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321102 General Mathematics Dr Wattana Toutip 6/1/09 1
321 102 General Mathematics
For the students from Pharmaceutical Faculty 1/2004
Instructor: Dr Wattana Toutip (ดร.วัฒนา เถาว์ทิพย์)
Chapter 1 Analytic geometry in the plane
Overview: The study of motion has been important since ancient times.
Calculus is the mathematical tool to describe it.
Conic sections are the paths traveled by planets, satellites and other
bodies (even electrons).
Figure 1 (Conic sections)
Topics:
1.1 Conic Sections
1.2 Translation of axis
1.3 Rotation of axis
321102 General Mathematics Dr Wattana Toutip 6/1/09 2
1.1 Conic Sections
Circles
Definitions
A circle is the set of points in a plane whose distance from a given fixed
point in the plane is constant. The fixed point is the center of the circle;
the constant distance is the radius.
■ How to draw a circle.
- Compass
- String
■ How to find the equation for a circle.
■ The standard-form equation for the circle of radius a centered at the
origin is
2 2 2x y a
321102 General Mathematics Dr Wattana Toutip 6/1/09 3
Example 1 Find the center and radius of the circle 2 2 9x y . Then
sketch the circle. Include the center and radius in the sketch.
Solution
Parabolas
Definitions
A parabola is the set that consists of all the points in a plane equidistant
from a given fixed point and a given fixed line in the plane. The fixed
point is the focus of the parabola. The fixed line is the directrix.
■ How to draw a parabola.
321102 General Mathematics Dr Wattana Toutip 6/1/09 4
■ How to find the equation for a parabola.
■ The standard-form equations for parabolas with vertices at the origin
and 0p are shown in the table below.
Equation Focus Directrix Axis Opens
2 4x py (0, )p y p y-axis Up 2 4x py (0, )p y p y-axis Down 2 4y px ( ,0)p x p x-axis To the right 2 4y px ( ,0)p x p x-axis To the left
321102 General Mathematics Dr Wattana Toutip 6/1/09 5
Example 2 Find the focus and directrix of the parabola 2 10y x . Then
sketch the parabola. Include the focus and directrix in the sketch.
Solution
Ellipses
Definitions
An ellipse is the set of points in a plane whose distances from two fixed
points in the plane have a constant sum. The two fixed points are the foci
of the ellipse.
■ How to draw an ellipse.
321102 General Mathematics Dr Wattana Toutip 6/1/09 6
■ How to find the equation for an ellipse.
■ The standard-form equations for ellipses centered at the origin with
0a b and 2 2c a b
Equation Foci Vertices Major axis
2 2
2 21
x y
a b ,0c ,0a x-axis
2 2
2 21
x y
b a 0, c 0, a y-axis
321102 General Mathematics Dr Wattana Toutip 6/1/09 7
Example 3 Find the foci and vertices of the ellipse
2 2
116 9
x y . Then
sketch the ellipse. Include the foci and vertices in the sketch.
Solution
Remark: Consider the ellipse
2 2
2 21
x y
a b , with 2 2c a b
1. If 0c (so that )a b then the ellipse will be a circle.
2. If c a (so that 0b ) then the ellipse will be a line segment
Hyperbolas
Definitions
A hyperbola is the set of points in a plane whose distances from two
fixed points in the plane have a constant difference. The two fixed points
are the foci of the hyperbola.
■ How to draw a hyperbola.
321102 General Mathematics Dr Wattana Toutip 6/1/09 8
■ How to find the equation for a hyperbola.
■ How to graph a hyperbola.
■ The standard-form equations for hyperbolas centered at the origin with
0, 0a b and 2 2c a b
Equation Foci Vertices Asymptote
2 2
2 21
x y
a b ,0c ,0a
by x
a
2 2
2 21
y x
a b 0, c 0, a
ay x
b
321102 General Mathematics Dr Wattana Toutip 6/1/09 9
Example 4 Find the foci and asymptotes of the hyperbola
2 2
116 9
x y .
Then sketch the hyperbola. Include the foci , vertices and asymptotes in
the sketch.
Solution
321102 General Mathematics Dr Wattana Toutip 6/1/09 10
■■ How to classify conic sections by Eccentricity. ■■
Eccentricity
Definition
An eccentricity of a conic section is the constant ratio of the distance
between the conic section and the focus to the distance between the conic
section and the directrix.
■ How to classify conic sections by Eccentricity.
Theorem
If e is the eccentricity of a conic section, then the conic section is:
(a) parabola if 1e
(b) ellipse if 1e
(c) hyperbola if 1e
Outline proof (a)
321102 General Mathematics Dr Wattana Toutip 6/1/09 11
Outline proof (b)
321102 General Mathematics Dr Wattana Toutip 6/1/09 12
Outline proof (c)
Remark: In both ellipse and hyperbola, the eccentricity is the ratio of the distance
between the foci and to the distance between the vertices.
c
Eccentricitya
321102 General Mathematics Dr Wattana Toutip 6/1/09 13
Example 5 Find the equation and sketch the graph for an ellipse of the
eccentricity 1
2e whose foci lie at the points (1,0) and ( 1,0) .
Example 6 Find the equation and sketch the graph for a hyperbola of
eccentricity 5
3e whose vertices locate at the points (0,3)and (0, 3) .
321102 General Mathematics Dr Wattana Toutip 6/1/09 14
1.2 Translation of axis
■ How to translate ( or shift) a graph of ( )y f x
Example 1.2.1 Sketch the graphs of the following equations:
2y x
2 1y x
2 2y x
2( 1)y x
2( 2)y x
321102 General Mathematics Dr Wattana Toutip 6/1/09 15
■ Rules for translating of axis
On the system of rectangular coordinate XY with the origin O, we can
construct a new system X’Y
’ with the origin O
’ as in the figure.
The Relations between ( , )x y and ( , )x y are the following:
x x h
y y k
(1.1)
or equivalently,
x x h
y y k
(1.2)
■ Applying equation (1.2) we obtain some conclusions as the following:
- To shift the graph of ( )y f x straight up k unit, we add –k to y
- To shift the graph of ( )y f x down k unit, we add k to y
- To shift the graph of ( )y f x to the right k unit, we add –k to x
- To shift the graph of ( )y f x to the left k unit, we add k to x
Example 1.2.2 Change the equation 2y x in order to shift its graph
straight up 4 units. The sketch then graph.
(h, k)
O X
X’
Y Y’
O’ k
h
321102 General Mathematics Dr Wattana Toutip 6/1/09 16
Example 1.2.3 Change the equation 2 2 4x y in order to shift its
graph down 3 units. Then sketch the graph.
Consequently, we obtain the standard-form equations for
conic sections as the followings:
■ The standard-form equation for the circle of radius a centered at the
point ( , )h k is
2 2 2( ) ( )x h y k a
■ The standard-form equations for parabolas with vertices at the point
( , )h k and 0p are shown in the table below.
Equation Focus Directrix Axis
2( ) 4 ( )x h p y k ( , )h k p y k p x h 2( ) 4 ( )x h p y k ( , )h k p y k p x h 2( ) 4 ( )y k p x h ( , )h p k x h p y k 2( ) 4 ( )y k p x h ( , )h p k x h p y k
321102 General Mathematics Dr Wattana Toutip 6/1/09 17
■ The standard-form equations for ellipses centered at the point ( , )h k
with 0a b and 2 2c a b
Equation Foci Vertices Major axis
2 2
2 2
( ) ( )1
x h y k
a b
,h c k ,h a k y k
2 2
2 2
( ) ( )1
x h y k
b a
,h k c ,h k a x h
■ The standard-form equations for hyperbolas centered at the point
( , )h k with 0, 0a b and 2 2c a b
Equation Foci Vertices
2 2
2 2
( ) ( )1
x h y k
a b
,h c k ,h a k
2 2
2 2
( ) ( )1
y k x h
a b
,h k c ,h k a
321102 General Mathematics Dr Wattana Toutip 6/1/09 18
Example 1.2.4 Find the center and radius of the circle 2 2 4 6 3 0x y x y . Then sketch the graph.
Example 1.2.5 Find the focus and the vertex of the parabola 2 6 8 25 0x x y . Then sketch the graph.
321102 General Mathematics Dr Wattana Toutip 6/1/09 19
Example 1.2.6 Find the standard form of the conic section 2 24 9 8 36 4 0x y x y . Then sketch the graph.
Example 1.2.7 Find the standard form of the conic section 2 29 4 18 16 29 0x y x y . Then sketch the graph.
321102 General Mathematics Dr Wattana Toutip 6/1/09 20
■ Quadratic Equations
A general form of quadratic equation may be written as
2 2 0Ax Bxy Cy Dx Ey F (1.3)
in which A, B and C are not all zero.
In this section we have seen that if the axis of a conic section parallel to
the coordinate axis then the equation of the conic section is in the form
2 2 0Ax Cy Dx Ey F (1.4)
in which the cross product term, Bxy , did not appear.
We can apply completing the squares to identify the equation.
We may have noticed that the graph of equation (1.4) is a (or an)
a) parabola if 0AC
b) ellipse if 0AC c) hyperbola if 0AC
Let’s consider the graph of the hyperbola
2 9xy
Note that, the graph is rotated through an angle of 4
radians from the x-
axis about the origin.
In the next section we will discuss on rotating of axes.
321102 General Mathematics Dr Wattana Toutip 6/1/09 21
1.3 Rotation of axes
Let a new coordinate XY be a counterclockwise rotation through angle about the origin of the coordinate XY as in the figure.
■ The relations between ( , )x y and ( , )x y are as follow:
cos sinx x y (1.5)
sin cosy x y (1.6)
321102 General Mathematics Dr Wattana Toutip 6/1/09 22
Example 1.3.1 The x- and y-axes are rotated through an angle of 4
radians about the origin. Find an equation for the hyperbola 2 9xy in the
new coordinates.
321102 General Mathematics Dr Wattana Toutip 6/1/09 23
■ Notice that the equation of rotation can be solved to obtain the inverse
relation as
cos sinx x y (1.7)
cos siny y x (1.8)
… How come? …
321102 General Mathematics Dr Wattana Toutip 6/1/09 24
Example 1.3.2 Find an equation for the ellipse whose foci located at the
point ( 6, 6) and ( 6, 6) with the constant sum 2 14 in the original
coordinate.
321102 General Mathematics Dr Wattana Toutip 6/1/09 25
■ Notice that if we apply the equation of rotation in the general quadratic
equation
2 2 0Ax Bxy Cy Dx Ey F (1.9)
then we obtain a new quadratic equation in the coordinate X Y as
2 2 0A x B x y C y D x E y F (1.10)
The new and old coefficients are related by the equations
2 2cos cos sin sinA A B C
cos2 ( )sin 2B B C A
2 2sin cos sin cosC A B C
cos sinD D E
sin cosE D E
F F
…Why ?…
321102 General Mathematics Dr Wattana Toutip 6/1/09 26
Consequently, we obtain a wonderful theorem for moving the cross
product term from the new quadratic equation as the following:
Theorem 1.3.1 Given2 2 0Ax Bxy Cy Dx Ey F be a
quadratic equation in the coordinate XY and 0B . If the coordinate
X Y is rotated through an angle and cot 2A C
B
then the equation
in the new coordinate is reduced in the form
2 2 0A x C y D x E y F
in which the cross product term did not appear.
Proof
321102 General Mathematics Dr Wattana Toutip 6/1/09 27
Example 1.3.3 The coordinate axes are to be rotated through an angle
to produce an equation for the curve
2 22 3 10 0x xy y
that has no cross product term. Find and the new equation. Identify the
curve.
321102 General Mathematics Dr Wattana Toutip 6/1/09 28
Example 1.3.4 Identify the graph of the equation
2 22 3 2 25 0x xy y
321102 General Mathematics Dr Wattana Toutip 6/1/09 29
Since axes can always be rotated to eliminate the cross product term, then
we might be consider any quadratic equations in the form
2 2 0Ax Cy Dx Ey F
This form represents
a) a circle if 0A C (special cases: a point or no graph)
b) a parabola if 0A and 0C or otherwise 0A and 0C
c) an ellipse if A and C are both positive or both negative (special
cases: a circle, a single point or no graph at all)
d) a hyperbola if A and C have opposite signs (special case: a pair
of intersection lines)
e) a straight line if 0A C and at least D and E is different from
zero
f) one or two straight line if the left hand side of the equation can
be factored into the product of two linear factors
However, we do not need to eliminate the xy-term from the equation
2 2 0Ax Bxy Cy Dx Ey F
to tell what kind of conic section the equation represents. We can use the
discriminant test as stated in the following theorem:
321102 General Mathematics Dr Wattana Toutip 6/1/09 30
Theorem 1.3.2 Let 2 4B AC be the discriminant of a quadratic equation
2 2 0Ax Bxy Cy Dx Ey F
Then the curve of the equation is
a) a parabola if2 4 0B AC
b) an ellipse if 2 4 0B AC
c) a hyperbola if 2 4 0B AC
Proof
Example 1.3.5 Fill in the blanks
(a) 2 23 6 3 2 7 0x xy y x represents a …………… because
…………………………………………………………………….
(b) 2 2 1 0x xy y represents a …………… because
…………………………………………………………………….
(c) 2 5 1 0xy y y represents a …………… because
…………………………………………………………………….
321102 General Mathematics Dr Wattana Toutip 6/1/09 31
Chapter 2 Vectors
2.1 Review of vectors
2.2 Linear combination and linearly independent
2.3 Vectors in three dimensional space
Chapter 3 Limit and continuity of functions
Chapter 4 Derivative of functions
Chapter 5 Applications of derivative and differentials
Chapter 6 Integration
Chapter 7 Applications of integration
Chapter 8 Differential equations