29.2) Let f (x) = cos x which is continuous and dierentiable on R from known facts. Consider x, y R.
By the Mean Value Theorem, there exists a c between x and y such that
f (x) f (y)
cos x cos y
= f (c)
= sin c
xy
xy
Taking the absolute values of both si
MAT 125A
HW #4
(1)
(2)
(3)
(4)
(5)
Problem 23.4 from Ross book
Problem 21.6 from Ross book
Explain why there are no continuous functions mapping [0, 1] onto (0, 1) or R.
Problem 24.1 from Ross book
Consider g C(S) where S R. Let F (t) = tf + (1 t)g. Show
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MAT 125A
HW #2
(1) Prove that f(x) = m3 is continuous Vac E R by verifying the e — 6 property.
(2) Let f E C(a,b ) Show that if f(7‘) : 0 for each rational 7" 6 (a,b) then f(a:) = O
for all real cc 6 (1,1)
(3) The postage-stamp function P is deﬁned by P(:
MAT 125A
HW #1
1) Prove that 12 + 22 + - - - + n2 = énm +1)(2n + 1) for all n E N
2) Prove that 11" — 4“ is divisible by 7 when n is positive ineger
3) Show that (5 + Viv/3 is not a rational number
4) Show that the following irrational—looking expressions
MAT 125A
HW #6
(1) Problem 31.2 from Ross
(2) Compute Taylor expansion of
(0.1)
f (x) = exp(sin log(1 + 3x)
around x = 0 up to third order in x.
(3) Compute Taylor series of
4 + x3
.
3 x3
near x = 0. For which x does the series converge to f (x)?
(4) Find
MAT 125A
HW #8
(1)
(2)
(3)
(4)
Problem 33.3 from Ross (3 points)
Problem 33.5 from Ross (3 points)
Problem 33.8 from Ross (3 points)
Prove that if f : [a, b] R is a continuous function and not everywhere zero, then
Rb
f (x)2 dx > 0. Using that, prove that
MAT 125A
HW #2
(1) Prove that f (x) = x3 is continuous x R by verifying the property.
(2) Let f C(a, b). Show that if f (r) = 0 for each rational r (a, b) then f (x) = 0
for all real x (a, b)
(3) The postage-stamp function P is defined by P (x) = A for 0
MAT 125A
TRIAL MIDTERM
(1) Let h(x) = x for rational numbers x and h(x) = 0 for irrational numbers. Show
that h is continuous at x = 0 and at no other point.
(2) Suppose f is continuous on [0, 2] and f (0) = f (2). Prove there exist x, y [0, 2]
such that
MAT 125A
HW #3
(1) Problem 13.3 from Ross textbook
(2) Show that the interior of each of the following sets is the empty set:
a) Q set or rational numbers
b) Cantor set defined in the lecture (see Example 5 of Paragraph 13 in the book)
(3) Show that every
MAT 125A
HW #5
(1) Problem 25.10 from the book (2 points)
(2) Prove that (5 points)
X
1
7
(0.1)
<
2
k
4
k=1
(3) Given nonzero
real numbers x1, x2, . . . define a notion of convergence for the infinite
Q
x
product
k=1 k (2 points). Prove that if the produ
Math 125A, Summer 2016
Homework 1
Due Date: Monday, June 27
Problem 1: Prove, using either Definition 17.1 or Theorem 17.2, that each function is continuous at x = c.
(a) f (x) = x4 and c = 3.
(b) f (x) = x and c = 0.
(
x sin( x1 ), if x 6= 0
(c) f (x) =
26.2) (a) Start with the geometric series
xn =
n=0
1
for |x| < 1.
1x
Taking the derivative to both sides, we obtain
nx(n1) =
n=1
1
for |x| < 1.
(1 x)2
Then, we multiply both sides by x
nxn =
n=1
x
for |x| < 1
(1 x)2
to obtain the desired result.
(b) Notic
24.2) For x [0, ), let the sequence of functions cfw_fn be dened by fn (x) =
(a) For x = 0, we have
x
n.
n
fn (0) = 0 n cfw_fn (0) 0.
For x (0, ), we have
x
= 0.
n
Thus, we choose to dene f (x) = 0 x [0, ) so that cfw_fn f on [0, ).
lim fn (x) = lim
n
n
25.2) Let the sequence of functions cfw_fn be dened by fn (x) =
limit f . Let x [1, 1]. Then we have
lim fn (x) = lim
n
n
xn
on [1, 1]. First we nd the pointwise
n
xn
.
n
Since |x| 1, the following inequality holds
1
xn
1
n.
n
n
n
Clearly, the following
28.2) (a) Let f (x) = x3 . By denition, we have
f (x) f (2)
x3 8
(x 2)(x2 + 2x + 4)
= lim
= lim
= lim x2 + 2x + 4 = 12.
x2
x2 x 2
x2
x2
x2
x2
f (2) := lim
(b) Let g(x) = x + 2. By denition, we have
x + 2 (a 2)
xa
g(x) g(a)
= lim
= lim
= lim 1 = 1.
xa
xa x
19.2) (a) Let
> 0 be given. Notice
|f (x) f (y)| = |3x + 11 (3y + 11)| = 3|x y|.
Then
|f (x) f (y)| < 3|x y| < |x y| < .
3
Choose = . Thus, if
3
|x y| < |f (x) f (y)| <
and f is uniformly continuous on R.
(b) Let
> 0 be given. Notice
|f (x) f (y)| = |x2 y
1
31.2) Let f (x) = sinh x = 2 (ex ex ). Then we have
1
f (x) = (ex + ex )
2
1
f (x) = (ex ex )
2
1
f (x) = (ex + ex )
2
1
f (4) (x) = (ex ex )
2
In general, the nth derivative becomes
f (n) (x) =
1 x
2 (e
1 x
2 (e
ex )
+ ex )
if n is even
if n is odd
Ev
1
20.8) Consider the function f (x) = x sin .
x
For lim f (x), we consider a sequence xn (1, ) where cfw_xn . By denition,
x
sin x1
sin
1
n
= lim
= lim
=1
lim f (x) := lim f (xn ) = lim xn sin
1
n
x
n
n
0
xn
x
n
from a well known trigonometric limit. Si
23.6) (a) Suppose that
an xn has a nite radius of convergence R < and an 0 n. Also, assume
the series converges at x = R, which by denition of convergence means
an Rn < . We want to show
n < . Notice that
an (R)
an (R)n =
(1)n an Rn
is an alternating seri
17.6) Homework assignment 17.5b gives us the following:
Proposition 1 Every polynomial is continuous on R.
Suppose f (x) =
p(x)
q(x)
on the domain cfw_x Rq(x) = 0 where p(x) and q(x) are polynomials. By Proposition
1, both p(x) and q(x) are continuous on
18.2) It would breakdown because the limit of subsequences x0 and y0 does not have to be in the
interval (a, b). Since we would only assume continuity of f on (a, b), we would lose it at the endpoints a
and b. This could cause the function to be unbounded
Math 125A, Summer 2016
Homework 6
Due Date: Wednesday, July 27, 2016
Problem 1: Let (X, d) be a metric space, and each Ui is an open subset of X for i = 1, , k.
Show that the intersection
ki=1 Ui = cfw_x X : x Ui for all i = 1, k
is also an open subset of
Due Date: Monday, July 11, 2016
Math 125A, Summer 2016
Homework 3
Problem 1: Determine the radius of convergence of the series
(
5n , if n is odd
an = 1
, if n is even.
3n
Problem 2: Consider a power series
P
n=0
P
n=0
an xn , where
an xn with radius of c
Math 125A, Summer 2016
Homework 2
Due Date: Tuesday, July 5, 2016
Problem 1: Let f be a uniformly continuous function. Prove that if (xn ), (yn ) are sequences
such that |xn yn | 0, then |f (xn ) f (yn )| 0 also.
Problem 2: Determine if the function is un