Determine the first four terms of the Maclaurin series for sin 2x (a) by using the definition of Maclaurin series. (b) by replacing x by 2x in the series for sin 2x. (c) by multiplying 2 by the series for sin x by the series for cos x, because sin 2x = 2 sin x cos x

Question
Series
asked 2021-03-11
Determine the first four terms of the Maclaurin series for sin 2x
(a) by using the definition of Maclaurin series.
(b) by replacing x by 2x in the series for sin 2x.
(c) by multiplying 2 by the series for sin x by the series for cos x, because sin 2x = 2 sin x cos x

Answers (1)

2021-03-12
(a)
Calculate the derivative for f(x) = sin(2x) and the the value for the derivatives till 8th order.
\(f(x)=\sin2x\)
\(f_1(x)=2\cos2x\)
\(f_2(x)=-4\sin2x\)
\(f_3(x)=-8\cos2x\)
\(f_4(x)=16\sin(2x)\)
\(f_5(x)=32\cos(2x)\)
\(f_6(x)=-64\sin(2x)\)
\(f_7(x)=-128\cos(2x)\)
\(f_8(x)=256\sin(2x)\)
At x=0, \(f(0)=\sin(2\times0)=0\)
\(f_1(x)=2\cos(2\times0)=2\)
\(f_2(x)=-4\sin(2\times0)=0\)
\(f_3(x)=-8\cos(2\times0)=-8\)
\(f_4(x)=16\sin(2\times0)=0\)
\(f_5(x)=32\cos(2\times0)=32\)
\(f_6(x)=-64\sin(2\times0)=0\)
\(f_7(x)=-128\cos(2\times0)=-128\)
\(f_8(x)=256\sin(2\times0)=0\)
Now, consider the formula for the maclaurin's series and express f(x) as Maclaurin's series.
Maclaurin's series expansion,
\(f(x)=f(0)+\frac{f_1(0)x}{1!}+\frac{f_2(0)x^2}{2!}+\frac{f_3(0)x^3}{3!}+\frac{f_4(0)x^4}{4!}+\frac{f_5(0)x^5}{5!}+\frac{f_6(0)x^6}{6!}+\frac{f_7(0)x^7}{7!}+\frac{f_8(0)x^8}{8!}+\frac{f_9(0)x^9}{9!}\)
\(=0+\frac{f_1(0)x}{1!}+0+\frac{f_3(0)x^3}{3!}+0+\frac{f_5(0)x^5}{5!}+0+\frac{f_7(0)x^7}{7!}+0\)
\(=\frac{2x}{1!}+\frac{(-8)x^3}{3!}+\frac{(32)x^5}{5!}+\frac{(-128)x^7}{7!}\)
So, \(f(x)=2x-\frac{4x^3}{3}+\frac{4}{15}x^5-\frac{8}{315}x^7\)
Hence, \(\sin2x=2x-\frac{4x^3}{3}+\frac{4}{15}x^5-\frac{8}{315}x^7\)
(b) Now, replace x by 2x and evaluate the series for \(\sin4x\).
\(\sin2x=2x-\frac{4x^3}{3}+\frac{4}{15}x^5-\frac{8}{315}x^7\)
\(\sin(2\times2x)=(2\times2x)-\frac{4(2x)^3}{3}+\frac{4}{15}(2x)^5-\frac{8}{315}(2x)^7\)
\(\sin4x=4x-\frac{8x^3}{3}+\frac{4\times32x^5}{15}-\frac{8\times128x^7}{315}\)
\(\sin4x=4x-\frac{32x^3}{3}+\frac{128x^5}{15}-\frac{1024x^7}{315}\)
Hence, \(\sin4x=4x-\frac{32x^3}{3}+\frac{128x^5}{15}-\frac{1024x^7}{315}\)
(c)
Now, determine the product for the 2, maclaurin's series fro sin x and cos x and compare it with the maclaurin's series for sin2x.
The series is same for both the expressions.
\(\sin x=x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040}\)
\(\cos x=1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}\)
\(2\sin x\cos x=2\times(x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040})\times(1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720})\)
\(=2(x-\frac{2x^3}{3}+\frac{2x^5}{15}-\frac{4x^7}{315})\)
\(=2x-\frac{4x^3}{3}+\frac{4x^5}{15}-\frac{8x^7}{315}\)
\(\sin2x=2x-\frac{4x^3}{3}+\frac{4x^5}{15}-\frac{8x^7}{315}\)
Hence, \(2\sin2x\cos x=2x-\frac{4x^3}{3}+\frac{4x^5}{15}-\frac{8x^7}{315}\)
0

Relevant Questions

asked 2021-03-11
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
b. Write the power series using summation notation.
c. Determine the interval of convergence of the series.
\(f(x)=\log_3(x+1),a=0\)
asked 2020-11-26
Taylor series
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
b. Write the power series using summation notation.
\(f(x)=\cos x,a=\pi\)
asked 2021-01-19
Taylor series
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
b. Write the power series using summation notation.
\(f(x)=2^x,a=1\)
asked 2021-02-25
Taylor series
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
b. Write the power series using summation notation.
\(f(x)=x\ln x-x+1,a=1\)
asked 2021-02-05
Any method
a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients.
b. Determine the radius of convergence of the series.
\(f(x)=\cos2x+2\sin x\)
asked 2021-01-24
a. Another Maclaurin series we examined is for sin x: a. Find the series for \(\frac{\sin(x^{2})}{x}\) . Express the result in the summation notation, not just as a partial listing of the series terms.
b. Another Maclaurin series we examined is for sin x:
d. Find the series for integral \(\frac{\sin(x^{2})}{x}\)
asked 2021-03-07
Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series
\((1+x)^{-2}=1-2x+3x^2-4x^3+...,for\ -1
\((1+4x)^{-2}\)
asked 2020-11-22
Any method
a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients.
\(f(x)=x^2\cos x^2\)
asked 2021-02-04
Calculate the following:
a. Find the Maclaurin series of cos(x) and find the radius of this series, without using any known power or Maclaurin series, besides geometric.
b. Find exactly the series of \(\cos(-2x)\)
asked 2020-12-17
Find the Maclaurin series for using the definition of a Maclaurin series. [Assume that has a power series expansion.Do not show that Rn(x) tends to 0.] Also find the associated radius of convergence. \(f(x)=(1-x)^{-2}\)
...