MATH 401, FALL TERM 2015, MODEL
SOLUTIONS TO PRACTICE EXAM 1
Note that the rst exam is on Wednesday 16th September, at 1:25 in
Room 207 Sackett.
1. (i) 6T2 sinx + (211:3 1) cos 2:. (ii) lJx lm11% -
2. The derivative is g = 15.1:2 142: + 3. (a) g5 17:1 = 4
Math 401 Spring 2009 - Homework # 7
Due on March 18, 2009
1. Problems from the textbook:
Problem set 4 (page 348) # 2, 3, 4.
Problem set 5 (page 349) # 2
2. Problem . Using the denition of limit show that
lim (x 3)2 = 0 and lim 1 x2 = 0.
x 3
x 1
3. Proble
Math 401 cfw_ Spring 2009 - Homework # 6
Due on February 25, 2009
Problem 1.
(6 points) Are the following series convergent or divergent?
Justify
your answer.
(i)
1
X
n2 n
2
3n + 1
n=1
1
X
n!)3
n
(21)
(3n)!
n=1
(ii)
(
(iii)
You may use the comparison test
Math 401 Spring 2009 - Homework # 9
Due on April 15, 2009
1. From the textbook: Page 137 # 1 only (i), (ii) and (iv).
2. Let f be a continuous function on R and dene
cos x
F (x) =
f (t) dt for x R.
/2
Compute the derivative of F .
12
3. Consider the parti
Math 401 Spring 2009 - Homework # 9
Due on April 22, 2009
1. From the textbook: Page 147 # 1, only (1)-(iv)
2. Problem Determine the radius of convergence and the interval of convergence
of the following series:
n=0
(n!)3 (x 1)n
(3n)!
3. Problem Let
ak =
MATH 401 - Introduction to Real Analysis
Solutions to Quiz # 3
Problem 1. Let f : R R be the function dened by f (x) = x3 + 2x. Prove
that f has an inverse function f 1 : R R. Calculate the value of the derivative Df 1 (y ) at y = f (1) = 3.
Solution: We
SPRING 2009
MATH 401 - NOTES
Sequences of functions
Pointwise and Uniform Convergence
Previously, we have studied sequences of real numbers. Now we discuss
sequences of real-valued functions. By a sequence cfw_fn of real-valued functions on D , we mean a
MATH 401 - Introduction to Real Analysis
Integration: the Riemann Integral
Let f be a function dened on an interval I = [a, b] and let
P = cfw_ a = y0 < y1 < < yn = b
be a partition of [a, b].
The mesh of P is the maximum length of the intervals [yk1 , y
Math 401 Spring 2009 - Homework # 11
Due on Friday May 1, 2009
1. Find the radius of convergence and the exact interval of convergence of the
following power series:
n=1
n 2n+1
x
4n
and
sin(3n + 1)xn
n=0
2. Problem Show that, for all x (1, 1),
1
(1) (n +
MATH 401 INTRODUCTION TO ANALYSIS,
FALL TERM 2013, PRACTICE FINAL EXAM
Note that the nal exam for the course will be on Wednesday 1st May,
6:50pm - 8:40pm, Room 108 Tyson. The location for the nal exam can be checked at
http:/www.geog.psu.edu/print-campus
MATH 401 INTRODUCTION TO ANALYSIS-I,
SPRING TERM 2013, PROBLEMS 12
Return by Monday 8th April
Throughout dene an = (1 + 1/n)n (n N), bn = (1 1/n)n (n = 2, 3, 4, . . . ) and
n
cn = 1 +
1
m!
m=1
( n N) .
1. (i) Prove that for m N, 2m1 m!.
(ii) Prove that
n
MATH 401 INTRODUCTION TO ANALYSIS-I,
SPRING TERM 2013, PROBLEMS 11
Return by Monday 1st April
1. Prove that for all n N we have
1
1
1
1
+ + + + n.
n
1
2
3
2. Prove, using the denition of a limit, that
n2 + 1
= 1.
n n2 + 3
lim
3. Let c be a xed positive n
Math 401 Spring 2009 - Homework # 8
Due on April 8, 2009
1. Problems from the textbook:
Page 105: # 6
Problem set 6: # 1 and 2 from page 350.
2. Suppose f, g : [a, b] R are continuous functions on [a, b] and dierentiable on
(a, b) with f (a) = f (b) = 0.
Math 401 cfw_ Spring 2009 - Homework # 5
Due on February 18, 2009
(4 points) Let
following sequences:
Problem 1.
bn
Show that fbn g and
Justify your answer.
Problem 2.
=
f ng be a sequence of real numbers.
a
1
and
1 + jan j
f ng are bounded.
cn
Consider t
MATH 401 INTRODUCTION TO ANALYSIS-I,
FALL TERM 2015, SOLUTIONS 8
1. Let A = cfw_a7 : 1 + m + :02 x3 > O. Prove that this set
(i) is nonempty
(ii) and is bounded above.
(iii) Is it bounded below?
(
i) 1+0i0203 = 1 > 0 so 0 E A. (ii) Suppose that as 2 3. Th
MATH 401, FALL TERM 2015,
SOLUTIONS TO PRACTICE EXAM 2
Note that Exam 2 is on Wednesday 14th October in Room 207 Sackett, 1:252:15
1. Suppose that :r is a real number with at > 1. (i) Prove that :c < x3. (ii) Prove
that 1 < x5 < 337. (i) Wea have m < 1,
MATH 401 INTRODUCTION TO ANALYSIS-I,
FALL TERM 2015, SOLUTIONS 5
0 S545! 4'- ) I
1. Find all real values of x such that % < 1. $#- \ $Q
There are three cases. (i) 1 S 3;, (ii) 0 S at < 1, (iii) a: < 0.
(i) 23(8 1) > 0. Thus the inequality holds iff m2 + a
MATH 401 INTRODUCTION TO
ANALYSIS, FALL 2015, SOLUTIONS 7
1. Let a be any element of the open interval (1,2). (i) Show that there is another
b 6 (1,2) with b > (1. (ii) Prove that (1,2) has no maximum.
(i) Letb: 7+2. Sincel<a<2wehavel <a< g+=b Ontheotherh
MATH 401 INTRODUCTION TO ANALYSIS-I, '
FALL TERM 2015, SOLUTIONS 3 .
Summary of order axioms (slightly different from the textbook): There is a relation
< which satises the following axioms. a, b, 0 denote real numbers.
01. Exactly one of a < b7 0. = b,
Math 401 Introduction to Analysis, Fall Term 2015, Solutions 2
1. An open rectangular box is made of very thin sheet metal. Its volume is 400 cm3, its
width is at cm, and its length is 42: cm. Obtain an expression for its depth in terms of :3.
Show that t
MATH 401 INTRODUCTION TO ANALYSIS-I,
FALL TERM 2015, SOLUTIONS 1
1. Differentiate the following with respect to 2:. (i) (2.15 1)(33: + 2). (ii) m% (it: + 1).
3 - - I. .,2+1
(111) a: lnx. (1v)es1nvt. (V 2+1.
m2wx+m+smx1y:nx+i(mngW+%rV?(m)&?m$+x?aw
l 2 _
ex
MATH 401 INTRODUCTION TO
ANALYSIS, FALL 2016, SOLUTIONS 7
1. Suppose that A is bounded above, B A and B is non-empty. Prove that sup B
exists and sup B sup A.
Proof. Let a = sup A. Then for every x B we have x A so that x a.
Hence B is bounded above by a.
MATH 401 INTRODUCTION TO ANALYSIS,
FALL TERM 2013, PROBLEMS 13
Return by Monday 15th April
1. Decide the convergence of the each of the following series, in each case proving
your assertion.
3
(i)
3+2
n
n=1
(n!)2
5n
(iv)
(2n 1)!
n=1
2. Prove that
4
(ii)
3
MATH 401 INTRODUCTION TO ANALYSIS-I,
FALL TERM 2013, PROBLEMS 14
Return by Monday 22nd April
1. Using the denition of limit, prove that
(i) limx0 (1 + x)1/2 = 1 (The identity b1/2 a1/2 = (b a)/(b1/2 + a1/2 ) could
be helpful in this question),
(ii) limx0+
MATH 401 INTRODUCTION TO ANALYSIS-I,
SPRING TERM 2013, PRACTICE EXAM 2
Note that Exam 2 is on Wednesday 27th February in Room 322 Sackett, 11:25-12:05
1. (25 points) Suppose that x is a real number with x > 1.
(i) Prove that x < x3 .
(ii) Prove that 1 < x
MATH 401, Spring 2011
Practice for Mid-Term #2
Exercise 1.
(a) Let (an )nN and (bn )nN be two sequences of real numbers. Write down the
precise meaning of each of the following two statements:
lim an = 401
n
and
lim bn = .
n
(b) Is the following statement
MATH 401, Spring 2011
Practice for Mid-Term #2
Exercise 3. Consider the sequence (xn )nN dened inductively by x1 = 1 and xn+1 =
n 1.
1 + xn for
(a) List the rst ve terms of the sequence. (Calculator allowed and recommended.)
(b) Prove that the sequence is