DATA 605 : Week 14 Discussion
Debabrata Kabiraj
May 02, 2020
QUESTION
Using R, provide the solution for any exercise in either Chapter 4 or Chapter 7 of the calculus textbook. If you are unsure of your solution, post your concerns.
Apex CALCULUS v4.0
Chapter 4. Applications of the Derivative
4.1 Newton’s Method
Exercise 11
Page 173
use Newton’s Method to approximate all roots of the given functions accurate to 3 places after the decimal. If an interval is given, find only the roots that lie in that interval. Use technology to obtain good initial approximations.
- \(f(x) = x^2 cos x + (x-1)sin x\) on (-3,3)
ANSWER
Solution
func <- function(x) {
x^2 * cos(x) + (x-1) * sin(x)
}
curve(func, col = 'blue', lwd = 2, from = 0, n = 100, xlim=c(0, 5), ylab='f(x)')
abline(a=0, b=0, lty = 5)
## $root
## [1] 0
##
## $f.root
## [1] 0
##
## $iter
## [1] 0
##
## $init.it
## [1] NA
##
## $estim.prec
## [1] 0Newton’s Method
#set initial x
x <- 2
while(TRUE){
if (x> -3 && x<3 ){
oldx = x
x = x^2 * cos(x) + (x-1) * sin(x)
print(x)
if (abs(abs(x)-abs(oldx))< 0.001){
break
}
}
}## [1] -0.7552899
## [1] 1.618589
## [1] 0.492721
## [1] -0.02605988
## [1] 0.02741486
## [1] -0.02590865
## [1] 0.02724797
## [1] -0.02576006
## [1] 0.02708408
## [1] -0.02561403
## [1] 0.0269231
## [1] -0.02547049
## [1] 0.02676495
## [1] -0.02532937
## [1] 0.02660954
## [1] -0.0251906
## [1] 0.0264568
## [1] -0.02505412
## [1] 0.02630665
## [1] -0.02491986
## [1] 0.02615902
## [1] -0.02478776
## [1] 0.02601384
## [1] -0.02465777
## [1] 0.02587103
## [1] -0.02452983
## [1] 0.02573055
## [1] -0.02440388
## [1] 0.02559232
## [1] -0.02427988
## [1] 0.02545629
## [1] -0.02415777
## [1] 0.02532239
## [1] -0.02403751
## [1] 0.02519058
## [1] -0.02391905
## [1] 0.0250608
## [1] -0.02380235
## [1] 0.02493299
## [1] -0.02368736
## [1] 0.02480711
## [1] -0.02357404
## [1] 0.02468312
## [1] -0.02346235
## [1] 0.02456096
## [1] -0.02335225
## [1] 0.02444058
## [1] -0.0232437
## [1] 0.02432195
## [1] -0.02313667
## [1] 0.02420503
## [1] -0.02303113
## [1] 0.02408977
## [1] -0.02292703
## [1] 0.02397614
## [1] -0.02282435
## [1] 0.02386409
## [1] -0.02272305
## [1] 0.02375359
## [1] -0.0226231
## [1] 0.02364461
## [1] -0.02252448
## [1] 0.02353711
## [1] -0.02242715
## [1] 0.02343105
## [1] -0.02233108
## [1] 0.02332641Use the animation to show newton’s method
ani.options(interval = 1, nmax = 50)
par(pch = 20)
showAnimation <- newton.method(FUN = function(x) x^2 * cos(x) + (x-1) * sin(x), init = 2, rg = c(-3, 10), tol = 0.001, interact = FALSE, col.lp = c("blue", "red", "red"))
## [1] 1.81328Combining the results above, we can see the roots are 0.023 and 1.813.