AArora_Posting13

library(rootSolve)

Code: pg 161 Book: Calculus Exercises Section 4.1 8-11

Problem: 8

equation <- function(x){
  return(x**3 + 5 * x**2 - x - 1)
}

plot(equation, from = -10, to = 10)
abline(0, 0, col = 'blue')

g <- function(x) {
    x^3 + 5*x^2 - x - 1
    }

gPrime <- function(x) {
    3*x^2 + 10*x -1
    }



tolerance <- .00001

root <- function(g, gPrime, guess, tolerance) {
    x = guess
    while (abs(g(x)) > tolerance) {
        x = x - g(x)/gPrime(x)
        }
    x
    }

root(g, gPrime, 5, tolerance)

## [1] 0.525429

root(g, gPrime, 0, tolerance)

## [1] -0.3691024

root(g, gPrime, -5, tolerance)

## [1] -5.156325

Verify with R:

coefficient <- c(-1, -1, 5, 1)
result <- polyroot(coefficient)
result

## [1]  0.5254276-0i -0.3691024+0i -5.1563252-0i

Problem: 9

equation <- function(x){
  return(x**4 + 2*x**3 - 7*x**2 - x - 1)
}

plot(equation, from = -4, to = 3, n = 1000, ylim=c(-40,40))
abline(0, 0, col = 'blue')

g <- function(x) {
    x^4 + 2*x^3 - 7*x^2 - x - 1
    }

gPrime <- function(x) {
    4*x^3 + 6*x^2 -14*x -1
    }



tolerance <- .00001

root <- function(g, gPrime, guess, tolerance) {
    x = guess
    while (abs(g(x)) > tolerance) {
        x = x - g(x)/gPrime(x)
        }
    x
    }

root(g, gPrime, 3, tolerance)

## [1] 1.961383

#root(g, gPrime, 1, tolerance)

root(g, gPrime, -3, tolerance)

## [1] -3.793903

Verify with R:

coefficient <- c(-1, -1, -7, 2, 1)
result <- polyroot(coefficient)
result

## [1] -0.0837397+0.3568934i -0.0837397-0.3568934i  1.9613824-0.0000000i
## [4] -3.7939030-0.0000000i

Problem: 10

equation <- function(x){
  return(x**17 - 2*x**13 - 10*x**8 + 10)
}

plot(equation, from = -4, to = 3, n = 1000, ylim=c(-40,40))
abline(0, 0, col = 'blue')

g <- function(x) {
    x**17 - 2*x**13 - 10*x**8 + 10
    }

gPrime <- function(x) {
    17*x^16 - 26*x**12 -80*x**7 + 10
    }



tolerance <- .00001

root <- function(g, gPrime, guess, tolerance) {
    x = guess
    while (abs(g(x)) > tolerance) {
        x = x - g(x)/gPrime(x)
        }
    x
    }

root(g, gPrime, 2, tolerance)

## [1] 1.39341

root(g, gPrime, 1, tolerance)

## [1] 0.9883117

root(g, gPrime, -2, tolerance)

## [1] -1.013403

Verify with R:

coefficient <- c(10,0,0,0,0,0,0,0,-10,0,0,0,0,-2,0,0,0,1)
result <- polyroot(coefficient)
result

##  [1]  0.6879488+0.7471924i -0.7018552+0.6786703i -0.7018552-0.6786703i
##  [4]  0.6879488-0.7471924i  0.0124521+0.9991391i -1.0134032-0.0000000i
##  [7]  0.0124521-0.9991391i  0.9883117+0.0000000i  0.1724880+1.3570560i
## [10] -1.2615042+0.3619573i -0.5516665-1.1339576i  0.9579780+0.7115178i
## [13] -0.5516665+1.1339576i -1.2615042-0.3619573i  0.9579780-0.7115178i
## [16]  1.3934096-0.0000000i  0.1724880-1.3570560i

Problem: 11

equation <- function(x){
  return(x**2 *cos(x) + (x-1)*sin(x))
}

plot(equation, from = -3, to = 3, n = 1000)
abline(0, 0, col = 'blue')

g <- function(x) {
    x**2 *cos(x) + (x-1)*sin(x)
    }

gPrime <- function(x) {
    -x**2*sin(x) + 3*x*cos(x) + sin(x) - cos(x)
    }



tolerance <- .00001

root <- function(g, gPrime, guess, tolerance) {
    x = guess
    while (abs(g(x)) > tolerance) {
        x = x - g(x)/gPrime(x)
        }
    x
    }
root(g, gPrime, 5, tolerance)

## [1] 4.87404

root(g, gPrime, 4, tolerance)

## [1] 7.96335

root(g, gPrime, 3, tolerance)

## [1] 1.813278

#root(g, gPrime, 2, tolerance)
root(g, gPrime, 1, tolerance)

## [1] 0.5245027

root(g, gPrime, 0, tolerance)

## [1] 0

#root(g, gPrime, -1, tolerance)
#root(g, gPrime, -2, tolerance)
root(g, gPrime, -3, tolerance)

## [1] -2.164774

root(g, gPrime, -4, tolerance)

## [1] -7.993809

root(g, gPrime, -5, tolerance)

## [1] -4.950579

Verified using the following website:

http://www.wolframalpha.com/widgets/view.jsp?id=a7d8ae4569120b5bec12e7b6e9648b86