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SMN 3023 ADVANCED CALCULUS SEM 2 SESSION 2011/12 ASSIGNMENT 1 i) What does it mean by conic sections? In mathematics, a conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic defined as a plane algebraic curve of degree 2. There are a number of other geometric definitions possible. One of the most useful, in that it involves only the plane, is that a conic consists of those points whose distances to some point, called a focus, and some line, called a directrix, are in a fixed ratio, called the eccentricity. There are three types of conic section are the hyperbola, the parabola, and the ellipse. The circle is a special case of the ellipse, and is of sufficient interest in its own right that it is sometimes called the fourth type of conic section. The type of a conic corresponds to its eccentricity, those with eccentricity less than 1 being ellipses, those with eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 being hyperbolas. In the focus- directrix definition of a conic the circle is a limiting case

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Page 1: Assignment Advance

SMN 3023

ADVANCED CALCULUS

SEM 2 SESSION 2011/12

ASSIGNMENT 1

i) What does it mean by conic sections?

In mathematics, a conic section (or just conic) is a curve obtained as the intersection of a

cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a

conic defined as a plane algebraic curve of degree 2. There are a number of other geometric

definitions possible. One of the most useful, in that it involves only the plane, is that a conic

consists of those points whose distances to some point, called a focus, and some line, called a

directrix, are in a fixed ratio, called the eccentricity.

There are three types of conic section are the hyperbola, the parabola, and the ellipse. The

circle is a special case of the ellipse, and is of sufficient interest in its own right that it is

sometimes called the fourth type of conic section. The type of a conic corresponds to its

eccentricity, those with eccentricity less than 1 being ellipses, those with eccentricity equal to

1 being parabolas, and those with eccentricity greater than 1 being hyperbolas. In the focus-

directrix definition of a conic the circle is a limiting case with eccentricity 0. In modern

geometry certain degenerate cases, such as the union of two lines, are included as conics as

well.

ii) What are the conic sections?

The graph of a first- degree equation in two variables,

Ax + By = C

Page 2: Assignment Advance

Where A and B are not both 0, is a straight line, and every straight line in a rectangular

coordinate system has an equation of this form. While the graph of the general second- degree

equation in two variables,

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 (1)

If there are any ordered pairs satisfying such an equation, it can be shown that (with some

exceptions) the graph will be one of the following four figures: a circle, a parabola, an ellipse,

or a hyperbola. There figures are called Conic sections, because they describe the intersection

of a plane and a double- napped cone as illustrated below.

We begin by concentrating on the cases where B = 0 in the general second-degree equation. In

other words, we focus on equation of form (1) that do not contain anxy-term and where A and

C are not both zero.

Conic Section

Definition

Circle

Each of the geometric figures are obtained by intersecting a double-napped right circular cone

with a plane. Thus, the figures are called conic sections or conics. If the plane cuts completely

across one nappe of the cone and is perpendicular to the axis of the cone, the curve of the

section is called a circle.

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.

Ellipse

If the plane isn't perpendicular to the axis of the cone, it is called an ellipse.An ellipse is the

set of all points in a plane, the sum of the distances from two fixed points in the plane is

constant.Many comets have elliptical orbits.

Parabola

If the plane doesn't cut across one entire nappe or intersect both nappes, the curve of the

intersection is called a parabola.

A parabola is the set of all points in a plane equidistant from a fixed point and a fixed line in

the plane.

Page 4: Assignment Advance

Hyperbola

If the plane cuts through both nappes of the cone, the curve is called a hyperbola.

The hyperbola is the set of all points in a plane. The difference of whose distance from two

fixed points in the plane is the positive constant.

iii) How do the conic sections form?

When you slice up a cone, each conic section has its own standard form of an equation

with x and y variables that you can graph on the coordinate plane. You can write the equation

of a conic section if you are given key points on the graph, or you can graph the conic section

from the equation. There are various ways that you can alter the shape of each of these graphs,

but the general graph shapes still remain true to the type of curve that they are.

It is important to be able to identify which conic section is which by just the equation because

sometimes that’s all you will be given (you won’t always be told what type of curve you are

graphing). Certain key points are common to all conics (vertices, foci, and axes to name a

few), so you start by plotting these key points and then identifying what kind of curve they

form.

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The above figure illustrates how a plane intersects the cones to create the conic sections.

The equations of conic sections are very important because they not only tell you which conic

section you should be graphing, but they tell you what the graph should look like. There are

trends in the appearance of each conic section based on the values of the constants in the

equation. Usually these constants are referred to as a, b, h, v, f, and d. Not every conic will

have all of these constants, but conics that do have them will be affected in the same way by

changes in the same constant. Conic sections can come in all different shapes and sizes: big,

small, fat, skinny, vertical, horizontal, and more.

An equation has to have x-squared and/or y-squared to create a conic. If neither x nor y is

squared, then the equation will be of a line (not considered a conic section). None of the

variables of a conic section may be raised to any power higher than two.

There are certain characteristics you will find unique to each type of conic that hint to you

which of the conic sections you are graphing. In order to recognize these characteristics the

way they are written, it is important that the x-squared term and the y-squared term are on the

same side of the equal sign.

Page 6: Assignment Advance

Circle

A circle is the set of all points a given distance (the radius, r) from a given point (the center).

To get a circle from the right cones, the plane slice occurs parallel to the base of either cone,

but does not slice through the element of the cones.

You can identify the equation for a circle when x and y are both squared, and the coefficients

on them are the same — including the sign. For example, take a look at

Notice that the x-squared and y-squared have the same coefficient (positive 3). That’s all the

info you need to recognize that you’re working with a circle.

Ellipse

An ellipse is the set of all points where the sum of the distances from two points (the foci) is

constant, and you may be more familiar with the term oval. In order to get an ellipse from the

two right cones, the plane must cut through one cone, not parallel to the base, and not through

the element.

You can identify the equation for an ellipse when x and y are both squared and the coefficients

are positive but different. For example, the equation

is one example of an ellipse. The coefficients on x-squared and y-squared are different, but

both are positive.

Parabola

A parabola is a curve where every point on the curve is equidistant from one point (the focus)

and a line (the directory). It looks a lot like the letter U, although it may be upside down or

sideways. To form a parabola, the plane slices through parallel to the side of the cones (any

side works, but the bottom and top are forbidden).

You can identify the equation for a parabola when either x or y is squared — not both. For

example, the equations

are both parabolas. In the first equation, you see an x-squared but no y-squared, and in the

second equation, you see a y-squared but no x-squared. Nothing else matters — sign and

coefficients will change the physical appearance of the parabola (which way it opens or how

fat it is) but won’t change the fact that it’s a parabola.

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Hyperbola

A hyperbola is the set of points where the difference of the distances between two points is

constant. The shape of the hyperbola is difficult to describe without a picture, but it looks

visually like two parabolas (although they are very different mathematically) mirroring one

another with some space between the vertices. To get a hyperbola, the slice cuts the cones

perpendicular to their bases (straight up and down), but not through the element.

You can identify the equation for a hyperbola when x and y are both squared and exactly one

of the coefficients is negative (coefficients may be the same or different). For example, the

equation

is an example of a hyperbola. This time, the coefficients on x-squared and y-squared are

different, but one of them is negative, which is a requirement to get the graph of a hyperbola.

iv) Elaborate on each of the conic section: definition, technique of graphing, etc.

You need to give example of questions in your elaboration.

Circle

A circle is a set of points in a plane that are equidistant from a fixed point. The distance is

called the radius of the circle, and the fixed point is called the center.

Suppose a circle has center (h,k) and radius r > 0.

Then the distance between the center (h,k) and any point (x, y) on the circle must equal r.

Center – Radius Form of the Equation of a Circle

The center – radius form of the equation of a circle with center (h, k) and radius r is

( x−h )2+( y−k )2=r2

Page 8: Assignment Advance

Example 1:

Find the center – radius form of the equation of a circle with radius 6 and center (-3,4). Graph

the circle, the give the domain and range of the relation.

Solution: Using the center – radius form with h = -3, k = 4, and r = 6, we find the equation of

the circle is

[ x−(−3)]2+( y−4)2=62or[ x−(−3)]2+( y−4)2=36.

The graph is shown in below. Because the center is (-3,4) and the radius is 6, the circle must

pass through the pour points (-3±6, 4) and (-3, 4±6), as illustrate in the table. Using this four

points, we see the domain is [-9, 3] and the range is [-2, 10].

Equation of a Circle with Center at the Origin

A circle with center (0,0) and radiurr has equation

Example 2:

Find the equation of a circle with center at the origin and radius 3. Graph the relation, and

state the domain and range.

x2+ y2=r2

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Solution: Using the form with r = 3, we find that the equation of the circle is

. The domain and range are [-3, 3].

Ellipses

An ellipse is the set of all points (x,y) in a plane, the sum of whose distance from two distinct

fixed points (foci) is constant.

The line through the foci intersects the ellipse at two points called vertices. The chord joining

the vertices id the major axis, and its midpoint is the center of the ellipse. The chord

perpendicular to the major axis at the center is the minor axis.

The standard form of the equation of an ellipse with center at the origin and major and

minor axes of lengths 2a and 2b, respectively (where 0 < b < a), is given by

x2

a2 + y2

b2 =1 orx2

b2 + y2

a2 =1

The vertical and foci lie on the major axis, a and c units, respectively, from the center, as

shown in figure below. Moreover, a, b, and c are related by the equation c2=a2+b2.

x2+ y2=9

x2+ y2=r2

Page 10: Assignment Advance

Example 3:

Find the standard form of the equation of the ellipse shown in figure below.

Solution:

From figure, the foci occur at (-2,0) and (2,0). So, the center of the ellipse is (0,0), the major

axis is horizontal, and the ellipse has an equation of the formx2

a2 + y2

b2 =1 .

Also from the figure, the length of the major axis is 2a = 6. So, a = 3.

Moreover, the distance from the center to either focus is c = 2. Finally,

b2=a2−c2=32−22=9−4=5

Substitute a2=32∧b2=(√5)2 yield the equation in standard form.

x2

32 + y2

(√5)2 =1

Example 4:

Sketch the ellipse given by 4 x2+ y2=36, and identify the vertices.

Solution:

Page 11: Assignment Advance

4 x2

36+ y2

36=36

36

x2

32 + y2

62 =1

Because the denominator of the y2-term is larger than the denominator of the x2-term, you can

conclude that the major axis is vertical. Moreover, because a = 6, the vertices are (0, -6) and

(0, 6). Finally because b = 3, the endpoints of the minor axis are (-3,0) and (3,0) as shown in

figure below.

Parabolas

A parabola is the set of all points (x,y) in a plane that are equidistant from a fixed line, the

directrix, not on the line. The midpoint between the focus and the vertex is the axis of the

parabola.

The standard from of the equation of a parabola with vertex (0,0) and directric y = -p is

given by

Page 12: Assignment Advance

x2=4 py , p≠0. Vertical axis

For directrix x= -p, the equation is given by

y2=4 px, p≠0. Horizontal axis

The focus is on the axis p units (directed distance) from the vertex.

Notice that parabola can have a vertical or a horizontal axis and that a parabola is symmetric

with respect to its axis. Examples of each are shown in figure below.

Example 5:

Find the focus of the parabola whose equation is y = -2x2.

Solution:

Because the squared term in the equation involves x, you know that the axis is vertical, and

the equation is of the form x2=4 py . You can write the original equation in this form as

follows.

x2=−12

y x2=4 (−12

) y

So, p = −18

. Because p is negative, the parabola opens downward, and the focus of the

parabola is

(0, p) = (0, 18

)

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Example 6:

Find the standard form of the equation of the parabola with vertex at the origin and focus at

(2, 0).

Solution:

The axis of the parabola is horizontal, passing through (0, 0) and (2, 0), as shown in figure

below, so, the standard form is y2=4 px . Because of the focus is p = 2 units from the vertex,

the equation is y2=4 (2 ) x y2=8 x .

The equation y2=8 x does not define y as a function of x. so, to use a graphing utility to graph

y2=8 x , you need to break the graph into two equations, y1=2√2 x and y2=−2√2x , each

of which is a function of x.

Hyperbolas

A hyperbola is the set of all points (x, y) in a plane, the difference of whose distances from

two distinct fixed points (foci) is a positive constant.

Page 14: Assignment Advance

The definition of a hyperbola is similar to that of an ellipse. The difference is that for an

ellipse, the sum of the distances between the foci and a point on the ellipse is constant,

whereas for a hyperbola the different of the distances between the foci and a point on the

hyperbola is constant.

The standard from of the equation of a hyperbola with cebter at the origin (where a ≠ 0

and b ≠ 0) is given by

x2

a2 −y2

b2 =1 ory2

a2 −x2

b2 =1 .

The vertices and foci are a and c units from the center, respectively. Moreover, a, b, and c are

related by the equation b=c2−a2.

Example 7:

Find the standard form of the equation of the hyperbola with foci at (-3, 0) and (3, 0) and

vertices at (-2, 0) and (2, 0), as shown in figure below.

Solution:

Page 15: Assignment Advance

From the graph, you can determine that c=3 because the foci are three units from the center.

Moreover, a = 2 because the vertices are two units from the center. So, it follows that

b=c2−a2

= 32 – 22

= 9 – 4

= 5.

Because the transverse axis is horizontal, the standard form of the equation is

x2

a2 −y2

b2 =1

Finally, substitute a2=22 and b2=(√5)2 to obtain

x2

22 −y2

(√5)2 =1

Example 8:

Sketch the hyperbola whose equation is 4 x2− y2=16 .

4 x2− y2=16

4 x2

16− y2

16=16

16

x2

22 −y2

42 =1

Because the x2=term is positive, you can conclude that the transverse axis is horizontal and

that the vertices occur at (-2, 0) and (2, 0). Moreover, the endpoints of the conjugate axis

occur at (0, -4) and (0, 4), and you can sketch the rectangle shown in figure below. Finally, by

drawing the asymptotes through the corners of this rectangle, you can complete the sketch

shown in figure below. Note that the equations of the asymptotes are y = 2x and y = –2x.

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v) Find application of each of the conic section

Two conic sections have inspired two useful navigational systems. Long Range Navigation

(Loran) was first used during World War II, in the early 1940’s. Loran uses hyperbolic

branches and chains of stations to aid ships and aircrafts in navigation. The second is the

global positioning system or GPS. GPS uses satellites and circles to locate a receiver

anywhere in the world. GPS was fully functional in July of 1995 and did become a more

versatile alternative to Loran, however, Loran is still in use today as a support for the new

global positioning position.

Hyperbola

Loran is referred to as a hyperbolic system. In order for the Loran system to work effectively,

the Loran receiver must be connecting with at least three transmitting stations. This set of

three stations is called chain. One of these station is designated as the master station and the

other two are secondary station.

Each station sends out repeated signals, traveling at the speed of light, pulsing at specific time

intervals. The chain of stations has unique time delays to distinguish chains from one another.

These signals reach the Loran receiver, located on the ship. By analyzing these time delays,

we are able to calculate the difference in distance from the ship to the master station and the

from the ship to one of the secondary stations.

The hyperbola has a very important distance property that helps locate the ship. The

hyperbola is the set of all points where the difference in distance to each of the foci of the

hyperbola is constant. This special property is the foundation for Loran.

Imagine that the world is flat and that the coverage area if a chain can be shown on Cartesian

plane. The master station and two secondary station are plotted on the grid. To find the

location of the Loran receiver, presumably on a ship, we need to find the difference in

distance from each master/ secondary pair to the ship. These distances can be converted to

longitude and latitude coordinates and the master/secondary pair become the foci of a

Page 17: Assignment Advance

hyperbola. These particular foci generate several different hyperbolas. Because the special

distance property, we can find which hyperbola has a difference of distance that coincides

with the constant our receiver calculated. Somewhere on this hyperbola will be the ship.

This same process is repeated for the other secondary station and the master station. Where

the two corresponding hyperbolas meet is the location of the ship. Because we are able to find

which stations the ship is closest to, there is only one possible location for the ship.

In the future above, M and S1 are the focal points of one hyperbola. M and S2 are the focal

points of the second hyperbola. Where these hyperbola meet is the location of the ship.

Circles

Similar to the Loran system, GPS works successfully with the transmission of signals,

travelling at the speed of light. In the case of GPS, we are working with satellites instead of

stations. GPS uses twenty-four satellites that orbit the entire earth, and hence it can provide

important navigational information anywhere in the world.

Also similar to Loran, three satellites are needed to find the altitude, longitude and latitude of

GPS receiver on earth. The satellites send signals that are picked up by the GPS receivers, just

like the Loran system. The receiver is then able to find the distance from each of the three

Page 18: Assignment Advance

satellites to the receiver. Because we know how fast the signals are travelling (the speed of

light), the simple distance formula provides us with this information (distance = speed x

time). Instead of using the hyperbola , each of the three satellites becomes the centre for a

sphere, with a radius equal to the distance we calculated using the formula. The three spheres

will interect at two points, one of which is usually unrealistic, perhaos below sea level. If

there are two plausible points, a fourth satellite distance calculation may be needed. The

appropriate point of intersection is the location of the receiver. The process of using three ( or

more ) sphere to locate a point is called trilateration.

The picture above shoes three circles of varying distances, centered around satellites (S1 , S2 ,

and S3 ). The point where the three circles (or spheres, as is the case with GPS) meet is the

location of the GPS receiver.

GPS and the Loran system are built on the foundation of fairly common mathematical

concepts-conic sections and distances formulas. Although the actual mathematics involved

requires rather complec calculations, it is helpful to understands, even broadly, how these

systems work.

Ellipse

Page 19: Assignment Advance

The ellipse is nevertheless the curve most often “seen” in everyday life. The reason is that

every circle, viewed obliquely, appears elliptical. Any cylinder sliced on an angle will reveal

an ellipse in cross-section (as seen in the Tycho Brahe Planetarium in Copenhagen)

Tilt a glass of water and the surface of the liquid acquires an elliptical outline. Salami is often

cut obliquely to obtain elliptical slices which are larger

The early Greek astronomers thought that the planets moved in circular orbits about an

unmoving earth, since the circle is the simplest mathematical curve. In the 17 th century,

Johannes Kepler eventually discovered that each planet treavels around the sun in an elliptical

orbit with the sun at one of its foci.

The orbit of the moon and of artificial satellites of the earth are also elliptical as the paths of

comets in permanent orbit around the sun.

Page 20: Assignment Advance

Halleys’s Comet takes about 76 years to travel abound our sun. Edmund Halley saw the

comet in 1682 and correctly predicted its return in 1759. Although he did not live long enough

to see his prediction come true, the comet is named in his honour.

On a far smaller scale, the electrons of an atom move in an approximately elliptical orbit with

the nucleus at one focus.

The ellipse has an important property that is used in the reflection of light and sound waves.

Any light or signal that starts at one focus will be reflected to the othe focus. The principles is

used in lithotrispsy, a medical procedure for treating kidney stones. The patient is placed in an

elliptical tank of water, with the kidney stone at one focus. High-energy shock waves

generated at the other focus are concentrated on the stone, pulverizing.

Page 21: Assignment Advance

The principle is also used in the construction of “ whispering galleries” such as in St. Pauls’s

Cathedral in London. If a person, whispers nears one focus, he can be heard at the other focus,

although he cannot be heard at many places in between.

Statuary Hall in the U.S. Capital building is elliptic. It was in this room that John Quincy

Adams, while a member of the House of Representatives, discovered this acoustical

phenomenon. H e situated his desk at a focal point of the elliptical ceiling, easily

eavesdropping on the private conversations of other House members located near the other

focal point.

The ability of the ellipse to rebound an object starting from one focus to the other focus can

be demonstrated with an elliptical billiard table. When a ball is placed at one focus and is

thrust with a cue stick, it will rebound to the other focus. If the billiard table is live enough,

the ball will continue passing through each focus and rebound to the other.

Page 22: Assignment Advance

Parabolas

One of the applications of parabolas involves the concept of a 3D parabolic refelector in

which parabola is revolved about its axis (the line segment joining the vertex and focus). The

shape of car headlights, mirrors in reflecting telescopes, and television and radio antennae

(such as the one below) all utilize this property.

Antenna of a Radio Telescope

All incoming rays parallel to the axis of the parabola are reflected through the focus.

Flashlights & Headlights

Page 23: Assignment Advance

In terms of a car headlight, this property is used to reflect the light rays emanating from the

focus of the parabola (where the actual light bulb is located) in parallel rays.

Parabolic Reflector

Parabolic reflectors work in much the same way as flashlights and antennas.

Path of a Projectile

Galileo Galilei found that all objects thrown form a parabolic path, no matter what. He

deduced this by the simple observation of watching objects being thrown. Galileo is

responsible for the modern concept of velocity and acceleration to explain projectile motion

that us studied today :

A projectile which is carried bya uniform horizontal motion compounded with a naturally

accelerated vertical motion describes a path which is a semi-parabola

Page 24: Assignment Advance

vi) A report

We found that a conic section is intersection of a plane and a cone.

Circle Ellipse Parabola Hyperbola

Ellipse (h) Parabola (h) Hyperbola (h)

Ellipse (v) Parabola (v) Hyperbola (v)

Page 25: Assignment Advance

By changing the angle and location of intersection, we can produce a circle, ellipse, parabola

or hyperbola, or in the special case when the plane touches the vertex, a point, line or 2

intersecting lines

Point Line Double Line

The General Equation for a Conic Section :

Ax2 + Bxy + Cy2 + Dx +Ey + F = 0

The type of section can be found from the sign of : B2 – 4AC

If B2 – 4AC is…. Then the curve is

< 0 Ellipse, circle, point or no curve

= 0 Parabola, 2 parallel lines, 1 line or no curve

> 0 Hyperbola or 2 intersecting lines

Page 26: Assignment Advance

The conic section : For any of the below with center (j,k) instead of (0,0), replace each x term

with (x-j) and each y term with (y-k)

Cicle Ellipse Parabola Hyperbola

Equation

(horiz,vertex)

x2+ y2 = r2 x2

a2 + y2

b2 =14px = y2 x2

a2 −y2

b2 =1

Equation of

Asymtotes : Y + (b/a)x

Equation (vert,

vertex)

x2+ y2=r2 y2

a2 + x2

b2 =14py = x2

y2/a2− x2

b2 =1

Equation of

Asymptotes

X = + (b/a)y

variable r = circle radius a = major radius

(=1/2 length

major axis)

b = minor radius

(= ½ length minor

axis)

c = distance

center to focus

p = distance from

vertex to focus (or

directrix)

a = ½ length

major axis

b = ½ length

minor axis

c = distance

center to focus

Eccentricity 0 c/a c/a

Relation to focus p = 0 a2−b2=c2 p = p a2+b2=c2

Definition : is the

locus of all points

which meet the

condition

Distance to the

origin is constant

Sum of distances

to each focus is

constant

Distance to focus

= distance to

directrix

Difference

between distance

to each foci is

constant

Page 27: Assignment Advance

Reference

en.wikipedia.org/wiki/Conic_section

http://math.about.com/library/blconic.htm

Forseth, K. R., Burger, C., Gilman, M. R., and Rumsey, D. (2007).How to Identify the Four

Conic Sections.Diperolehpada Mei 2, 2012 daripadahttp://www.dummies.com/how-

to/content/how-to-identify-the-four-conic-sections.html

University of Regina, Canadian Mathematical Society, and Imperial Oil Foundation (ESSO).

(2012).Navigation.Diperoleh pada Mei 6, 2012 daripada

http://mathcentral.uregina.ca/beyond/articles/LoranGPS/Navigation.html

Jill Briton.(2012).Occurrence of The Conics. Diperoleh pada Mei 6, 2012 daripada

http://britton.disted.camosun.bc.ca/jbconics.htm

Goodman, A., and Hirsch, L. (2004).Precalculus: understanding functions. United States:

Thomson.

Hornsby, J., Lial, M. L., and Rockswold, G. K. (2007). A graphical approach to precalculus

with limit: a unit circle approach (4th ed). United States: pearson.

John W. Coburn (2010). Precalculus (2nded). New York: McGraw-Hill.

Larson, R., Hostetler, R. P., and Edwards, B. H. (2005).Precalculus: a graphing approach

(4thed). New York: Houghton Mifflin.

Raymond A. Barnett, Micheal R. Ziegler & Karl E. Byleen (2000). Precalculus : A graphing

approach. United States: McGraw-Hill.