(due Monday, February 8, 2010)
(1) Let (V, ) be a normed vector space.
(a) Prove that
| x y | xy
for all x, y V
and use this to show that the mapping
(b) Let (xn )nN and (yn )nN be se
(due Tuesday, March 23, 2010)
(1) (a) Prove that the nite union and the nite intersection of
compact sets (in a metric space) is compact.
(b) Is the same true for innite unions or intersections. Justify
(due Tuesday, April 6, 2010)
(1) (a) Show that there is an inner product on the space of all
n n matrices given by
A, B = Tr(AB ) =
aij bij ,
where Tr(A) = i aii denotes the trace of the matrix A.
Calculus I Review, Part 4
Derivatives of Trigonometric Functions
22. The circular sector shown here has radius 1 and angle radians.
(a) What is the length of the circular arc that comprises part of the
boundary of the sector?
(b) What is the area of the
Calculus I Review, Part 3, Revised
16. Inverse Functions. Let f be a one-to-one dierentiable function. Then
f has an inverse, which is continuous since f itself is continuous. Given a
point y in the domain of f , let x = f (y). Then y = f 1 (x).
Taylor Remainder Formulas
Every smooth function f has Taylor polynomials
f (n) (a)
= f (a) + f (a)(x a) +
(x a)2 + +
(x a)k .
For example, the simplest Taylor polynomials are the constant function T0 (x) =