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AMA INTERNATIONAL UNIVERSITY – BAHRAIN
Salmabad, Kingdom of Bahrain
COLLEGE OF ENGINEERING
B
Department of Math & Science
2nd Trimester SY 2019-2020
MATH 406 – Differential Calculus with Analytic Geometry
Special Project
Submission Date
Leader
Members
Submission Deadline
Section
ID#:
1.
ID#:
2.
ID#:
3.
ID#:
April 22, 2020 at 5:00PM ( use this file to put all your answers)
Activity 1
In “Hardball,” the man is searching for an amateur mathematician who has gone into hiding. It can
be hypothesizes that people who disappear and are still alive are like satellites that have lost their
planet. It explains that if the planet vanishes, the satellites would not have a focus to their orbit, and
would travel in another direction, likely to another source of gravity.
One such satellite orbiting the Earth is the Geostationary Operational Environmental Satellite
(GOES). The GOES-10, also called the GOES-EAST, satellite is a high altitude geosynchronous orbiting
satellite used for weather observations. If the Earth vanished, this satellite would maintain the same
velocity but would travel at a path tangent to its orbit. In this activity we will focus on the direction
of that travel by calculating the slope of the tangential path.
The GOES-EAST appears to be stationary in the sky because its nearly circular orbit is located in the
Earth’s equatorial plane and matches the rotation of the Earth. Because of this circular orbit, its path
can be expressed as the equation of a circle with the center of the Earth as the origin.
1. What is the radius of the circle the GOES-EAST traces?
2. If the center of the Earth is the origin (0, 0), what is the equation for the orbit?
If the Earth were to vanish, the path of the satellite would be a straight line tangent to the circular
orbit of the satellite. To find the slope of this tangent line at any point on the orbit, calculate the
derivative of the equation of the orbit. Rather than explicitly solving the equation for a variable
before calculating the derivative, use the implicit form of the equation and find the derivative of the
equation with respect to x (dy/dx).
Prepared by :
Dr. Lina S. Calucag
Course Coordinator &Member Teachers
Date :
Reviewed by:
Verified by:
Approved by:
Dr. Belen T. Lumeran
Department head
Dr. Noaman M. Noaman
Associate Dean
Dr. Beda T. Aleta
Dean
Date :
Date :
Date :
AMA INTERNATIONAL UNIVERSITY – BAHRAIN
Salmabad, Kingdom of Bahrain
COLLEGE OF ENGINEERING
B
Department of Math & Science
2nd Trimester SY 2019-2020
MATH 406 – Differential Calculus with Analytic Geometry
Special Project
3. Find the derivative of the equation for the GOES-EAST orbit.
The National Oceanic and Atmospheric Administration (NOAA) operates polar orbiting weather
satellites that travel in circular orbits around the Earth. One such satellite is NOAA-15.
4. Without knowing the altitude of the NOAA-15, calculate the derivative of the circular orbit.
5. Explain why the radius of the circular orbit does not affect the slope of the line tangent to the
orbit.
Circular orbits are special cases of elliptical orbits. In fact, the paths of all of the satellites around the
Earth are elliptical orbits.
6. Find the derivative of this general case using implicit differentiation.
Activity 2
Using the information from the activity 1, calculate the current linear velocity of the GOES-EAST
satellite and express your answer in meters/second.
If the Earth does not vanish, then in order for the GOES-EAST satellite to leave its orbit it would have
to have a great enough velocity to overcome the pull of gravity back to the Earth. This velocity is
called the escape velocity, Ve. This is represented by the equation
where G is the gravitational constant ( 6.673 x 10-11 m3 /kg s2 ), M is the mass of the Earth
(6 x 1024kg), and R is the radius of the satellite from the center of the Earth (in meters).
2. Calculate the escape velocity for the GOES-EAST satellite.
3. How much greater would the velocity of the GOES-EAST need to be to escape the Earth’s orbit?
4. What would happen to the GOES-EAST satellite if the velocity were less than the velocity required
to maintain a circular orbit?
Prepared by :
Dr. Lina S. Calucag
Course Coordinator &Member Teachers
Date :
Reviewed by:
Verified by:
Approved by:
Dr. Belen T. Lumeran
Department head
Dr. Noaman M. Noaman
Associate Dean
Dr. Beda T. Aleta
Dean
Date :
Date :
Date :
AMA INTERNATIONAL UNIVERSITY – BAHRAIN
Salmabad, Kingdom of Bahrain
COLLEGE OF ENGINEERING
B
Department of Math & Science
2nd Trimester SY 2019-2020
MATH 406 – Differential Calculus with Analytic Geometry
Special Project
5. Describe the orbit of the GOES-EAST satellite if the velocity was greater than the velocity required
for a circular orbit, but less than the escape velocity.
Activity 3: Applications of the Derivative
To determine the titles of the picture below, solve the 18 application of derivatives below. Then
replace each numbered blank with the letter corresponding to the answer for that problem. Show all
solutions on the answers given below.
Here is the title right-side-up:
" __ __ __ __ __ __ __
16 10 18 4 16 18 7
__ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __
17 5 9 2 10 13 8 8 16 7 6
12 16 3 3 18
__ __ __ __ __"
14 13 11 6 5
Here is the title upside-down:
"__ __ __ __ __ __ __
17 4 13 8 9 11 12
__ __
13 2
__
18
__ __ __ __ __ __ __
17 18 15 15 18 6 9
__ __ __ __ __
12 18 10 17 5
__ __ __ ."
1 16 14
Derivative Application Problems:
1. Find the equation of the line normal to the curve f(x) = x3 – 3x2 at the point (1, -2).
2. Find the equation of the line tangent to the curve x2 y – x = y3 – 8 at the point where x = 0.
3. Determine the point(s) of inflection of f(x) = x3 – 5x2 + 3x + 6.
4. Determine the relative minimum point(s) of f(x) = x4 – 4x3.
5. A particle moves along a line according to the law s = 2t3 – 9t2 + 12t – 4, where t≥0 . Determine
the total distance traveled between t = 0 and t = 4.
6. A particle moves along a line according to the law s = t4 – 4t3, where t≥0. Determine the total
distance traveled between t = 0 and t = 4.
7. If one leg, AB, of a right triangle increases at the rate of 2 inches per second, while the other leg,
AC, decreases at 3 inches per second, determine how fast the hypotenuse is changing (in feet per
second) when AB = 6 feet and AC = 8 feet.
Prepared by :
Dr. Lina S. Calucag
Course Coordinator &Member Teachers
Date :
Reviewed by:
Verified by:
Approved by:
Dr. Belen T. Lumeran
Department head
Dr. Noaman M. Noaman
Associate Dean
Dr. Beda T. Aleta
Dean
Date :
Date :
Date :
AMA INTERNATIONAL UNIVERSITY – BAHRAIN
Salmabad, Kingdom of Bahrain
COLLEGE OF ENGINEERING
B
Department of Math & Science
2nd Trimester SY 2019-2020
MATH 406 – Differential Calculus with Analytic Geometry
Special Project
8. The diameter and height of a paper cup in the shape of a cone are both 4 inches, and water is
leaking out at the rate of ½ cubic inch per second. Determine the rate (in inches per second) at
which the water level is dropping when the diameter of the surface is 2 inches.
9. For what value of y is the tangent to the curve y2 – xy + 9 = 0 vertical?
10. For what value of k is the line y = 3x + k tangent to the curve y = x3 ?
11. Determine the slopes of the two tangents that can be drawn from the point (3, 5) to the
parabola y = x2 .
12. Determine the area of the largest rectangle that can be drawn with one side along the x-axis and
two vertices on the curve y = e-x2
13. A tangent drawn to the parabola y = 4 – x2 at the point (1, 3) forms a right triangle with the
coordinate axes. What is the area of this triangle?
14. If the cylinder of largest possible volume is inscribed in a given sphere, determine the ratio of
the volume of the sphere to that of the cylinder.
15. Determine the first quadrant point on the curve y2x = 18 which is closest to the point (2, 0).
16. Two cars are traveling along perpendicular roads, car A at 40 mph, car B at 60 mph. At noon
when car A reaches the intersection, car B is 90 miles away, and moving toward it. At 1PM, what is
the rate, in miles per hour, at which the distance between the cars is changing?
17. A 26-foot ladder leans against a building so that its foot moves away from the building at the
rate of 3 feet per second. When the foot of the ladder is 10 feet from the building, at what rate is
the top moving down (in feet per second)?
18. A rectangle of perimeter 18 inches is rotated about one of its sides to generate a right circular
cylinder. What is the area, in square inches, of the rectangle that generates the cylinder of largest
volume?
Prepared by :
Dr. Lina S. Calucag
Course Coordinator &Member Teachers
Date :
Reviewed by:
Verified by:
Approved by:
Dr. Belen T. Lumeran
Department head
Dr. Noaman M. Noaman
Associate Dean
Dr. Beda T. Aleta
Dean
Date :
Date :
Date :
AMA INTERNATIONAL UNIVERSITY – BAHRAIN
Salmabad, Kingdom of Bahrain
COLLEGE OF ENGINEERING
B
Department of Math & Science
2nd Trimester SY 2019-2020
MATH 406 – Differential Calculus with Analytic Geometry
Special Project
Activity 4 – Prepare a comprehensive report of the activities you have done, in a paragraph form.
Include the following:
*how you start the activity
*who participated in the activity
*how many hours did you do per activity.
*What did you learn from the activities you have done
* Enumerate math topics you use in the activities
*You may add photos of you doing the activity, label the photos per activity
Project Evaluation Rubric
Student NameID#_____________________
Project
Title____________________________________
DIRECTIONS: All projects must be submitted/presented by the assigned deadline. This time needs to
be scheduled well in advance to avoid conflicts.
Planning and Time Management: This score indicates the amount of time spent planning the case/problem
and the effectiveness of time management skills throughout the completion of the case/problem (meeting
assigned deadlines, turning in paperwork, etc.).
Indicator
Level 0
Level 1
Level 2
Level 3
Level 4
Level 5
PLANNING &
TIME
MANAGEMEN
T
No
evidenc
e of
planning
Circle one
0
Procrastination
lead to
incomplete
case/problem
2
2.5
Little
planning or
forethought
Basic
planning and
time
management
needs
necessary for
case/problem
completion
met
6
6.5
Case/problem
hastily
completed for
deadline
4
4.5
Planning and
time
management
exhibited
enhance the
overall
case/problem
8
Exhibits a
professional
level of
planning
and time
managemen
t
8.5
10
Comments:
Points Possible: 10
Score
Time and Effort: This score indicates the amount of time and effort the student expended completing the
case/problem.
Indicator
Level 0
Level 1
Level 2
Level 3
Level 4
TIME AND
EFFORT
Circle one
No
evidence of
effort
Little or no
“authentic”
time spent on
case/problem
0
Prepared by :
Dr. Lina S. Calucag
Course Coordinator &Member Teachers
Date :
4
5
Minimal effort
Met minimum
time
requirements
and didn’t
complete
case/problem
8
9
Meets basic
time and effort
required to
complete
case/problemt
12
Time and
effort
expended on
case/problem
enhances the
overall
case/problem
13
16
Level 5
Exhibits a
professional
level of time
and effort
expended on
case/problem
17
Reviewed by:
Verified by:
Approved by:
Dr. Belen T. Lumeran
Department head
Dr. Noaman M. Noaman
Associate Dean
Dr. Beda T. Aleta
Dean
Date :
Date :
Date :
20
AMA INTERNATIONAL UNIVERSITY – BAHRAIN
Salmabad, Kingdom of Bahrain
COLLEGE OF ENGINEERING
B
Department of Math & Science
2nd Trimester SY 2019-2020
MATH 406 – Differential Calculus with Analytic Geometry
Special Project
Comments:
Points Possible: 20
Score
Evidence of Learning and Risk Factor: This score indicates the level of knowledge gained by the student evident
through the case/problem, and the extent to which the student was “stretched” or took risks through the case/problem
experience.
Indicator
Level 0
Level 1
Level 2
Level 3
Level 4
Level 5
EVIDENCE
OF
LEARNING
AND RISK
FACTOR
No
evidence of
genuine
learning
Circle one
Student never
stretched their
knowledge/cap
abilities
0
4
Little
demonstration
of genuine
learning; limited
risks taken
5
8
Case/problem
demonstrates
genuine
learning/risks
were taken for
expand-ing
knowledge and
skills
9
12
13
Case/Proble
m and
case/problem
experience
clearly
“stretched”
student
knowledge
and skills
16
17
Student took
several risks
to achieve a
superior level
of knowledge
and skills
through the
case/problem
process
20
Comments:
Points Possible: 20
Score
Degree of Difficulty: This score indicates the variety and complexity of the components to completing the
project.
Indicator
Level 1
Level 2
Level 3
Level 4
Level 5
Level 0
Case/proble
m
DEGREE OF
DIFFICULTY
incomplete
Circle one
0
Not ageappropriate
difficulty
Little degree
of difficulty
evident
One
dimensional
case/problem
4
5
8
9
Case/problem
comprised of
more than
one
component of
appropriate
difficulty
12
13
Case/problem
comprised of
multiple
components
or
components
exhibit great
difficulty
16
17
Case/proble
m
complexity
approaches
professional
quality
20
Comments:
Points Possible: 20
Score
Portfolio Preparation: This score indicates the quality of the portfolio.
Prepared by :
Dr. Lina S. Calucag
Course Coordinator &Member Teachers
Date :
Reviewed by:
Verified by:
Approved by:
Dr. Belen T. Lumeran
Department head
Dr. Noaman M. Noaman
Associate Dean
Dr. Beda T. Aleta
Dean
Date :
Date :
Date :
AMA INTERNATIONAL UNIVERSITY – BAHRAIN
Salmabad, Kingdom of Bahrain
COLLEGE OF ENGINEERING
B
Department of Math & Science
2nd Trimester SY 2019-2020
MATH 406 – Differential Calculus with Analytic Geometry
Special Project
Indicator
Level 0
Level 1
Missing
case/proble
m report
Incomplete
case/problem
report
PROJECT
REPORT
PREPARATI
ON
Some
required
sections
missing
Circle one
0
2
2.5
Level 2
Case/problem
report has
major
formatting
and/or many
spelling errors
4
4.5
Level 3
Level 4
Level 5
Case/probl
em report
complete
with
several
minor
errors
Case/problem
report
complete
with very few
minor errors
Case/proble
m report
clear,
concise,
accessible,
with unique
content
6
6.5
Contents
concise and
accessible
8
8.5
No
formatting
or spelling
errors
10
Comments:
Points Possible: 10
Score
Quality of Final Project: This score indicates the actual quality of the physical product or quality of the
case/problem experience, with “professional” quality being a score of 10..
Indicator
QUALITY OF
FINAL
PROJECT
Level 0
Level 1
No physical
case/problem
or
documentation
of
case/problem
experience
Little
concern
for
case/probl
em quality
or
incomplete
case/probl
em
4
5
Circle one
0
Level 2
Case/problem
completed but
demonstrates
low quality
8
9
Level 3
Level 4
Level 5
Case/probl
em
demonstrat
es
appropriate
quality
High quality
case/problem
illustrates
student work
ethic
Professiona
l quality
product or
case/proble
m
experience
12
13
16
17
20
Comments:
Points Possible: 20
Score
Total Score________
Exemplary
Proficient
Developing
Emerging
Needs Major Support
Final Points________
Case/Problem Ranking Key
Strong evidence of meeting or exceeding learning goals
Evidence suggests adequate meeting of learning goals.
Evidence suggests some learning goals are met.
Evidence suggests some partially met learning goals.
Evidence suggests failure to meet desired learning goals.
Prepared by :
Dr. Lina S. Calucag
Course Coordinator &Member Teachers
Date :
95-100%
85-94%
65-84%
50-64%
Below 50%
Reviewed by:
Verified by:
Approved by:
Dr. Belen T. Lumeran
Department head
Dr. Noaman M. Noaman
Associate Dean
Dr. Beda T. Aleta
Dean
Date :
Date :
Date :
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