DATA 605 - Homework 14

This week, we’ll work out some Taylor Series expansions of popular functions.

\[f (x) = (1−x)\]

\[f (x) = e^x\]

\[f (x) = ln(1 + x)\]

\[f(x)=x^{(1/2)}\]

For each function, only consider its valid ranges as indicated in the notes when you are computing the Taylor Series expansion. Please submit your assignment as an R- Markdown document.

library(tidyverse)

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library(calculus)

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Question 1

\[f (x) = (1−x)\]

taylor("1-x", 0, var = "x", order = 5)

## $f
## [1] "(1) * 1 + (-1) * x^1"
## 
## $order
## [1] 5
## 
## $terms
##   var coef degree
## 0   1    1      0
## 1 x^1   -1      1
## 2 x^2    0      2
## 3 x^3    0      3
## 4 x^4    0      4
## 5 x^5    0      5

Question 2

\[f (x) = e^x \]

taylor("exp(1)^x", 0, var = "x", order = 5)

## $f
## [1] "(1) * 1 + (1) * x^1 + (0.5) * x^2 + (0.166666666666667) * x^3 + (0.0416666666666667) * x^4 + (0.00833333333333333) * x^5"
## 
## $order
## [1] 5
## 
## $terms
##   var        coef degree
## 0   1 1.000000000      0
## 1 x^1 1.000000000      1
## 2 x^2 0.500000000      2
## 3 x^3 0.166666667      3
## 4 x^4 0.041666667      4
## 5 x^5 0.008333333      5

Question 3

\[f (x) = ln(1 + x)\]

taylor("log(1+x)", 0, var = "x", order = 5)

## $f
## [1] "(1) * x^1 + (-0.5) * x^2 + (0.333333333333333) * x^3 + (-0.25) * x^4 + (0.2) * x^5"
## 
## $order
## [1] 5
## 
## $terms
##   var       coef degree
## 0   1  0.0000000      0
## 1 x^1  1.0000000      1
## 2 x^2 -0.5000000      2
## 3 x^3  0.3333333      3
## 4 x^4 -0.2500000      4
## 5 x^5  0.2000000      5

Question 4

\[f(x)=x^{(1/2)}\]

taylor("x^(1/2)", 0, var = "x", order = 5)

## $f
## [1] "(Inf) * x^1 + (-Inf) * x^2 + (Inf) * x^3 + (-Inf) * x^4 + (Inf) * x^5"
## 
## $order
## [1] 5
## 
## $terms
##   var coef degree
## 0   1    0      0
## 1 x^1  Inf      1
## 2 x^2 -Inf      2
## 3 x^3  Inf      3
## 4 x^4 -Inf      4
## 5 x^5  Inf      5