321 102 general mathematics - khon kaen university

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


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


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.


■ 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


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.


■ 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


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.


■ 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


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


■■ 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)


Outline proof (b)


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


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) .


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


■ 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


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


■ 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


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.


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.


■ 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.


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)


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.


■ 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? …


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.


■ 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 ?…


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


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.


Example 1.3.4 Identify the graph of the equation

2 22 3 2 25 0x xy y


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:


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

…………………………………………………………………….


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

321 102 general mathematics - khon kaen university

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