Solutions for Problems for Chapter 3.
8. Use both the definition of limit and a sequence approach to establish lim
x2
1 1 11 x
1 |x 2 |
1x
1x
1x
5
for x close to 2, say for |x 2 | 1
i. e. , 3
x
,
2
2
2
1 1
then
2|x 2 |
if |x 2 | min
,
1x
2
1
1
x
1
scrat
M 317 sec 2
Exam on Chapters 1 and 2
Solutions
1.Let a n denote a bounded sequence that does not converge. Prove that a n contains
at least two subsequences that converge to different limits.
Solution: Since the sequence is bounded, it must have at least
Chapter 23: Amino Acids
Amino acids are molecules that contain both a carboxylic acid and an amino group
Of biological importance are -amino acids
small molecules that components of proteins (20 naturally occurring)
O
NH 2
HO
R
Amino acids can also be fu
1. How is the number 0 defined by the axioms governing the reals? What do the axioms
imply about division by 0 ?
x0 x
Axiom 3 says:
x
Axiom 4 says:
x
Axiom 9 says:
x y
y
x0
x y
y
xy
Axiom 8 says:
x
R,
x 0
R, x
then
z yx
and
z y0
0,
x
xy 0
1
so
such that
Additional Problems
Sequences
3. Find an N such that |a n L | 10 3 for n N
2
2
a an
L 0 |a n 0 |
if 2
n1
n1
4 n1
i.e.,
Then |a n 0 |
if n 42 N
2
1
n3
n 1/3
1
b an 1
i.e.,
c an 2 2
n
1 | 13
n
Then |a n 1 |
if
L 1 |a n
L2
log n log 2
i.e.,
|a n
2| 2
T
Exam 2
Chapter 3
1.
For each of the following cases, either give an example or explain (using
definitions) why no such example is possible:
a. p is an accumulation point for A and is also an interior point for A
b. p is an accumulation point for A and is
Exam 3
Chapter 4
Solutions
1. Use the difference quotient to show that if f x is differentiable at all x, then
g x f x 1 f x 1 is also differentiable at all x.
Dhg x 1 f x h 1 f x h 1 f x 1 f x 1
h
1 f xh1 f xh 1 f xh1 f x 1 f xh1 f x 1
h
1 f xh 1 f x 1
Exam 4
Chapter 5
1. Suppose f x is defined and bounded on I 0, 1
a) State a condition on f which implies that f is integrable on I
f is continuous on I, or f is monotone on I
b) State a condition in terms of f, P , and f, P and partitions P
which implies
Exam 2
Continuity
1. Use the
definition of continuity to show that that f x x 2 1 is continuous at x 1.
|f x
Then
f 1 | |x 2 1
0,
2 | |x 2
2 /5
1 | |x 1|x
such that |f x
1|
5
2
|x
f1 |
1|
if
1|
|x
when
1
2
1 | min
|x
1
2
, 2 /5
2. Use whatever method
Sample Final Exam
Part A
1. State the completeness axiom. What is the point of this axiom; what is it designed to
achieve?
2. The nested interval theorem and the Balzano-Weierstrass theorem can be stated as
theorems about the real numbers. What do they ea
14. Suppose f
D R . Show that g x f x
Dhg x 1 f x h
h
1 f xh
h
2
fx
2
1 f xh
h
2
DR
f xh f x f xh f x
2
f x f xh 1 f xh
h
fx
2
fx fx
Dhf x f x h Dhf x f x
lim D h g x 2 f x f x
Then
h0
15. see exam solutions
22. Show that if 0 a b then 1
By the mean va
M 317
Problems Chapter 5
1. For f x 1 x on I 0 x 2 , let P denote a uniform partition of I with N 4. Then
compute each of the quantities, m|I |, f, P , RS f, P , f, P , and M|I |, where the tag points
for the Riemann sum are the midpoints of the intervals
MAT 142 Course Syllabus | Spring 2016 B
MAT 142: College Mathematics
Spring 2016 iCourse/oCourse Syllabus
Instructor: Beth Jones
E-mail: [email protected]
Office Hours:
In Person:
9 am 10 am Tuesday and Thursday
12 noon 1 pm Thursday
Virtual:
12 noon 1 pm