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
6 par sol9.pdf Openwith v
MATH 314 Assignment #9
1. Let f be the function on [0,13 given by
_ 1 ifze[0,1]nQ,
f(I)'cfw_ ifxe[0.1]\Q.
(a) Suppose that- 0 g c C. d 31. Find supcfw_f(.7:) :I E [c.d] and i
MATH 314
Assignment #8
due on Wednesday, November 23, 2016
1. For each of the following functions, determine the interval(s) where the function is
increasing or decreasing, and nd all maxima and minim
MATH 314
Assignment #3
due on Friday, September 30, 2016
1. (a) Prove that limn
n = and limn
1
n
= 0.
(b) Prove that if limn an = a, then limn |an | = |a|. Is the converse true?
Justify your answer.
2
MATH 314
Assignment #7
due on Wednesday, November 16, 2016
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.
(b) Find an expl
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
U (f, P ) L(f, P )
[f (ti ) f (ti1 )](ti t
MATH 314
Assignment #4
due on Friday, October 7, 2016
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.
(b) Let bn := [
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
MATH 314
Assignment #2
due on Wednesday, September 21, 2016
1. (a) Prove that there is no rational number r such that r2 = 3.
(b) Prove that a + b 2 is an irrational number for all rational numbers a
MATH 314
Assignment #10
1. Calculate the following integrals.
1
(a)
e x dx.
0
1/e
3
x2 cos x dx.
(c)
(d)
arctan x dx.
0
1
2. Find the following integrals.
2
(a)
x3 4 x2 dx.
ln x dx.
e
(b)
0
x2 (1
MATH 314
Assignment #1
due on Wednesday, September 14, 2016
1. Let A, B, C, and X be sets. Prove the following statements:
(a) A (B C) = (A B) (A C).
(b) X \ (A B) = (X \ A) (X \ B).
2. Use the princi
MATH 314
Assignment #6
due on Friday, October 21, 2016
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
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.
Soluti
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
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,
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+
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
PRACTICE PROBLEMS
Ordinary annuity: A series of equal payments or receipts occurring over a specified number of
periods with the payments or receipts occurring at the end of each period.
Annuity due:
Nominal and Real GDP
1000000
900000
800000
700000
600000
Nominal
Real (92)
500000
400000
300000
200000
100000
0
1988
1989
1990
1991
1992
1993
year
1994
1995
1996
1997
1998
Graphing Example - Canadian
6
Fur sallpdf
Open with v
1. Let f(:) := : 2 0 and fs :=U for 2: <5. 0.
[5.) Use the denition of derivative to show that f is differentiable at 0.
Proof. We have
_ 2
m HI) rm) = m I_ = 1m Fa
340+ x
6
Fur noi3.pdf
Open with v
lin:|.l_,no = 00. By Theorem 1.3, it follows that limnnm 71; = 0.
(h) Prove that if lin'i.,_.r_.,o :1.1 = a, then link-,4; Ian] = |a|. Is the converse true?
Justify your
6 Fur 30'8.pdf
[2.1 pen wit h v
i r swarm
increasing or decreasing, and nd all maxima and minirna.
(a) HI) := 4m :54, I E R.
Solution. We have f'(a: = 4(1 3:3), I E IR. Clearly, f'(.7:) > U for
Open with v
_1 .,. - ii 1 [iii'
Solution. We have
an = 1: G4k+2 = 3: 11mm = 3! G4k+4 = _1! k E -
sequences that converge to different limits.
(h) Let b := [1 + (1)]n + 1003(11 for n E IN. Find an incr
6 par so|6.pdf
. Iii-"1:14:39-
for any real number M, there exists a positive real number K such that x) .> M
whenever |z| 2 K.
[5.) Fix a point In E B. Prove that there exists a positive real number
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
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
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.
University of Alberta
ECONOMICS 299: Quantitative Methods in Economics
Fall 2015
Instructor: Junaid Jahangir
Office: T 8-26
LEC A1: TR 9:30 10:50
T 1 93
LAB D1: T 3:00 3:50 T B39
LAB D2: W 3:00 3:50 T