Page 496 Exercise 27

Use the Taylor series given in Key Idea 8.8.1 to create the Taylor series of the given functions.

\(f(x)=sin(2x+3)\)

Solution using R

library(pracma)

f <- function(x) sin(2*x + 3)

taylor_series <- taylor(f, 0)  

print(taylor_series)
## [1]  0.09407953  1.31998927 -0.28224001 -1.97998499  0.14112001

Solution by Hand

Calculate the first few terms:

\[ f(x) = sin(2x+3)\\ f'(x) = 2cos(2x+3)\\ f''(x) = -4sin(2x+3)\\ f'''(x) = -8cos(2x+3)\\ f''''(x) = 16sin(2x+3)\\ \]

Evaluate at x=0: \[ f(0) = sin(3)\\ f'(0) = 2cos(3)\\ f''(0)= -4sin(3)\\ f'''(0)=-8cos(3)\\ f''''(0)=16sin(3)\\ \]

Use the Taylor Series formula to find the terms:

\[ a_0 = f(0) = sin(3)\\ a_1 = \frac{f'(0)}{1!} = 2cos(3)\\ a_2 = \frac{f''(0)}{2!} = -2sin(3)\\ a_3 = \frac{f'''(0)}{3!} = -\frac{4}{3} cos(3)\\ a_4 = \frac{f''''(0)}{4!} = \frac{2}{3}sin(3) \]

The Taylor Series up until the fourth derivative is:

\[ sin(3) + 2cos(3) -2sin(3) - \frac{4}{3} cos(3) + \frac{2}{3} sin(3) \]