MATH1500X
Caluclus
R. Smirnov
Assignment #5
due 29.10.12
Read the pages 53-56 in the textbook and the notes.
1. Use the language to prove that the following functions are continuous
in their respective domains: (a) ax + b, (b) x, (c) cos x, (d) tan1 x.
2.
MATH1500X
Caluclus
R. Smirnov
Assignment #1
due 21.09.12
Read the pages 1-18 in the textbook and notes.
1. Let x, y, z R and x = 0, moreover xy = xz. Prove that y = z. [Hint:
Use the fact that R is a eld.]
2. Let a, b Q, a = b. Prove that there exists a n
MATH1500X
Caluclus
R. Smirnov
Assignment #2
due 28.09.12
Read the pages 25-37 in the textbook and notes.
1. Use the denition of limit of a sequence to show that limn un = 0,
n
where un = (1) .
n
2. Argue by contradition to prove that the sequence cfw_un ,
MATH1500X
Caluclus
R. Smirnov
Assignment #8
practice, dont hand in
Read the pages 97-103, 108-111 in the textbook and the notes.
1. Solve the equation
x+1
(y 3 + 4)dy =
x
11
.
4
2. Evaluate the area of the region bounded by the parabola y = 2x2 + x + 1,
t
MATH1500X
Caluclus
R. Smirnov
Assignment #7
due 26.11.12
Read the pages 71-88 in the textbook and the notes.
1. 4.75 (a), (c), (d), (e), (g), (j), (n) in the textbook
2. Prove that for any positive real numbers a and b, such that b > a and for
any natural
MATH1500X
Caluclus
R. Smirnov
Assignment #4
due 19.10.12
Read the pages 43-49, 53-56 in the textbook and the notes.
1. Prove that among all of the triangles with the same perimeter, the
equilateral triangle has the greatest area. [Hint: Use Herons formula
MATH1500X
Caluclus
R. Smirnov
Assignment #3
due 10.10.12
Read the pages 27- 37 and 43-45 in the textbook and and the lecture
notes.
1. Evaluate the limit
1
n+2
lim
sin 2
.
n n + 1
n + 4n
2. Prove that
lim q n = 0,
n
where 0 < |q| < 1.
3. Evaluate the lim
MATH1500X
Caluclus
R. Smirnov
Assignment #6
due 14.11.12
Read the pages 71-88 in the textbook and the notes.
1. Use the denition (refer to p. 72 in the textbook) to prove that (a)
d
1
ex ; (b) dx tan x = cos2 x .
d x
dx e
=
2. Use the rule for dierentiati