Math 142B
Final Examination
September 2, 2011
Instructions
1. You may use any type of calculator, but no other electronic devices during this exam.
2. You may use one page of notes, but no books or other assistance during this exam.
3. Write your Name, PI
Midterm 1
Math 142A, Lecture C
Fall 2016
Time allowed: 50mins
(40 points total)
Name:
PID:
1
1. (3 points) State the Completeness Axiom for R.
Every nonempty subset S R which is bounded above has a
least upper bound.
2. Carefully state the following defin
MATH 142B
Homework 2 Solutions
Jonathan Conder
Section 6.2
6. (a) Given n N, let Pn be the regular partition of [a, b] into n intervals. Dene f : R R by f (x) = x. Now
n
U (f, Pn ) =
a+i
i=1
ba
=
n
n
i=1
ba
n
ba
n
(b a)2
a+
n2
n
i
i=1
(b a)2 n(n + 1)
n2
2
HOMEWORK #2 SOLUTIONS
Solution to Problem 6.2.2. Assume that P1 = cfw_x0 , x1 , . . . , xn and P2 = cfw_y0 , y1 , . . . , ym . Then, there exist
xi1 and xi such that xi xi1 = gap P1 . Since P1 is a refinement of P2 , so there exists yj1 and yj such that
HOMEWORK #6 SOLUTIONS
Solution to Problem 8.5.2
A quick calculation shows that
Then (1 + x)1 =
P
k=0
1
k
1 k
k x
=
= (1)k
k
k=0 (x)
P
= (by the geometric series formula)
1
1+x
Solution to Problem 8.5.4. By the extreme value theorem, there is xm , xM [a, b
HOMEWORK #9 SOLUTIONS
Solution to Problem 9.2.1. If |x| < 1, then limn |x|n = 0, so
f (x) = lim fn (x) =
n
10
= 1.
1+0
If |x| > 1, then |x|n as n , so
1
|x|n
n 1n
|x|
f (x) = lim fn (x) = lim
n
1
+1
=
01
= 1.
0+1
For |x| = 1, fn (1) = 0 for all n, so f (1
HOMEWORK #4 SOLUTIONS
Solution to Problem 6.6.1. a. By the product rule and the second FTC, we have that
Z x
Z x
3
Z x
d
d
x
5x4
2 2
2
2
2 2
2
.
x t dt =
x
+ x4 =
t dt + x (x ) = 2x
t dt = 2x
dx
dx
3
3
0
0
0
b.
d
dx
Z
ex
ln t dt
1
= ln(ex )
d x
(e ) = x
HOMEWORK #8 SOLUTIONS
Solution to Problem 9.1.1. a) We use the ratio test:
ak+1
(k+1)p
lim ak
k
kp
= lim a (
k
k p
) = a.
k+1
Thus if a < 1, the series converges, regardless of the values of p; if a > 1, the series diverges, regardless of the values
of p.
HOMEWORK #1 SOLUTIONS
Solution to Problem 6.1.3.
Let P = cfw_x0 , .x1 , ., xn be a partition of [a, b]. Given any partition interval [xi1 , xi ], there is a rational number
q [xi1 , xi ] (here we use that Q is dense in R), and we know that f (q) = 0. Thu
HOMEWORK #5 SOLUTIONS
Solution to Problem 8.2.1
(1 + x)1/2 1 + 12 x 81 x2 +
going to zero for small x near 0)
1 3
16 x
+ , by the Lagrange Remainder Theorem (note the remainders are clearly
1 + 21 x 18 x2 < (1 + x)1/2 < 1 + 12 x
Letting x = 1 gives 1.375
HOMEWORK #10 SOLUTIONS
Solution to Problem 9.5.1. We will apply the ratio test to all series.
a. The ratio is
kx x
= <1
lim
k 5(k + 1)
5
P
Thus, the series converges when |x| < 5 and diverges when |x| > 5. For x = 5, we have
k=1
P
(1)k
the p-test. For
Homework #7 Solutions
June 2, 2016
Solution to Problem 8.7.4
One argument to show this: Any polynomial defined on (a,b) can be extended
continuously to the closed interval [a,b], hence is bounded on [a,b], hence can
1
, which is unbounded on (a,b)
not uni
HOMEWORK #3 SOLUTIONS
Solution to Problem 6.4.3. By the extreme value theorem, there is xm , xM [a, b] such that f (xm ) f (x)
f (xM ), x [a, b]. From this we obtain
Z b
Z b
Z b
f (xM ) = f (xM )(b a).
f (x)
f (xm )
f (xm )(b a) =
Since
Rb
a
a
a
a
f (x
Math 142B
Summer 2011 Midterm Exam 1 Solution
1. Let f : [0, 1] R be dened by
f (x) =
1
x if x = n for some n N,
0 otherwise.
Show the f is integrable on [0, 1] and determine the value of
1
0
f.
1
1
Given a natural number n, f (x) = 0 for each x in ( k ,
Math 142B
Midterm Exam 2 Solution
1. Let f : R R be continuous. Dene
x
(x t) f (t) dt for all x.
G(x) =
0
Use the Second Fundamental Theorem to show that G (x) = f (x) for all x. (Hint:
Use the linearity property of the integral to rewrite it in a more co
Math 142B
Midterm Exam 2
August 25, 2011
Instructions
1. You may use any type of calculator, but no other electronic devices during this exam.
2. You may use one page of notes, but no books or other assistance during this exam.
3. Write your Name, PID, an
Math 142B
Midterm Exam 1
August 11, 2011
Instructions
1. You may use any type of calculator, but no other electronic devices during this exam.
2. You may use one page of notes, but no books or other assistance during this exam.
3. Write your Name, PID, an
MATH 142B
Homework 3 Solutions
Jonathan Conder
Section 6.4
1. (a) This is false. For example f could be nonzero at a single point.
(b) This is false, with the same counterexample as part (a).
(c) This is true, by monotonicity.
(d) This is false. For examp