Exercise 8.8

Problems Key Idea 8.8.1 gives the \(n^{th}\) term of the Taylor series of common functions. In Exercises 3 – 6, verify the formula given in the Key Idea by finding the first few terms of the Taylor series of the given function and identifying a pattern.

  1. \(f(x) = e^x; c = 0\)
library(calculus)
myf=function(x) exp(x)
taylor(myf, var=c(x=0), order=6)
## $f
## [1] "(1) * 1 + (0.999999999996052) * x^1 + (0.499999999979692) * x^2 + (0.166666665585433) * x^3 + (0.0416666593124569) * x^4 + (0.00833328285918537) * x^5 + (0.00138886379238021) * x^6"
## 
## $order
## [1] 6
## 
## $terms
##   var        coef degree
## 0   1 1.000000000      0
## 1 x^1 1.000000000      1
## 2 x^2 0.500000000      2
## 3 x^3 0.166666666      3
## 4 x^4 0.041666659      4
## 5 x^5 0.008333283      5
## 6 x^6 0.001388864      6
  1. \(f(x) = sin x ; c = 0\)
myf=function(x) sin(x)
taylor(myf, var=c(x=0), order=6)
## $f
## [1] "(1) * x^1 + (-0.166666665593812) * x^3 + (0.00833328307142597) * x^5"
## 
## $order
## [1] 6
## 
## $terms
##   var         coef degree
## 0   1  0.000000000      0
## 1 x^1  1.000000000      1
## 2 x^2  0.000000000      2
## 3 x^3 -0.166666666      3
## 4 x^4  0.000000000      4
## 5 x^5  0.008333283      5
## 6 x^6  0.000000000      6
  1. \(f(x) = tan x; c= 0\)
myf=function(x) tan(x)
taylor(myf, var=c(x=0), order=6)
## $f
## [1] "(1) * x^1 + (0.333333040935881) * x^3 + (0.132895350565712) * x^5"
## 
## $order
## [1] 6
## 
## $terms
##   var      coef degree
## 0   1 0.0000000      0
## 1 x^1 1.0000000      1
## 2 x^2 0.0000000      2
## 3 x^3 0.3333330      3
## 4 x^4 0.0000000      4
## 5 x^5 0.1328954      5
## 6 x^6 0.0000000      6