Math 125A, Summer 2016
Homework 4
Due Date: Monday, July 18, 2015
Problem 1: Determine whether the statement is true or false. If the statement is true, then
prove it. Otherwise, provide a counterexam
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 (
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 (
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
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
Co
Q7. )MFW K6452. W 51 Cuba? Sfdx
Xé-[chéﬂ ad (“WCHSK 7M3 f9 mf
pwbk SMce, {A16)6Wa5\s KIM/WM}? WW5)
NI”. #S.
(9». NW W 40602 Sam 97D (MW/ks 9f (He/Mg
Cmvw‘S‘h? Fm f5 (ii—Y" M F7 5 B . So
:3 (aué) C F
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
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)
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.
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 everywh
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
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 (y
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
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 = cf
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
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
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 i
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
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
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
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
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.
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