ADDITIONAL NOTES AND EXERCISES FOR 6.3
Newton’s method does not always work. The following result gives sufficient
conditions for when it does.
Theorem 1 (Newton).Let f: [a, b]→Rbe continuous. Suppose that f(c) = 0
for some cin (a, b). Suppose that f00 exists and is bounded on (a, b); that is, there
exists M > 0such that
f00(x)≤M
for every x∈(a, b).
Suppose that f0is bounded away from zero; that is, there exists ² > 0such that
f0(x)≥²
for every x∈(a, b). Then there is a closed interval I:= [d, e]⊂(a, b)containing c
such that the sequence given recursively by
xn=xn−1−f(xn−1)
f0(xn−1)
converges to c. Moreover, xnis in Ifor each n.
For this to make sense it helps if we have a notion of convergence of sequences.
We will do this in second semester calculus. For now, we will just use the intuitive
statement “as ngets large, xngets arbitrarily close to c” and write lim
n→∞
xn=c.
Let us now analyze the various conditions in the statement.
Bounding f0away from zero should seem reasonable from the definition of xn
but it is not necessary. The following exercise show that Newton’s method may
work even if f0is not bounded away from zero.
Exercise 2. Let f(x) = x2. Adopt the notation of Newton’s theorem. Show that
if xn−16= 0 then xn=xn−1
2. Suppose that x0= 1.What is x1? How about x2?
How about xn? Argue that xnconverges to zero as ngoes to ∞when x0= 1.
In the above example, any x0except zero will do.
The next exercise illustrates what can go wrong if we do not bound f0away from
zero.
Exercise 3. Let f(x) = x4−4x−11. Then f(0) <0< f(3) so by the intermediate
value theorem there exists a root between x= 0 and x= 3. What happens if we
apply Newton’s method when we start at x0= 1? Sketch the graph of fon the
interval [0,3].
We also require a bound on the second derivative. The following exercise shows
what can go wrong.
Exercise 4. Let f(x) = x1
3. Observe that x= 0 is a root of f. Compute f00(x)
and explain why f00 is unbounded near x= 0. Attempt to use Newton’s method
with x0= 1. What happens? Illustrate your conclusion with a diagram.
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