517 HW8
1. Find the third order Taylor polynomial of f (x, y) = (x + y)3 at (1, 1).
2. Let f (x) = (x1 + . . . + xn )k and assume j1 + . . . + jn = k.
(a) Show that D1j1 . . . Dnjn f (x) = k!.
(b) Show that if i1 + . . . + in = k, then
D1j1 . . . Dnjn xi11 . . . xinn
(
j1 ! . . . jn !,
=
0,
if i1 = j1 , . . . , in = jn
else.
(c) Conclude that
k
j1 . . . jn
=
k!
.
j1 ! . . . jn !
3. Let f : U → Rm be of class C 1 , with U ⊂ Rn an open set containing the line segment L from
a to a + h. Show that if T : Rn → Rm is linear with matrix A, then
|f (a + h) − f (a) − T (h)| ≤ |h| max ||f 0 (x) − A||.
x∈L
4. Let f : Rm → Rm be of class C 1 on the unit ball B1 (0). Use Problem 3 to show that if
f (0) = 0, f 0 (0) = I and ||f 0 (x) − I|| < for all x ∈ B1 (0), then f (B1 (0)) ⊂ B1+ (0).
5. Let f : Rn → Rm be of class C 1 at a and suppose dfa : Rn → Rm is injective. Use Problem
3 to show that f is injective in a neighborhood of a.
1