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Taylor and MacLaurin: Polynomials

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Wize University Calculus 1 Textbook > Applications of Differentiation

Taylor and MacLaurin and Polynomials

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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)?
You must justify your answer.
More Taylor and MacLaurin and Polynomials Questions:
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
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}.