Mdulo 1
Tema
Divisibilidad
Qu son los numeros primos?
Qu son los numeros
compuestos?
Criba de Erasttenes
Factorizacin prima Mtodo del
rbol
Factorizaci prima Mtodo de
divisores
Nmero Enteros
Orden de operaciones
Enlace
http:/www.educatina.com/matematicas/a
Chapter 1: Functions and Limits
August 13, 2012
()
Chapter 1: Functions and limits
August 13, 2012
1 / 12
Functions
It is common that the values of one variable depend on the values of
another.
Examples:
The area of a circle depends on the radius A = r 2
MA 1505 Mathematics I
Tutorial 6 Solutions
1. (i) As the temperature function is only valid within the hotel room, its domain is
cfw_(x, y ) : 0 x 10, 0 y 10.
(ii) The heater is at the location where the temperature is highest. It is clear that the larges
MA 1505 Mathematics I
Tutorial 5 Solutions
1. Rewrite the function:
1
f (x) = (x + |x|) =
2
0 < x < 0
x 0<x<
The Fourier series of f (x) is given by
a0 +
(an cos nx + bn sin nx).
n=1
a0 =
an =
1
1
2
bn =
1
x dx =
0
.
4
1 x sin nx cos nx
+
n
n2
x cos nx dx
MA 1505 Mathematics I
Tutorial 4 Solutions
1.
Let a = 3i + 2j + k.
Then u = projw a is parallel to w
and v = a projw a is perpendicular to w
and a = u + v.
We compute
u=
and
2.
3+6+4
aw
1
w=
(i + 3j + 4k) = (i + 3j + 4k)
2
w
1 + 9 + 16
2
1
1
v = a u = (3i
MA 1505 Mathematics I
Tutorial 3 Solutions
n
1. (a) Let un = (1)n (x+2) .
n
lim
n
n
un+1
(x + 2)n+1
= lim
= |x + 2|.
n
un
n+1
(x + 2)n
By ratio test, the power series is convergence in |x + 2| < 1.
So the radius of convergence is 1.
(b) Let un =
(3x2)n
.
MA 1505 Mathematics I
Tutorial 2 Solutions
1. (a)
lim
x/2
1
1 sin x
cos x
sin x
= lim
= lim
=.
1 + cos 2x x/2 2 sin 2x x/2 4 cos 2x
4
a sin ax
2
ln(cos ax)
cos ax = lim a sin ax cos bx = a .
(b) lim
= lim
x0 b sin bx
x0 b sin bx cos ax
x0 ln(cos bx)
b2
c
MA 1505 Mathematics I
Tutorial 1 Solutions
1. Note that (g f )(x) =
2. (a) y =
ax + b
,
cx + d
y=
6
|3 x | and (f g )(x) =
6
.
|3x|
ad bc
a(cx + d) c(ax + b)
=
(use quotient rule)
2
(cx + d)
(cx + d)2
(b) y = sinn x cos mx , y = n sinn1 x cos x cos mx m
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA 1505 Mathematics I
Tutorial 11
1. Evaluate S f (x, y, z ) dS and S F dS , where f (x, y, z ) = x + y + z and F = x2 i + y 2 j + z 2 k
and S is the surface dened parametrically by
r(u, v ) = (2u
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA 1505 Mathematics I
Tutorial 10
1. The base of a circular fence with radius 10 m is given by x = 10 cos t, y = 10 sin t. The
height of the fence varies from 3 m to 5 m such that, at position (x,
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA 1505 Mathematics I
Tutorial 9
1. Let F(x, y, z ) = 2xy i + (x2 + 2yz )j + y 2 k. Show that F is a conservative vector eld. Find a
function f such that
f = F.
Ans: f (x, y, z ) = x2 y + y 2 z +
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA 1505 Mathematics I
Tutorial 8
1. Find the volume of the solid whose base is the region in the xy -plane that is bounded by the
parabola y = 4 x2 and the line y = 3x, while the top of the solid
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA 1505 Mathematics I
Tutorial 7
1. Calculate the following iterated integrals:
a
b
(a)
0
0
1
2
(x2 + y 2 ) dxdy
(b)
1
1
2
3 ab(a
Ans: (a)
+
0
(b) 3 3 3/2
b2 )
xy
dxdy .
4 x2
2. Evaluate the follo
Chapter 1: Functions and Limits
August 13, 2012
()
Chapter 1: Functions and limits
August 13, 2012
1 / 10
Functions
It is common that the values of one variable depend on the values of
another.
Examples:
The area of a circle depends on the radius A = r 2
Chapter 2: Dierentiation
August 15, 2012
()
Chapter 2: Dierentiation
August 15, 2012
1 / 50
The derivative
Let f (x ) be a given function. The derivative of f at the point a,
denoted by f (a), is dened to be
f (x ) f (a)
x a
x a
f (a) = lim
(provided that
Chapter 3: Integration
August 22, 2012
()
Chapter 3: Integration
August 22, 2012
1 / 41
The indenite integral
Integration and dierentiation are linked by the Fundamental
Theorem of Calculus (more later). One can think of indenite
integration as an inverse
Chapter 2
Differentiation
Formal Derivative
f(x) is a function.
At a point a (x = a), the derivative of f is
provided the limit exists.
equivalent formulation (set x = a + h)
Notation
Geometrical Meaning
=
Tangent Line
As Q approaches P, secant lines app
Chapter 1
Functions: limits and continuity
Terminology (review)
The symbol y = f(x) denotes the statement y is
a function of x.
A function represents a rule that assigns a
unique value y to each value x.
Input x and output y. Input-output process.
x i
Textbook chapters from Calculus by Thomas, 12th ed.
The chapter numbering in the current edition of the textbook is different to that in earlier editions (on
which the syllabus is based). Here is an updated list of chapter numbers for the 12th edition (th
MATHEMATICS 1505 Mathematics I
National University of Singapore
Semester 1, 2012/13
Graeme Wilkin
Instructor.
E-mail.
Graeme Wilkin, Office S17 05-08, ext 67803
[email protected] (please include MA1505 in the subject line)
Office hours. TBA.
Text. Thomas
MA1505 Lecture Group C: Suggested problems on graphs of multivariable functions
Taken from Thomas, Calculus, 12th ed.
The second half of the course involves the study of multivariable functions. Every question you
encounter will be easier if you have a go
Chapter 9. Line Integrals
9.1
9.1.1
Introduction
Work Done I
(i) Let F be a constant force acting on a particle
in the displacement direction as shown in gure
(i) above. Suppose the distance moved by the
particle is s. The work done is given by
W = F s.
(
Chapter 3: Integration
August 22, 2012
()
Chapter 3: Integration
August 22, 2012
1 / 41
The indenite integral
Integration and dierentiation are linked by the Fundamental
Theorem of Calculus (more later). One can think of indenite
integration as an inverse
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA 1505 Mathematics I
Tutorial 6
1. Imagine you are visiting a country in the winter season. Let T (x, y ) = 36 1 [x2 + (y 5)2 ]
5
be the temperature at location (x, y ) in a 10ft 10ft hotel room
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA 1505 Mathematics I
Tutorial 5
1. Show that the Fourier series for
1
f (x) = (x + |x|), < x < ; f (x + 2 ) = f (x)
2
is given by
(1)n+1
(1)n 1
cos nx +
+
sin nx .
4
n2
n
1
2. Find the Fourier se