High School
SAT
ACT
University
MCAT
LSAT
DAT

Find the second order Taylor polynomial T_2(x) for f(x)=√x at x=16. Use this to…

Related Topics

Wize University Calculus 1 Textbook > Applications of Differentiation

Taylor Polynomial Remainder

3 Activities

Wize University Calculus 1 Textbook > Applications of Differentiation

Taylor and MacLaurin and Polynomials

4 Activities

checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
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?
More Taylor Polynomial Remainder Questions:
Let f(x)=cos⁡(3x)+x−1f(x)=\cos{(3x)}+x-1. Approximate f(π3+15)f(\frac{\pi}{3}+\frac{1}{5}) using linear approximation. Is this overestimation or underestimation of the actual value? Estimate the max error bound for this approximation.
Use the second order Taylor polynomial for f(x)=xf(x)=\sqrt{x} at x=16x=16 to approximate 17\sqrt{17}. What is a good bound on the error in this approximation?
Estimate the size of the error made in the linear approximation to estimate 99.9\sqrt{99.9}.
Let f(x)=cos⁡(3x)+x−1f(x)=\cos{(3x)}+x-1. Approximate f(π3+15)f(\frac{\pi}{3}+\frac{1}{5}) using linear approximation. Is this overestimation or underestimation of the actual value?
Consider f(x)=1+xf(x) = \sqrt{1+x}.
Taylor polynomials
Find the degree 4 Taylor polynomial for f(x) = sin(3x) centered at x=π/6.x=\pi/6. What is the lowest error bound within [0,π/6][0,\pi/6]?
Concept Clarifier
Find the fifth order Taylor polynomial for y=sin⁡ xy=\sin\ x centred about x0=πx_0=\pi. What is the minimum error bound for such Taylor polynomial in the interval x∈[0,3π]x \in [0,3\pi]?
If f(x)=e−2x+3x2f(x)=e^{-2x}+3x^2 approximate f(0.1)f(0.1) ? Find the maximum error for approximating positive numbers?
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?
Let f(x)=cos⁡(3x)+x−1f(x)=\cos{(3x)}+x-1. Approximate f(π3+15)f(\frac{\pi}{3}+\frac{1}{5}) using linear approximation. Is this overestimation or underestimation of the actual value? Estimate the max error bound for this approximation.
Consider f(x)=1+xf(x) = \sqrt{1+x}.
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
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}.
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}.