Curve Graphing, Newton's Method

Department of Mathematics, UTA

To find x [Log [x + 5] - Log [x ]] one may graph the expression first:

which suggests the limit at ∞ is 5.

To apply L'Hospital's rule notice that the expression can be written as = with at ∞ .

Consequently the limit of the original expression is

Critical numbers = {x | f ' [x] = 0 or f ' [x] does not exist} constitute a key element for studying maxima or minima (extrema) of f [x].

Pick the real solutions only (use cut & paste )

To verify that f [x] achieves a local max at x = .902652 with max = 5.1841 and local min at x = 2.56238 with

min = f [2.56238] = -105.931 one can utilize the second derivative test as follows:

which shows that f '' is negative (as required) at x = .902652 and f '' is positive (as required) at x = 2.56238.

Another way to find the min or max of f [x] is to use the built-in numerical algorithms as follows (provided that

some initial approximation to the critical numbers is established by plotting f [x ] ).

so taking (5.1841) = 5.1841 gives the correct maximum attained at x = 0.902652.

Continuous function on a closed bounded interval attains its absolute min and absolute max, quite often at the endpoints of the interval [a,b]. Suspects for absolute min or max are among critical numbers augmented by points a, b.

To solve f [x] = 0 numerically (which is always the case for any polynomial of degree higher than 5, except

in the special cases) the Newton's algorithm is as follows : identify the starting point near the solution

and use = . Then lim = x is the root, i.e., f [x]=0. The initial guess should be found

from the graph of f [x].

To find all roots of = 0, graph first over large interval [-a, a].

As seen from the graph (zoom if needed) there are 2 real roots, namely around -3 and around 10.

To find the positive solution take the initial guess as 9:

To streamline computations define a new function g[x] which evaluates numerically the first 25 digits

of approximation sequence:

Then

To compute n iterations starting at x0 one uses the NestList[g, x0, n] as follows:

9 |

10.76857749469214437367303609341825902336`25 |

10.216955306206155083503611723443462405`24.1064 |

10.12058299258635588182831319643859182795`23.2099 |

10.117806239312983280218683208819809`22.3144 |

10.11780397413295562844228212398523`21.4188 |

10.117803974131449032521296575722705`20.5233 |

Compare the above results to built-in functions such as NRoots[expr[x] = = 0, x], NSolve[ expr[x] = = 0], FindRoot[expr[x] = = 0,{x, }],

where is the initial approximation chosen in the vicinity of the exact solution.

As one can see the 4-th iteration in Newton's scheme agrees up to the first 7 digits with the exact solution,

while the 6-th iteration provides the exact first 21 digits! Numerical analysts, "Go to dust!"

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