{2!}+\frac{\left(x-\ln\sqrt{3}\right)^3}{3!}
Taylor and MacLaurin: Polynomials
Now let Tn(x)T_n(x) be the nnth degree Taylor polynomial centred at x=1x = 1 for the function
f(x)=ln⁡(x)f(x) = \ln(x)
For which value(s) of nn will Tn(1.1)T_n(1.1) give an underestimate of ln⁡(1.1)\ln(1.1)?
Taylor and MacLaurin: Polynomials
Let T3(x)T_3(x) be the third degree Taylor polynomial centred at x=0x = 0 for
f(x)=(x+1)cos⁡(x)f(x) = (x + 1)\cos(x)
Write down T3(x)T_3(x). Make sure that you simplify the coefficients.
Taylor and MacLaurin and Polynomials
Find the 2nd degree Taylor polynomial of f(x)=e2xf\left(x\right)=e^{2x}centered at a=ln⁡3a=\ln\sqrt{3}.
(i.e. find the first 3 terms (i.e.n=0,1,2n=0,1,2) of the Taylor polynomial of f(x)f\left(x\right) with a=ln⁡3a=\ln\sqrt{3})
Find the 2nd degree Taylor polynomial of f(x)=e2xf\left(x\right)=e^{2x}centered at a=ln⁡3a=\ln\sqrt{3}.
(i.e. find the first 3 terms (i.e.n=0,1,2n=0,1,2) of the Taylor polynomial of f(x)f\left(x\right) with a=ln⁡3a=\ln\sqrt{3})
Find the 2nd degree Taylor polynomial of f(x)=e2xf\left(x\right)=e^{2x}centered at a=ln⁡3a=\ln\sqrt{3}.
(i.e. find the first 3 terms (i.e.n=0,1,2n=0,1,2) of the Taylor polynomial of f(x)f\left(x\right) with a=ln⁡3a=\ln\sqrt{3})
Determine the second order Taylor approximation of 3+x2\sqrt{3+x^2} about x=1x=1.
3+x2  about x=1  [second order approximation]\sqrt{3+x^2}\ \ about\ x=1\ \ [\text{second\ order\ approximation]}
f(0)=f(x)=3+x2f^{(0)}=f(x)=\sqrt{3+x^2}
f(1)(x)=12(3+x2)−12.2xf^{(1)}(x)=\frac{1}{2}(3+x^2)^{-\frac{1}{2}}.2x
Taylor and MacLaurin and Polynomials
If f(x)f(x) has the following Taylor series approximation find the 10th derivative of f(x)f(x)at x=0.x=0.
f(x)=1−x22!+x44!−x66!+…f(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\ldots
Taylor Polynomials
Find quadratic approximation for x\sqrt x at x=4x=4
Taylor and MacLaurin and Polynomials
Find the 2nd degree Taylor polynomial of f(x)=e2xf\left(x\right)=e^{2x}centered at a=ln⁡3a=\ln\sqrt{3}.
Find the second order Taylor polynomial T2(x)T_2(x) for f(x)=xf(x)=\sqrt{x} at x=16x=16. Use this to approximate 17\sqrt{17}. What is a good bound on the error in this approximation?
Consider f(x)=1+xf(x) = \sqrt{1+x}.
Find the 2nd degree Taylor polynomial of f(x)=e2xf\left(x\right)=e^{2x}centered at a=ln⁡3a=\ln\sqrt{3}.
(i.e. find the first 3 terms (i.e.n=0,1,2n=0,1,2) of the Taylor polynomial of f(x)f\left(x\right) with a=ln⁡3a=\ln\sqrt{3})
Find that second order Taylor series polynomial used to approximate f(x)=e2xf(x)=e^{2x}
at x = 0
If f(x)f(x) has the following Taylor series approximation find the 10th derivative of f(x)f(x) at x=0.x=0.
f(x)=1−x22!+x44!−x66!+…f(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\ldots
Find the Maclaurin polynomial of degree 2019 of the function f(x)=20x2+16f(x) = 20x^2+16.
Find second order Taylor series approximation for x\sqrt{x} about x=2x=2.
Find the following Taylor polynomials
Find the Maclaurin series for f(x)=cos⁡3xf(x)=\cos{3x} up to and including x4x^4 terms.
Let
F3(x)=c0+c1x+c2x2+c3x3F_3(x) = c_0 + c_1x + c_2x^2 + c_3x^3
be the third order Taylor polynomial for the function
Taylor polynomials
In the Maclaurin polynomial of degree 30 of the function f(x)f(x), the coefficient of x10x^{10} is given by 1/720001/72000. Find f(10)(0)f^{(10)}(0).
Taylor and MacLaurin and Polynomials
Find the 2nd degree Taylor polynomial of f(x)=e2xf\left(x\right)=e^{2x}centered at a=ln⁡3a=\ln\sqrt{3}.
(i.e. find the first 3 terms (i.e.n=0,1,2n=0,1,2) of the Taylor polynomial of f(x)f\left(x\right) with a=ln⁡3a=\ln\sqrt{3})
Find the following Taylor polynomials
Find the following Taylor polynomials (just type the answer, do NOT type "T2(x)=T_2\left(x\right)=" in front of your answer).
Find the following Taylor polynomials
a) 2nd degree for f(x)=xf\left(x\right)=\sqrt{x}
b) 3rd degree for g(x)=sin⁡xg\left(x\right)=\sin x
Taylor Polynomials
Find the following Taylor polynomials
T4(x)T_4(x) of a function ff about point x=−1x=-1 is given as follows:
T4(x)=2+5(x+1)+12(x+1)2+25(x+1)4T_4(x)=2+5(x+1)+12(x+1)^2+\frac{2}{5}(x+1)^4
Find
Consider f(x)=1+xf(x) = \sqrt{1+x}.