MAA 4212, Spring 2002—Homework #6 non-book problems
Hand in only the second exercise below.
1. Prove the following version of Taylor’s Theorem. The difference between this version
and the ones proven in class is that this one uses the Taylor polynomial of the same
degree of differentiability as the function, rather than one less. The cost of this apparent
improvement is that one does not get any sort of formula for the remainder, but just
knowledge of the size of the remainder. For simplicity, I’ve based everything at the origin,
but you should be able to translate this to the more general statement if you ever need
to.
Theorem. Let Ube a ball in Rncentered at 0. Let fbe an mtimes continuously
differentiable function. For x∈B, define
Pm(x) = f(0) + X
i
xifi(0) + 1
2X
i,j
xixjfij(0) + · · · +1
m!X
i1,i2,...im
xi1xi2· · · ximfi1i2···im(0),
and
Rm(x) = f(x)−Pm(x).
(Above, fij means ∂2f/∂ xi∂xjetc, and a sum over kindices means a k-fold sum.) Prove
that
lim
x→0
Rm(x)
kxkm= 0.
Remark. Using the “big-oh, little-oh” terminology we introduced once in class, this
says that Rm(x) is o(kxkm) (“little-oh of kxkm”) as x→0. Thus, if a function is mtimes
continuously differentiable at a point, then its mth-order Taylor polynomial at that point
is a good approximation to order m.
2. p. 214/ 16. This is a generalization of the “second partials test” you learned in
Calculus III. To do the problem you will (probably) need problem 1 above, and the
following definition.
Definition. Let Abe an n×nsymmetric matrix of real numbers, and for all x∈Rn,
let hA(x) = Pi,j Aijxixj. (This is called a quadratic form.) We say that Ais
positive-definite if hA(x)>0,∀x6=0
positive-semidefinite if hA(x)≥0,∀x
negative-definite if hA(x)<0,∀x6=0
negative-semidefinite if hA(x)≤0,∀x.
We also call Adefinite if Ais either positive-definite or negative-definite, semidefinite if
Ais either positive semidefinite or negative semidefinite, and indefinite if Ais neither
positive-semidefinite nor negative-semidefinite. The identity matrix is an example of a
matrix that is positive-definite; minus the identity is an example of a matrix that is
negative-definite. The matrix 1 0
0 0 !is positive-semidefinite but not positive-definite,
and the matrices 1 0
0−1!and 0 1
1 0 !are indefinite. Warning: One can show that a
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