Advanced Calculus

1. UNIVERSITI PENDIDIKAN SULTAN IDRIS
FAKULTI SAINS DAN MATEMATIK
Sem. 2 Session 2018/2019
Assignment
SMN 3023 Advanced Calculus
Instruction:
1. The assignment can be done in a group of 4 members
2. The assignment must be typed using Microsoft Office Word (Times Roman, 12 pt)
3. All mathematical sentences and symbols must be typed using Equation Editor.
4. You can use internet as your reference, but you also need to use at least two reference
books.
Name Matric.Num
Kerolin Blontharam D20181084225
Nur Aqilah Bt Alwi D20181084236
Fatin Nur Alya Bt Abd Aziz D20181084235
Jayasri Jeeva D20181084234

2. Part A:
1. What is L’Hospital Rule ?
Rule uses derivatives to help evaluate limits involving indeterminate forms.
2. When can you use L’Hospital Rule ? Give example of problem and show how you can
use the L’Hospital Rule
When can use L’Hospital Rule?
a.
b. 0
c. , ,
d.
Example:
 lim
sin
=
So, we have already established that this is a 0/0 indeterminate form so let’s
just apply L’Hospital’s Rule.
lim
sin
= lim
cos
= =1

3. Part B:
1. What does it mean by conic sections ?
A conic section is a curve obtained as the intersection of the surface of a cone with a plane.
 Conic sections are the sections of a double cone (i.e. two equal circular cones placed
with their vertices in contact and having the same axis) made by a plane.
 The mathematical definition of conic section:
A conic section is the locus of a point that moves in a plane so that its distance from
fixed point in the plane bears a constant ratio to its distance from a fixed straight line
in the plane.
2. What are the conic sections ?
a) A pair of intersecting straight lines
b) A circle
c) A parabola
d) An ellipse
e) A hyperbola
3. How do the conic sections form?
 A pair of intersecting straight lines is formed by a plane section of the double cone
through the common vertex, and these straight lines will be generators of the double cone
 A circle is formed by a plane section of the double cone perpendicular to the common
axis
 A parabola is formed by a plane section of the double cone parallel to a generator.
 An ellipse is formed by a plane section of the double cone, cutting only one half of the
double cone, but neither perpendicular to the common axis nor parallel to a generator.
 A hyperbola is formed by a plane section cutting both halves of the double cone, but not
passing through the common vertex.

4. 4. Elaborate on each of the conic section: definition, technique of graphing, etc. You
need to give example of questions in your elaboration.
a) Circle
 Definition
A circle is defined as the set of points in a plane whose distance from some fixed centre
point is a constant.
 Technique Of Graphing
We now let C ( 磨 be the fixed point and P (x,y) be a point that lies on the circle.
Figure 1.2: Circle with C ( 磨 as a fixed point
C ( 磨 is called the center of the circle and the distance between the center and
any point on the circle is called radius
C ( 磨
P (x,y)
C ( h
r
P (x,y)
Figure 1.3: Circle with the
center C(h,k) and radius r

5. If P (x , y) is any point on a circle with center C(h , k) and radius r, then the length CP = r.
Using the formula for the distance between the two points, we get
m 磨 m h = r
m 磨 m h = (1.1)
Equation (1.1) is the equation of a circle with center C(h,k) and radius r.
If the center is at origin ,then we get
磨 (1.2)
Expanding the equation (1.1), we obtain
m 磨 m h磨 h
m 磨 m h磨 h m
磨 磨 (1.3)
Where
D= -2h
E= -2k
F= h m
Equation (1.3) is the general form of the equation of a circle with centre C(h.k) and radius r.
 Example:
Find the equation of a circle with center (2,3) and radius 5
Solution:
From equation (1.1), m 磨 m h =
Is a circle with center (h,k) and radius r.
Hence ,the equation of a circle is m 磨 m െ =
If we expand this equation, we have
m 1 1 磨 m h磨
磨 m 1 m h磨 m

6. b) Parabola
Definition :
A parabola is the set of all points (x,y) that are equidistant from a fixed line, the directrix, and a
fixed point, the focus, not on the line. The midpoint between the focus and the directrix is the
vertex, and the line passing through the focus and the vertex is the axis of the parabola.
Parabolic Conic Section
Standard Equation Of A Parabola
The standard form of the equation of a parabola with vertex h k and
directrix y k m p is:
x m h 1p y m k Vertical axis
For directrix x h m p the equation is
y m k 1p x m h Horizontal axis
The focus lies on the axis p units directed distance from the vertex. The
coordinates of the focus are :
h k p Vertical axis
h p k Horizontal axis

7. All parabolas have the same set of basic features.The axis of symmetry is a line that is at the
same angle as the cone and divides the parabola in half.
Directrix.
A fixed, straight line. The parabola is the locus (series) of points in which any given point is of
equal distance from the focus and the directrix.
Axis of symmetry.
This is a straight line that passes through the turning point "vertex"of the parabola and is
equidistant from corresponding points on the two arms of the parabola.
Vertex.
The point where the axis of symmetry crosses the parabola is called the vertex of the parabola. If
the parabola opens upward or to the right, the vertex is a minimum point of the curve. If it opens
downward or to the left, the vertex is a maximum point.
Focus
A fixed point on the interior of the parabola that is used for the formal definition of the curve.
A right triangle is formed from the focal point of the parabola.
Technique of Graphing Parabola
Any quadratic equation of the form 磨 h t has curved called a parabola.

8. Step 1: Determine whether the parabola open upwards or downwards and if the vertex a
minimum or a maximum.
 If the coefficient a in the equation is positive,the parabola opens upward in a vertically
oriented parabola,so the vertex is a minimum.
 If the coefficient a in the equation is negative,the parabola opens downward in a
vertically oriented parabola,so the vertex is a maximum.
 If the equation has a squared y term instead of a squared x term, the parabola will be
oriented horizontally and open sideways, to the right or left, like a "C" or a backward
"C."
Step 2:Find the axis symmetry.
 The axis of symmetry is the straight line that passes through the turning point (vertex)
of the parabola.
 To find the axis of symmetry,use this formula m
t
h
Step 3:Find the vertex
 The axis of symmetry, you can plug that value in for x to get the y coordinate. These two
coordinates will give you the vertex of the parabola.
 The vertex of a parabola is m
t
h
m
t
h
Step 4: Set up a table with chosen values of x and calculate the values of corresponding y-
coordinates.
 Create a table with particular values of x in the first column. This table will give you the
coordinates you need to graph the equation.

9.  You should include at least two values above and below the middle value for x in the
table for the sake of symmetry.
 Calculate the values of corresponding y-coordinates. Substitute each value of x in the
equation of the parabola, and calculate the corresponding values of y.
 We found at least five coordinate pairs for the parabola graph to be plotted.
Step 5: Plot the table points on the coordinate plane.
 Each row of the table forms a coordinate pair (x, y) on the coordinate plane.
 Graph all points using the coordinates given in the table.
 The x-axis is horizontal; the y-axis is vertical.
Step 6: Connect the graphs
 To graph the parabola, connect the points plotted in the previous step.
 Connect the points using slightly curved lines.
 This will create the most accurate image of the parabola.
Example Of Questions
1. Find the focus of the parabola
m t m t
Solution :
th h ܿ hǡh hh h t磨 ݉ h th݁ h ܿh h.
磨 m m t݁thh h ܿh t h
磨 m m ܿ t݉ 磨 hh t h t磨

10. 磨 m ܿ݉ h
磨 m hh ܿt h h t݁ t h
m 磨
m 磨 m t h th hh h
݉h th݁ t h ܿh t h t :
m 1݉ 磨 m h
ܿ h hh h ܿ h h m h hh ݉ m .
h hܿ h ݉ t hh݁h tǡh h ݉h ht h ݉hh h h . h ܿ h ݉h ht h t ݉ ܿht
h ǡh h : h ݉ m
c) Ellipses
Definition:
Let be the two fixed points and h is a positive number such that h > (i.e, h is
greater than the distance between the two points hh . The set of all points 磨
for which
h (1.1)
Is called an ellipse; hh are called the foci of ellipse.
Figure 1.2: Ellipse

11. Figure 1.2 shows an ellipse in what is called standard position. hh are called foci, the
middle point between hh is named as centre, the major axis and the minor axis.
The equation of ellipse is derived from equation (1.1). assume that h = 2 , the centre is at the
origin and the major axis is along the -axis. and are the foci of ellipse.
Therefore, the point 磨 lies on the ellipse if only if
2
磨 m 磨
Transferring the term m 磨 to the right hand side and squaring both sides. We get
磨 = 4 m 磨 m 1 m 磨
Simplifying this equation, it is reduced to
1 m 磨 = 1 m
Cancelling the factor 4 and squaring both side again, we then have
m 磨 1
m
Be rearranging this equation, it is reduced to
m 磨 m
This equation can be written as
磨
m
(1.3)
Substituting h m we then have
磨
h
, with t h (1.4)
Equation (1.4) is the equation of an ellipse in the standard position.
The relation between h hh is shown in Figure 1.5.
Figure 1.5: the relation between h hh
c

12. From Figure 1.5, it is clear that (-c,0) and (c,0) are the foci of ellipse, the length of major axis is
2m and the length of minor axis is 2n.
By using similar argument, we get the equation of ellipse with y-axis as major axis as
磨
h
with m < n (1.6)
Equation (1.6) is the equation of (1.4) with x and y are interchanged. It represents the ellipse of
equation (1.4) reflected in the line y = x. the foci hh m are now on the 磨-axis.
(See Figure 1.7). the centre remains at the origin. The relation between h hh is given by
h m
Figure (1.7)

13. Example:
Identify the curve 1 m h 磨 磨 m
Solution:
1 m h 磨 磨 m
1 m 1 磨 磨
1 m m 1 磨 m
1 m 磨
1 m 磨
It is an ellipse with centre m . Here 1
hh h t 磨th h . We
observe that, t h. In this case, the major axis is is vertical and has length 10, and the minor
axis is horizontal and has length 5. Hence, the relation between h hh is given by h m
m 1
1
. ܿ
െ
. We therefore have the foci located at m m
െ hh m െ .
d) Hyperbola
Definition:
A hyperbola is the set of all points (x,y) for which the absolute value of the difference between
the distances from two distinct fixed points called foci is constant.
Standard Equation Of a Hyperbola
The standard form of the equation of a hyperbola with center at (h,k) is
m
h
m
磨 m h
t
hhǡh h h t t t h h
OR
磨 m h
h
m
m
t
hhǡh h h t t ǡh t h
The vertices are h units from the centre,and the foci are c units from the centre,where
h t .

14. The foci are two fixed points equidistant from the center on opposite sides of the transverse axis.
The vertices are the points on the hyperbola that fall on the line containing the foci. The line
segment connecting the vertices is the transverse axis. The midpoint of the transverse axis is
the center. The hyperbola has two disconnected curves called branches
Eccentricity of a Hyperbola. The eccentrity of a hyperbola is given by a ratio :
e
c
a
Because c t a for hyperbola,it follows that e t for hyperbolas.
If the eccentrity is large,then the branches of the hyperbola are nearly flat.
If the eccentrity is closer to 1,then branches of the hyperbola are more pointed.
Asymptotes Of A Hyperbola
For a horizontal transverse axis,the equations of the asymptotes are
磨 h
t
h
m hh 磨 h m
t
h
m
For a vertical transverse axis,the equations of the asymptotes are,
磨 h
h
t
m hh 磨 h m
h
t
m

15. Techniques of graphing
Step 1 : Determine which of the standard forms applies to the given equation.
x m h
a
m
y m k
b
Tranverse axis is horizontal
y m k
a
m
x m h
b
Tranverse axis is vertical
Step 2 : Use the standard form identified in Step 1 to determine the position of the transverse axis;
coordinates for the vertices, co-vertices, and foci; and the equations for the asymptotes.
if the equation is in the form of
x
a
m
y
b
then
 the transverse axis is on the x-axis
 the coordinates of the vertices h
 the coordinates of the co-vertices t
 the coordinates of the foci
 the equation of asymptotes 磨
t
h
x
if the equation is in the form of
y
a
m
x
b
then
 the transverse axis is on the y-axis
 the coordinates of the vertices h
 the coordinates of the co-vertices t
 the coordinates of the foci t
 the equation of asymptotes 磨
h
t

16. step 3 : Solve for the coordinates of the foci using the equation c a b
step 4: Plot the vertices, co-vertices, foci, and asymptotes in the coordinate plane, and draw a
smooth curve to form the hyperbola
Find the vertices,foci,eccentricity and asymptotes and sketch the graph for the following
equation,
磨
m
11
Solution :
a2
= 25 and b2
= 144,
So a = 5 and b = 12.
h t
11
െ
m hh 磨 磨 m
Thus, the center is at h
The vertices and foci are above and below the center,so,
foci are at m െ hh െ
vertices are at hh m
Eccentricity,e
h
h
h
െ
Because the y part of the equation is dominant,then the slope of the asymptotes,
t m h th m t m h
磨 m

17. 磨 m m
m
Asymptotes are 磨
5. Find application of each of the conic section.
a) Circle
A pizza delivery area can be represented by a circle, and extends to the
points (0,18) and (−6,8) (these points are on the diameter of this circle). Write an equation for
the circle that models this delivery area.
Solution:

18. We first plotted the two points that form a diameter of the circle that represents the pizza
delivery area: (0,18) and (−6,8).
Since the center of a circle is midpoint between any two points of the diameter, we can use the
Midpoint Theorem
磨 磨
to get the centre of the circle:
mh
m െ െ .
To get the radius of the circle , we can use the distance formula m 磨 m 磨 to get
the distance between the centre and one of the points, lets pick
(0,18): m m െ m െ െ1.
The equation of circle is (x + 3)² + (y – 13)² = 34.
b) Parabola
A satellite dish antenna is to be constructed in the shape of a paraboloid. The paraboloid is
formed by rotating the parabola with focus at the point (25, 0) and directrix x=−25 about the
x-axis, where x and y are in inches. The diameter of the antenna is to be 80 inches.
a. Find the equation of the parabola and the domain of x.
x= 磨
The domain is the positive real number set
b. Sketch the graph of the parabola, showing the location of the focus.

19. c. A receiver is to be placed at the focus. The designer has warned that a user or installer
should take care, the receiver would hit the ground and could be damaged if the antenna
were placed "face-down". Determine algebraically whether this observation is correct.
Yes, as discovered in part a and shown in the graph of part b, the focus is at x = 25,
while the dish only extends to x = 16.
c) Ellipse
An asteroid has elliptical orbit with the sun at one focus. Its distance from the sun
ranges from 1818 million to 182182 million miles. Write an equation of the orbit of
the asteroid.
Here we have
We also have
2ae=| m m h1
A= |C m

20. Now using the relation to find the value of b, we have
hh h m t
t m െ h
The required equation of the ellipse of the asteroid will be
h
磨
t
= 1
磨
െ h
= 1
d) Hyperbola
Two radar sites are tracking an airplane that is flying on a hyperbolic path. The first radar site is
located at (0,0), and shows the airplane to be 200 meters away at a certain time. The second radar
site, located 160 miles east of the first, shows the airplane to be 100 meters away at this same
time. Find the coordinates of all possible points where the airplane could be located. (Find the
equation of the hyperbola where the plane could be located).
Let’s draw a picture first and remember that the constant difference for a hyperbola is always
2a. The plane’s path is actually on one branch of the hyperbola; let’s create a horizontal
hyperbola, so we’ll use the equation
m
h
m
磨mh
t
We know that the distance from the “leftmost” focus to the plane (hyperbola) is 200 meters, and
the distance from the “rightmost” focus to the plane (hyperbola) is 100 meters.

21. This is actually the “constant difference” of the hyperbola, which is 2a.
200−100=2a
a=50.
Thus,
h =2500.
We also know that
2c (distance between foci) =160
so
c=80.
Since
h t
we can obtain
t :t m h m െ
In the model, the centre of the hyperbola is at (80,0), so the path of the airplane follows the
hyperbola
m
m
磨
െ

22. Reference
Howard Anton, lrl C_Bivens, & Steven Davis. (2012). Calculus Early
Transcendentals(10th ed., pp. 730 - 744). Wiley.
Introduction to Conic Sections | Boundless Algebra. (2019). Retrieved from
https://courses.lumenlearning.com/boundless-algebra/chapter/introduction-to-
conic-sections/
http://tutorial.math.lamar.edu/Classes/CalcI/LHospitalsRule.aspx
https://www.ck12.org/calculus/applications-of-parabolas/lesson/Applications-of-
Parabolas-MAT-ALY/
https://www.emathzone.com/tutorials/geometry/applications-of-ellipse.html
Mohd Nain Hj. Awang, & Umar Baba. (2012). Conic sections and applications of
integral. Kuala Lumpur: Utusan Publications & Distributors.