Download MAT 425: Introduction to Analysis I, Spring 2013 Review Problems

MAT 425: Introduction to Analysis I, Spring 2013
Review Problems
Set 1A → Quantifiers.
a) Each of the following true statements is in universal-existential form. Write the corresponding statement with the order of quantifiers reversed, and show why it is false.
(i) Every line segment has a midpoint
(ii) Every non-zero rational number has a rational reciprocal.
(iii) Every non-empty subset of the positive integers has a smallest element.
(iv) There is no largest prime.
Set 1B → Quantifiers. Rewrite each of the statements below using the symbols ∃ and ∀.
a) 1 + 2 + 3 + ... + n = n(n + 1)/2.
b) Between any two distinct real numbers there is a rational number.
√
c) If p is a prime number, then p is irrational.
d) Every polynomial of odd degree has a real root.
e) Every integer > 1 is a product of primes.
f ) If m and n are positive integers then m can be written as nq + r where 0 ≤ r < n.
Set 2A → Cardinality.
a) (Strichartz, §1.2.3, Exercise #5.) Let A1 , A2 , A3 , ... be countable sets, and let their Cartesian product A1 ×A2 ×A3 ×... be defined to be the set of all sequences (a1 , a2 , a3 , ...), where ak
is an element of Ak for some k ≥ 1. Prove that the Cartesian product is uncountable. Show
that the same conclusion holds if each of the sets A1 , A2 , A3 , ... has at least two elements.
b) Show that the open interval (0, 1) has the same cardinality as R. [Help: Use one of the
trig functions.]
c) Show that for any a < b, c < d ∈ R, the intervals [a, b] and [c, d] have the same cardinality.
d) Show that Q is countable.
Set 2B → Cardinality. Show that each of the following pairs of sets have the same
cardinality.
a) A = {1, 2, 3, ...} and B = {7, 8, 9, ...}.
–2–
b) A = {1, 2, 3, ...} and B = {1, 10, 100, 1000, 10000, ...}.
c) A = {1, 2, 3, ...} and B = {2, −4, 8, −16, 32, −64, ...}.
d) A is the closed interval [0, 1], B is the closed interval [0, 100].
e) A is the closed interval [0, 1], B is the closed interval [0, 1/100].
f ) A is the closed interval [1, 3], B is the closed interval [4, 9].
Set 3A → Proofs. Prove the following statements.
a) The equality (1 + 2 + 3 + ... + n)2 = 13 + 23 + 33 + ... + n3 holds for any integer n > 0.
√
b) If p is a prime number, then p is an irrational number.
c) For every integer n > 0, 5n + (2)3n−1 + 1 is divisible by 8.
d) Show that ∀a, b ∈ R, |a − b| ≤ |a| + |b|.
e) Show that ∀a, b ∈ R, |a − b| ≥ |a| − |b|.
Set 3B → Proofs. Prove the following statements.
√
a) 2 is an irrational number.
b) Between any two distinct real numbers there is a rational number.
c) Prove that there is always a rational number strictly between any two rational numbers
(i.e. ∀a, b ∈ Q s.t. a < b, ∃q ∈ Q s.t. a < q < b).
d) If p is prime and p|n2 (n is a positive integer), then p|n.
Set 4 → Rational Numbers. Prove the following statements (a, b ∈ Q).
a) |a − b| ≤ |a| + |b|.
b) |a| − |b| ≤ |a − b|.
c) |a| − |b| ≤ |a − b|.
Set 5 → Cauchy Sequences.
a) Let {xk } and {yk } be two equivalent Cauchy sequences. Show that equivalence of Cauchy
sequences respects the properties of
Reflexivity: Show that {xk } ∼ {xk };
Symmetry: Show that {xk } ∼ {yk } if {yk } ∼ {xk };
Transitivity: Show that {xk } ∼ {yk } and {yk } ∼ {zk } ⇒ {xk } ∼ {zk }.
–3–
b) What kinds of real numbers are representable by Cauchy sequences of integers?
*c) Suppose {xk } = x1 , x2 , x3 , ... and {yk } = y1 , y2 , y3 , ... are two sequences of rational
numbers. Define the shuffled sequence to be x1 , y1 , x2 , y2 , x3 , y3 , .... Prove that the shuffled
sequence is Cauchy if and only if {xk } and {yk } are equivalent Cauchy sequences
d) Can a Cauchy sequence of positive rational numbers be equivalent to a Cauchy sequence
of negative rational numbers?
e) Show that the rationals are dense in R by showing that given any real number x and error
1/n, there exists a rational number y such that |x − y| ≤ 1/n.
*f ) Show that there are an infinite number of rational numbers in between any two distinct
real numbers.
g) Give the definition of: A Cauchy sequence; Two equivalent sequences; A convergent
sequence.
h) Which of the following sequences of rational numbers are Cauchy? Which are equivalent
to another?
1) {xk } = (1, 21 , 13 , 14 , ..., k1 , ...);
−1
2) {yk } = (−1, −1
, −1
, −1
, ..., 2k+1
, ...);
3
5
7
, ...).
3) {zk } = ( 23 , 54 , 76 , ..., 2k+1
2k
i) If {xk } and {yk } are two Cauchy sequences of rationals, show that {cxk + dyk } is Cauchy,
where c, d ∈ Q. Hint: Show first that when {xk } is Cauchy, then {cxk } is Cauchy.
j) Is the Cauchy sequence (0.9, 0.99, 0.999, ...) equivalent to (1, 1, 1, ...)? Explain why.
k) Let {xk }, {yk }, and {zk } be Cauchy sequences of rationals. Show that if {xk } ∼ {yk }
and {yk } ∼ {zk }, then {xk } ∼ {zk }.
Set 6 → Limits and Completeness.
a) Let x1 , x2 , x3 , ... be a sequence of real numbers such that |xk | ≤ 1/2k , and set yk =
x1 + x2 + ... + xk . Show that the sequence y1 , y2 , y3 , ... converges.
q p
√
√ p √
√
*b) Prove that if 0 < a < 2, then a < 2a < 2. Prove that the sequence 2, 2 2, 2 2 2, ...
√
√
converges. What is the limit? (Help: if limn→∞ an = l, then limn→∞ 2an = 2l.)
c) Let {xk } and {yk } be two sequences of reals such that xk → x and yk → y. We know
that if xk ≥ yk for all k ≥ m for some m, then x ≥ y. Would that statement still be valid
with strict inequalities? (i.e. is the following statement true: if xk > yk for all k ≥ m for
–4–
some m, then x > y?) Explain why.
*d) We saw that R, and Z were complete sets of numbers, but that Q was not complete.
What about subsets? Is the subset of integers {0, 1} complete? Is the open interval of real
numbers (0, 1) complete? Explain why.
e) What is the Axiom of Archimedes?
f ) Which of the following statements is true?
1) If a sequence of reals is convergent, then it is Cauchy.
2) If a sequence of reals is Cauchy, then it is convergent.
3) A sequence of reals is convergent if and only if it is Cauchy.
4) All of the above
5) None of the above
*g) Which of the following sets of numbers are complete? 1) Z; 2) Q; 3) R.
Set 7A → Theory of limits.
a) Find the inf and the sup (if they exist) of the following sets. Also decide when the sup
and the inf happens to belong to the set.
(i) { n1 : n ∈ N}.
(ii) { n1 : n ∈ Z and n 6= 0}.
(iii) {x : x = 0 or x = 1/n for some n ∈ N}.
√
(iv) {x : 0 ≤ x ≤ 2 and x ∈ Q}.
(v) {x : x2 + x + 1 ≥ 0}.
(vi) {x : x2 + x + 1 < 0}.
(vii) {x : x < 0 and x + 2 + x − 1 < 0}.
(viii) { n1 + (−1)n : n ∈ N}.
b) For the following sequences, give sup, inf (if they exist) and lim, lim.
(i) (2, 1.9, 1.8, 1.7, ..., 2 − (k − 1)/10, ...)
(ii) (1, −1, 1, −1, ..., (−1)k , ...)
(iii) (1/2, −1/3, 1/4, −1/5, ..., (−1)k−1 /(k + 1), ...)
–5–
(iv) (0.6, 0.66, 0.666, ..., 2/3(1 − 1/10k ), ...)
(v) (−1, 2, −3, 4, −5, ..., (−1)k k, ...)
c) Suppose A is a nonempty set that is bounded below. Let −A denote the set of all −x for
x ∈ A. Prove that
(i) −A is nonempty;
(ii) −A is bounded above;
(iii) − sup(−A) = inf(A).
Set 7B → Theory of limits.
a) Give the first four terms for the following sequences:
√
k
(i) { k+1
}.
k+1
(ii) { (−1)k!
}.
k−1
(2x)
(iii) { (2k−1)
5 }.
(iv) { cos(kx)
}.
x2 +k2
b) Find the kth term for the following sequences:
(i) (−1/5, 3/8, −5/11, 7/14, −9/17, ...).
(ii) (1, 0, 1, 0, 1, ...).
(iii) (2/3, 0, 3/4, 0, 4/5, ...).
c) Find the sup, inf (if they exist) and lim, lim.
(i) (−1, 1/3, −1/5, 1/7, ..., (−1)k /(2k − 1), ...).
(ii) (2/3, −3/4, 4/5, −5/6, ..., (−1)k+1 (k + 1)/(k + 2), ...).
(iii) (1, −3, 5, −7, ..., (−1)k−1 (2k − 1), ...).
k
(iv) (1, 4, 1, 16, 1, 36, ..., k 1+(−1) , ...).
Set 8 → Open Sets, Closed Sets.
a) Give an example of subsets A and B of R such that all three of the following conditions
hold:
(i) Neither A nor B is open;
–6–
(ii) A ∩ B = ∅;
(iii) A ∪ B is open.
b) Show that B = {1/k : k = 1, 2, 3, ...} ∪ {0} is not a perfect set, but that B = B.
Set 9 → Compact Sets.
a) If B1 , B2 , ..., Bn is a finite open cover of a compact set A, can the union ∪nk=1 Bk equal A
exactly?
b) Show that the Cantor set is non-empty.
Set 10 → Limits
a) Using the definition of limit seen in class, show that if limx→x0 f (x) = l, then limx→x0 |f (x)| =
|l|.
b) Give an example of a function where limx→0 f (x2 ) exists but limx→0 f (x) does not.
c) Show that limx→0 x sin 1/x = 0.
d) Show that limx→0 x2 sin 1/x = 0.
p
e) Show that limx→0 |x| sin 1/x = 0.
Set 11 → Continuous Functions
a) What is the domain for which the following functions are continuous?
(i) f (x) =
x
x2 −1
(ii) f (x) =
(iii) f (x) =
1+cos x
3+sin x
1
(10+x)1/4
(iv) f (x) =
(v) f (x) =
−1
10 (x−3)2 ,
0,
x 6= 3
x=3
x−|x|
x
b) Using the definition of continuity seen in class, show that f (x) = x is continuous at x0
for any x0 ∈ R.
c) Suppose that f is a function satisfying |f (x)| ≤ |x| for all x. Using the definition of
continuity seen in class, show that f is continuous at 0. (Notice that f (0) must equal 0.)
d) Give an example of a function f such that f is continuous nowhere, but |f | is continuous
everywhere.
–7–
e) Let f (x) = 2x3 − 3x2 + 7x − 10.
(i) What is f (1)? What is f (2)?
(ii) Show that ∃x ∈ [1, 2] such that f (x) = 0.
f ) Let f be a continuous function with domain D = [a, b] (a closed and bounded interval of
real numbers, and thus a compact set). What can be said about f (D)?
Set 12 → Differential Calculus
a) Using the definition of the derivative seen in class, show that if f (x) = 1/x, then f 0 (x0 ) =
−1/x20 , for x0 6= 0.
b) Let f be a function such that |f (x)| ≤ x2 for all x. Prove that f is differentiable at 0.
(Note that we must have f (0) = 0.)
c) Give an example of a function f for which limx→∞ f (x) exists, but limx→∞ f 0 (x) does not
exist.
d) Prove that if
a0
1
+
a1
2
an
+ ... n+1
= 0, then a0 + a1 x + ... + an xn = 0 for some x ∈ [0, 1].
e) Prove that
1
1 √
< 66 − 8 < .
9
8
f ) Suppose h is a function such that h0 (x) = sin2 (sin(x + 1)) and h(0) = 3. Find (h−1 )0 (3).
*g) Find a formula for (f −1 )00 (x).
*h) Show that f is convex on an interval if and only if for all x and y in the interval we have
f (tx + (1 − t)y) < tf (x) + (1 − t)f (y),
for 0 < t < 1.
(This is just a restatement of the definition of convexity seen in class, but a useful one, which
is actually used more often in more advanced situations.)
x
i) 1) Find the Taylor polynomial of degree 3 at x0 = 0 for f (x) = ee . 2) Find the Taylor
polynomial of degree 4 at x0 = 0 for f (x) = x5 + x3 + x.
*j) Suppose that ak and bk are the coefficients in the Taylor polynomials at x0 of f and g,
respectively. In other words, ak = f (k) (x0 )/k! and bk = g (k) (x0 )/k!. Find the coefficients ck
of the Taylor polynomials at x0 of 1) f + g, and 2) f 0 , in terms of the ak ’s and the bk ’s.
k) Let fm (x) = x3 − 3x + m. Show that fm never has 2 roots in [0, 1] no matter what m.
l) Let f (x) = 2x2 − 7x + 10. Let a = 2, b = 5. Verify the Mean Value Theorem.
m) Show that
b−a
1+b2
< arctan b − arctan a <
b−a
1+a2
if a < b.
–8–
n) Show that π/4 + 3/25 < arctan 4/3 < π/4 + 1/6.
o) Let f 0 (x) = 1/x ∀x > 0 and f (1) = 0. Show that f (xy) = f (x) + f (y) ∀x, y > 0.
p) Give the Taylor polynomial of degree n for the following functions: 1) sin x at x0 = 0; 2)
cos x at xo = 0; 3) ex at xo = 0; 4) log x at xo = 1.
Set 13 → Integral Calculus
Rb
a) Prove that 0 x3 dx = b4 /4, by considering partitions into n equal subintervals, using the
P
3
4
3
2
formula N
k=1 k = N /4 + N /2 + N /4.
b) Decide which of the following functions are integrable on [0, 2], and calculate the integral
when you can:
x,
0≤x≤1
(i) f (x) =
x − 2, 1 ≤ x ≤ 2.
(ii) f (x) = x + [x], where [] means “integer part of”.
√
1,
x = a + b 2, where a, b ∈ Q
(iii) f (x) =
0,
elsewhere.
c) Find
Z bZ
a
Rb
d
f (x)g(y)dy dx
c
Rd
in terms of a f and c g. (This is an exercise in notation; it is crucial that you recognize a
constant when it appears.)
d) Prove that
(i) If f is integrable on [a, b] and f (x) ≥ 0 for all x ∈ [a, b], then
Rb
a
f ≥ 0.
(ii) If f and g are integrable on [a, b] and f (x) ≥ g(x) for all x ∈ [a, b], then
Rb
a
f≥
e) Find the derivatives of the following functions.
Z
x3
(i) F (x) =
sin3 tdt.
a
Z
x
y
Z
(ii) F (x) =
15
8
Z
(iii) F (x) =
a
f ) Find (f −1 )0 (0) if f (x) =
Rx
0
1
dt dy.
1 + t2 + sin2 t
b
1+
t2
x
dt.
+ sin2 t
1 + sin(sin t)dt. (Don’t try to evaluate f explicitely.)
Rb
a
g.
–9–
g) Suppose f is integrable on [a, b]. Prove that ∃x ∈ [a, b] such that
Rx
a
=
Rb
x
.