Taylor Remainder Formulas
Every smooth function f has Taylor polynomials
k
Tk (x)
=
f (n) (a)
(x a)n
n!
n=0
= f (a) + f (a)(x a) +
f (a)
f (k)
(x a)2 + +
(x a)k .
2!
k!
For example, the simplest Taylor polynomials are the constant function T0 (x) =
f (a)
5 2.1 EXERCISE SET | mommy/l
Practice Exercises
Ia Exorcist: I-IO. identify the linear equations in one variable.
1.x-9-l3
3.xz9-13
9
5. '- '13
.l
7.
Vix+1r303
9.1x+2|=5
2.
4.
B.
84
1 0.
x-15820
x215-2o
32:20
x
Vix+1r=05
Ix+5l=8
Solve each equalion in Exe
MyMathLab
Practice Exercises
In Exercises [10, identify the linear equations in one variable.
1. x 9 = 13 linear 2. x 15 = 20 linear
3. x2 9 = 13 not linear 4. x2 - 15 = 20 not linear
5. 2 = 13 not linear 6. E = 20 not linear
x
x
7. Vi! + 17 = 0.3- line
MATH 328
REAL ANALYSIS
WINTER 2010
Assignment 2
(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
:V R
x x
is continuous.
(b) Let (xn )nN and (yn )nN be se
MATH 328
REAL ANALYSIS
WINTER 2010
Assignment 4
(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
your
MATH 328
REAL ANALYSIS
WINTER 2010
Assignment 5
(due Tuesday, April 6, 2010)
(1) (a) Show that there is an inner product on the space of all
n n matrices given by
n
t
A, B = Tr(AB ) =
aij bij ,
i,j=1
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.
1
(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).
(a) Show