Math 125B, Winter 2015.
Homework 5 Solutions
9.3.2. (a) In all such problems, write (x, y) = r(a, b), where a2 + b2 = 1. Observe that |a|, |b| 1 and
one of |a|, |b| is at least 1/ 2 > 1/2.
If a and b are xed (that is, the point (x, y) lies on a xed line t
326
Chapter 9 Convergence in R"
of the interval [0. l]. we have by Theorems 8.30 and 9.30 that E is connected.
On the other hand. since E C R by the deﬁnition of R. it is easy to check that
Ur) : U H E and V0 2: V H E are nonempty sets which are relativel
HOMEWORK #3
Solutions
1. For each of the following, find all values of p R for which f is improperly
integrable on I:
(a) f (x) = 1/xp , I = (1, ),
(b) f (x) = 1/xp , I = (0, 1),
(c) f (x) = 1/(x logp x), I = (e, ).
Z d
d1p
c1p
, if p 6= 1, and
Solution.
Math 125B, Winter 2015.
Homework 8 Solutions
11.4.2(b). By the Chain Rule, D(f g)(a) = Df (g(a) Dg(a), and the determinant of the product of
two square matrices is the product of determinants, thus det D(f g)(a) = det Df (g(a) det Dg(a)
11.4.4. By the Cha
414 Chapter 11 Differentiability on R"
The Chain Rule can be used to Compute individual partial derivatives with-
: I, out writing out the entire matrices Df and Dg. For example, suppose that
i 3" f(u1, . . . , um) is differentiable from R’" to R, that g(
for improper inte-
reover. since M (x)
:rs A, B such that
lave
lf(.\‘, y)l dx
.1‘) dx < 8.
E
11.4.
“S, that c < d are
s 1f
is,
4. B such that
i
um.WW«J.A c
Section 11.1 Partial Derivatives and Partial Integrals 391
for all y e [c, d] which satisfy ly
both exist
at any such
I:
itives of f.
eﬁnition of
we crashed.
o compute
rmula:
cannot be
r
l6.
-_
Section 11.2 The Definition of Differentiability 401
Proof. If (x. v) # (0. 0), then we can use the one—dimensional Product Rule
to verify that both f.- and
Math 125B, Winter 2015.
Homework 6 Solutions
11.1.2. (a) For (x, y) = (0, 0),
fx (x, y) =
2x(x4 + 2x2 y 2 y 4 )
,
(x2 + y 2 )2
while f (x, 0) = x2 implies fx (0, 0) = 0. If (x, y) = r(a, b) with a2 + b2 = 1, we have
|fx (x, y)| = r |2a(a4 + 2a2 b2 b4 )| r
Math 125B, Winter 2015.
Homework 4 Solutions
5.1.0. (c) Yes. First, f is Riemann integrable on any closed interval on which f is integrable, by
composition with continuous function. Then, for any number y 0, y y + 1 (when y 1, this is
trivial, and when y
“Aﬁks ‘ ‘
the. 11 .’ L‘M
160 Chapter 5
Integrability on R
EXERCISES
5.3.0. Suppose that a < b. Decide which of the following statements are true
and which are false. Prove the true ones and give counterexamples for
the false ones.
a) If f is continuou
W
en. .3. ,.- s a“
168 Chapter 5
Integrability on R
To show that sin x /x is not absolutely integrable on [1, oo), notice that
f’mlsinxld >i/k” |sinx|d
x _ x
1 x k22 (k—l)7r x
II
for each n 6 Nn _>_ 2. Since
k+l 1
nl n 11—1-11
de=/2 de=log(n+l)—10g2-
150 Chapter 5 lntegrability on R
EXERCISES
5.2.0. Suppose that
and which ar
the false ones.
a) If f and g are Riemann integrable on [(1. b]
integrable on [(1, b]. »
b) If f is Riemann integrable on [(1, b] and P is any polynomial on R,
then P o f is Riema
Math 125B, Winter 2015.
Homework 3 Solutions
5.3.0. (a) Yes. By Fundamental Theorem of Calculus and the chain rule: F (x) = f (g(x)g (x). Both
factors are nonnegative: f 0 by the assumption, and g 0 as g is increasing.
In fact, we only need that f 0, f is
Math 125B, Winter 2015.
Feb. 4, 2015. ‘
MIDTERM EXAM 1
1< e ‘{
NAME(print in CAPITAL letters, ﬁrst name ﬁrst): _ _
NAME (sign): _ _.
ID#: _ _
Instructions: Each of the 5 problems has equal worth. Read each question carefully and answer it
in the space pro
Math 125B, Winter 2015.
Mar. 4, 2015.
MIDTERM EXAM 2
NAME(sign): _ _
ID#: _ _
Instructions: Each of the 4 problems has equal worth. Read each question carefully and answer it
in the space provided. You must show all your work for full credit. Carefully pr
PRACTICE MIDTERM #1
Solutions.
1. Let f : [a, b] R be a continuous monotonically increasing function, and f (a) =
c, f (b) = d. Let g: [c, d] [a, b] be the function inverse to f (that is, if f (x) = y, then
g(y) = x). Prove that
Z d
Z b
g(x) dx = bd ac.
f
HOMEWORK #2
Solutions
1. Let f be integrable on [a, b], and let x0 , x1 , x2 , . . . be a sequence of points in [a, b]
Z xn
X
such that a = x0 < x1 < x2 < . . . and lim xn = b. Prove that the series
f (t) dt
n
Z
n=1
xn1
b
converges and that its sum equal
MIDTERM #1
Solutions
Problem 1. Let f : [a, b] R be a continuous function. Suppose that for some
Z b
f (x) dx. Prove that f is constant.
partition P of [a, b], U (f, P ) =
a
Solution. Let P = cfw_a = x0 < x1 < . . . < xn = b. Obviously,
Z
b
f (x)dx =
a
n
HOMEWORK #1
Solutions
5.1.3. Let E = cfw_1/n : n N. Prove that the function
f (x) =
n
1, if x E,
0 otherwise
is integrable on [0, 1]. What is the value of
R1
0
f (x) dx?
Solution. It is obvious that for every partition P of [0, 1], L(f, P ) = 0 (indeed, a
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Real Analysis
Math 125B, Spring 2013
Midterm II: Solutions
1. Dene curves : R R3 , : R R2 and a function f : R3 R2 by
(t) = (t, t2 , t3 ),
(t) = (t5 , t7 ),
f (x, y, z) = (x3 y, y 2z).
(a) Compute the dierential matrices [d(1)] and [d(1)].
(b) Compute the
Math 125B, Winter 2015.
Homework 2 Solutions
5.1.9. As g(x) = x is a continuous function on [c, d] (which is true as soon as c 0, so you do not
need the strict inequality c > 0), and f is Riemann integrable, the composite function g f = f is
Riemann integ
Math 125B, Winter 2015.
Homework 1 Solutions
0
1
5.1.0. (a) No. For example f : [0, 1] R, given by f (x) =
x [0, 1)
, is integrable (with integral
x=1
0, as we showed in the lecture) but is not continuous.
1 x Q
, has U (f ) = 1 and L(f ) = 1, as
1
xQ
/
i
—-—————’—
138 Chapter 5 Integrability on R
to extend the integral to the case a > b. In particular, if f (x) is integrable
and nonpositive on [a, b], then the area of the region bounded by the curves
y = f(x), y = 0, x = a, andx = b is given by fb“ f(x)dx