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
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, 3|n if and only if 3|n2 .
Proof . Let n be an integer. By the division algorithm, n = 3k + r, where k is an
integer and r = 0, 1, 2. If 3|n, 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
Chapter 5. Integration
1. The Riemann Integral
Let [a, b] (a < b) be a closed and bounded interval in IR. By a partition P of [a, b]
we mean a finite ordered set cfw_t0 , t1 , . . . , tn such that
a = t0 < t1 < < tn = b.
The mesh of P is mesh(P ) := maxc
Chapter 1. Sets and Numbers
1. Sets
A set is considered to be a collection of objects (elements). If A is a set and x is an
element of the set A, we say x is a member of A or x belongs to A, and we write x A.
If x does not belong to A, we write x
/ A. A
Chapter 4. Dierentiation
1. Basic Properties of the Derivative
Let f be a real-valued 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
provided this limit exists. The derivative f at a is den
Chapter 2. Sequences
1. Limits of Sequences
By a mapping f from a set X to a set Y we mean a specific rule that assigns to each
element x of X a unique element y of Y . The element y is called the image of x under f
and is denoted by f (x). The set X is c
Chapter 3. Continuous Functions
1. Limits of Functions
Let E be a subset of IR and c a point of IR. We say that c is a limit point of E if there
exists a sequence (xn )n=1,2,. in E such that xn = c for all n IN and limn xn = c.
The set of all limit points
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.
(b) Show that p has one and only one real root.
1
2. (15 points) Let f be a rea
The Midterm Exam (Math 314 A1)
October 27, 2011
Name:
I.D.#:
1. (16 points)
(a) Let a, b, c, and d be positive real numbers. Prove that
c
a
<
b
d
implies
a+c
c
< .
b+d
d
(b) Use the interval notation to express the set cfw_x IR : |2x + 1| 5.
1
2. (16 poin
MATH 314
Assignment #6
1. Let f be a continuous function from IR to IR such that lim|x| 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 #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 #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
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 lim|x| 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
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
t
t
e (2t) dt
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
Chapter 1. Sets and Numbers
1. Sets
A set is considered to be a collection of objects (elements). If A is a set and x is an
element of the set A, we say x is a member of A or x belongs to A, and we write x A.
If x does not belong to A, we write x
/ A. A