MATH 314
Assignment #4
1. (a) Let an := 2(1)n+1 + (1)n(n+1)/2 for n IN. Find four subsequences of
(an )n=1,2,. such that they converge to dierent limits.
Solution. We have
a4k+1 = 1,
a4k+2 = 3,
a4k+3 = 3,
a4k+4 = 1,
k IN.
Thus (a4k+1 )k=1,2,. , (a4k+2 )k=
MATH 314
Assignment #4
1. (a) Let an := 2(1)n+1 + (1)n(n+1)/2 for n IN. Find four subsequences of
(an )n=1,2,. such that they converge to dierent limits.
Solution. We have
a4k+1 = 1,
a4k+2 = 3,
a4k+3 = 3,
a4k+4 = 1,
k IN.
Thus (a4k+1 )k=1,2,. , (a4k+2 )k=
MATH 314
Assignment #7
1. Let f (x) := x2 for x 0 and f (x) := 0 for x < 0.
(a) Use the denition of derivative to show that f is dierentiable at 0.
Proof . We have
lim+
x0
f (x) f (0)
x2
= lim+
= lim+ x = 0
x0
x
x0
x0
and
lim
x0
f (x) f (0)
= 0.
x0
This s
MATH 314
Assignment #6
1. Let f be a continuous function from IR to IR such that limx f (x) = , that is,
for any real number M , there exists a positive real number K such that f (x) > M
whenever x K.
(a) Fix a point x0 IR. Prove that there exists a p
MATH 314
1. (a) Prove that limn
Assignment #3
n = and limn
1
n
= 0.
Proof . For M > 0, let N = M 2 + 1. Then n > N implies
limn n = . By Theorem 1.3, it follows that limn
n > M 2 = M . Hence,
1
n
= 0.
(b) Prove that if limn an = a, then limn an  = a.
MATH 314
Assignment #2
1. (a) Show that for any integer n, 3n if and only if 3n2 .
Proof . Let n be an integer. By the division algorithm, n = 3k + r, where k is an
integer and r = 0, 1, 2. If 3n, then r = 0 and n = 3k. It follows that n2 = 9k 2 . So
3
MATH 314
Assignment #1
1. Let A, B, C, and X be sets. Prove the following statements:
(a) A (B C) = (A B) (A C).
Proof . Suppose x A (B C). Then x A or x B C. If x A, then x belongs
to both A B and A C; hence, x (A B) (A C). If x B C, then x B
and x C; he
The Final Exam (Math 314 A1)
December 20, 2010
Name:
I.D.#:
1. (10 points) Let p(x) := x3 + x 100, x IR.
(a) Prove that the function p is strictly increasing on the real line.
Proof . We have
p (x) = 3x2 + 1 > 0 x IR.
Hence p is strictly increasing on the
MATH 314
Assignment #9
1. Let f be an increasing function on [a, b] with < a < b < .
(a) Let P = cfw_t0 , t1 , . . . , tn be a partition of [a, b]. Prove
n
[f (ti ) f (ti1 )](ti ti1 ).
U (f, P ) L(f, P )
i=1
Proof . Let mi := infcfw_f (x) : ti1 x ti an
MATH 314
Assignment #10
1. Calculate the following integrals.
1
(a)
e x dx.
0
Solution. Use change of variable: x = t2 for 0 t 1. When t = 0, x = 0, and when
t = 1, x = 1. The substitution together with integration by parts gives
1
e
x
1
t
t
e (2t) dt
MATH 314
Assignment #8
1. For each of the following functions, determine the interval(s) where the function is
increasing or decreasing, and nd all maxima and minima.
(a) f (x) := 4x x4 , x IR.
Solution. We have f (x) = 4(1 x3 ), x IR. Clearly, f (x) > 0
MATH 314
Assignment #10
1. Calculate the following integrals.
1
(a)
e x dx.
0
Solution. Use change of variable: x = t2 for 0 t 1. When t = 0, x = 0, and when
t = 1, x = 1. The substitution together with integration by parts gives
1
e
x
1
dx =
0
0
e (2t
MATH 314
1. (a) Prove that limn
Assignment #3
n = and limn
1
n
= 0.
Proof . For M > 0, let N = M 2 + 1. Then n > N implies
limn n = . By Theorem 1.3, it follows that limn
n > M 2 = M . Hence,
1
n
= 0.
(b) Prove that if limn an = a, then limn an  = a.
MATH 314
Assignment #6
1. Let f be a continuous function from IR to IR such that limx f (x) = , that is,
for any real number M , there exists a positive real number K such that f (x) > M
whenever x K.
(a) Fix a point x0 IR. Prove that there exists a p
MATH 314
Assignment #8
1. For each of the following functions, determine the interval(s) where the function is
increasing or decreasing, and nd all maxima and minima.
(a) f (x) := 4x x4 , x IR.
Solution. We have f (x) = 4(1 x3 ), x IR. Clearly, f (x) > 0
MATH 314
Assignment #9
1. Let f be an increasing function on [a, b] with < a < b < .
(a) Let P = cfw_t0 , t1 , . . . , tn be a partition of [a, b]. Prove
n
[f (ti ) f (ti1 )](ti ti1 ).
U (f, P ) L(f, P )
i=1
Proof . Let mi := infcfw_f (x) : ti1 x ti an
MATH 314
Assignment #7
1. Let f (x) := x2 for x 0 and f (x) := 0 for x < 0.
(a) Use the denition of derivative to show that f is dierentiable at 0.
Proof . We have
lim+
x0
f (x) f (0)
x2
= lim+
= lim+ x = 0
x0
x
x0
x0
and
lim
x0
f (x) f (0)
= 0.
x0
This s
Math 314/414
ANALYSIS I and II
Dr. Rene Poliquin
<rene.poliquin@ualberta.ca>
http:/www.math.ualberta.ca/rpoliqui/Courses/314/index.html
http:/www.math.ualberta.ca/rpoliqui/Courses/414/index.html
April 2008
CONTENTS
Math 314/414
Contents
1 Real Numbers
1.1
Chapter 4. Dierentiation
1. Basic Properties of the Derivative
Let f be a realvalued function dened on an interval I in IR. The derivative of f
at a point a I is dened to be
lim
xa
f (x) f (a)
,
xa
if this limit exists as a real number. The derivative f
Chapter 5. Integration
1. The Riemann Integral
Let a and b be two real numbers with a < b. Then [a, b] is a closed and bounded
interval in IR. By a partition P of [a, b] we mean a nite ordered set cfw_t0 , t1 , . . . , tn such
that
a = t0 < t1 < < tn = b
Math 314 Fall 2013 Homework 4 Solutions
Due Wednesday Oct. 9 5pm in Assignment Box (CAB 3rd Floor)
There are 6 problems, each 5 points. Total 30 points.
Please justify all your answers through proof or counterexample.
You can use any theorem/lemma/proposi

4. Differentiation
4.1. Derivatives.
4.1.1. Denition.
Denition 4.1. Let f be a real function. At a point x0 inside its domain, if the limit
x
lim
x0
f (x) f (x0)
x x0
(4.1)
exists, we say f is dierentiable at x0 , and call the limit its derivative at x0
Math 314 Fall 2013 Homework 7 Solutions
Due Wednesday Nov. 6 5pm in Assignment Box (CAB 3rd Floor)
There are 6 problems, each 5 points. Total 30 points.
Please justify all your answers through proof or counterexample.
1
Question 1. Let g(x) be continuous
Math 314 Fall 2013 Homework 8 Solutions
Due Wednesday Nov. 13 5pm in Assignment Box (CAB 3rd Floor)
There are 6 problems, each 5 points. Total 30 points.
Please justify all your answers through proof or counterexample.
Question 1. Let f (x) = exp [x ln x]
I
The Final Exam (Math 314 A"1)
December 14, 2015
I.D.ff:
Name:
1. (10 points) (a) Prove that a +brt is an irrational number for all rational numbers a
and b with b * O. (lt is known that t/2 is an irrational number.)
Prry. /e/ c;: a*/,8 J":'rttr / / o, ,
The Midterm Exam (Math 314 Al")
October 27, 2016
I.D.ff:
Name:
1. (12 points) (a) Use the interval notation to
express the set
cfw_r e IR : l2  3rl > 8.
llot zttl zI cfw_an/*/ t/ tja<g orz4r?,.
[/1or!, zia, <g a Jz<sz q J,2,4 /oe x?#.
y'/oreor
MATH 314
Assignment #4
1. (a) Let an := 2(1)n+1 + (1)n(n+1)/2 for n IN. Find four subsequences of
(an )n=1,2,. such that they converge to dierent limits.
Solution. We have
a4k+1 = 1,
a4k+2 = 3,
a4k+3 = 3,
a4k+4 = 1,
k IN.
Thus (a4k+1 )k=1,2,. , (a4k+2 )k=
MATH 314
Assignment #1
1. Let A, B, C, and X be sets. Prove the following statements:
(a) A (B C) = (A B) (A C).
Proof . Suppose x A (B C). Then x A or x B C. If x A, then x belongs
to both A B and A C; hence, x (A B) (A C). If x B C, then x B
and x C; he