# MTH 227 Differential Equations REVIEW FOR QUIZ

Our quiz is moved to November 15,2013 due to international intellectual convention. As part of your review, solve the following exercises and problems. Refer to your notes and textbook for further reference.

Similar exercises and problems will be given.
Solve the following:

1. Solve

a) y’ + 2xy = 10x.

b) y’ – 3y = 12

c) y” + 4y’ – 5y = 0

d) y” -7y’ +12y =0

2. At Shoprite, a restaurant serves pepper soup at 190 F. After 4 minutes, the soup cooled to 170 F. If the room temperature is 68 F, find the time that the soup has cooled to 130F.

3. Find the total amount of N 70000.00 deposited at Mayfresh bank compounded continuously at 8.6 % nominal interest rate after 8 years.

4. Madonna university students are increasing at the rate equal to 10 % of its population each year. If in the beginning of this year there are 8,000 students, how many students will be expected after 3 years?

Good luck!

Don’t forget to bring with you the workbook. Final check on your assignment on chapters 5,6,and 7 will be conducted. Solve at least two exercises or problems at the end of the chapters.

DR. VICTOR E. ODARVE, BSME/MAED/MEP/EDD

# Nature Teaches Mathematics

By Vic Odarve

Not everybody loves mathematics! “I don’t like numbers. That’s why I did not take up engineering courses”, said one student. This summed up everything on his schooling! Since kinder up to high school, he has performed miserably in mathematics. But one thing is certain; whether he likes it or not, mathematics is a part of his everyday life. Unknown to many, math already existed in nature the moment we took our first breath of life. What we do is simply marvel mathematics!

The “petals” flower form spirals curve describing Fibonacci’s numbers.

Also nature teaches mathematics! Nature and math works in never-ending harmoniously. From the particle world of billions of stars spreading in our cosmic ocean, mathematical laws seem to work keeping the universe in order. From Newton’s law to Kepler’s Laws of planetary motion and Einstein’s law of relativity, math offers our deep understanding on things around us.

Nature has its own way of expressing mathematics! Patterns such as hexagonal beehives, spiraling Fibonacci flowering trees, fractals flowers and plants, fruits and leaves, and many more are actually mathematics!

FRACTALS, BEEHIVES, AND FIBONACCI’S NUMBERS

Fractals are repeating geometric pattern. They are rough fragmented geometric shapes that can be repeatedly subdivided into parts and that each small portion can be viewed as a replica of the whole. These are geometric designs we teach in classrooms. Since repeated, in mathematics is called iterations or algorithms. Newton’s method, simplex programming, Gauss Seidel, Jacobi, and many more algorithms are among few examples. Some plants exhibit simple recursive operations that after a few number repetitions or iterations their shape can be recognized as a repeated geometric pattern. It is mathematics! Fractals in nature are always pleasing to our eyes – clouds, plants leaves and fruits, galaxies, shells and many more. Plants such as sunflowers, ferns, cabbages, and papaya leaves are among a few of them.

Leaf arrangement suggests repetition

Honeycomb structure is also a nature mathematical exhibition. Look at its hexagonal shape. Its cross-section is made of the comb, a latticework of hexagonal cells made of beeswax. It is an expression of geometry in action. The design might be the most economical in terms of labor and wax; hence expressing least cost in economics and linear programming in mathematics!
Fibonacci number is a simple example illustrating the existence of sequence in mathematics. Just like arithmetic and geometric sequences and progressions. But in the Fibonacci sequence of numbers, each number is the sum of the previous two: 0,1,1,2,3,5,8,13,21, etc. And these numbers are exhibited in various “family trees” and patterns of spirals of leaves and seeds. The image of a nautilus shell approximates the sequence of the Fibonacci’s number in nature. Once again, nature teaches math!

Fractals, Fibonacci‘s shells, and beehives are iconic images that capture the beauty of nature displays in mathematical expressions. These displays are like ornaments to the eyes of the mathematicians. A living math in motion in order for us to love and appreciate nature! Indeed, nature teaches mathematics!

# Smartphones Boost Your Math Skills

By Vic Odarve

It is a wonderful world. Just a tip of your finger and the world will go round and round. Almost everybody loves this gadget… a cell phone! The cell phone is one of the most versatile electronic gadgets in the market today.This gadget becomes a necessity for everybody. Latest cell phone features include a camera, internet access, storage for pictures, audio, and video files, and many more. These things can help college students with all aspects of their studies. Just press the call button and you are connected to your loved ones and friends throughout the world. Talking and chatting! Young and old people alike… cell phones are their indispensable partners wherever they go. It seems our lives are difficult without this gadget.

Mathematical Formulas from these textbooks are stored on smartphones.

But college students use these cell phones in other means… boosting their math power! It is a learning tool to develop math skills. Take for example, a student in Calculus! While studying under the shade of the old oak tree at the university campus, he forgot to recall the formula in the Differential Calculus. The product and quotient rules of differentiation! What else did he do? With a cell phone, he simply scanned the picture files and refreshed the formulas on the screen with ease. That’s how it is. Instead of pictures, mathematical formulas are put there! And instead of audio for love and rock songs, recorded audio files on discussion or explanation of the mathematical formulas are put there. Storages for picture and audio files on his phone are filled with mathematical clips. Cell phones become a storage file for his studies. That keeps boosting his math understanding and skills! The student is the one who does the recordings and arranging the formulas for his easy access and reference.

It is his real world with phones… maximizing his gadget to boost his math power. “In my

spare time, Sir, I spent listening to my recorded audio and helped me a lot to remember the previous mathematical theorems, principles, and formulas which are often used in mathematics”, he added.

This way he familiarizes the mathematics by listening to the audio discussions and scanning thru pictures of formulas instead of memorization. No wonder this student is one of the top achievers in my class in Calculus!

It is now a different world. It is different from our college days. Technology really works wonders! Cell phones are now a partner with students in studying mathematics. Like Higgs particle that gives mass to everything that will interact with it, cell phones give boost your math power to those who make it as a tool in mathematics. It seems this gadget is inseparable to our lives. Perhaps someday society has come to a point where the idea of living without cell phones seems absurd. With this gadget, developing the student’s math skills becomes more exciting. We are living in this digital world. And undoubtedly, smartphones boost students’ math skills and understanding!

# LEARNING MATH AT CLASSROOM

LEARNING MATH AT CLASSROOM

Let us learn from plants! It is the nature’s way. Just like learning math at classroom! Plants make use of the quantum physics superposition theory. They harvested packets of light energy and converted to chemical energy with remarkable efficiency. Experts found that photons come in and see many energy paths and the effect called “coherence” helps determine which path is the most efficient to follow. That is it! Plants, the receiving end, maximize the conversion process!

Students studying mathematics are like plants harvesting the light energy. If students do not find the most efficient way to capture the wisdom of the classroom discussion, they are not converting efficiently the best knowledge that supposes to be imparted by the lecturers. Everything they do from the start to the end of the lectures has something to do with their own learning method.
Classroom lectures work on the same idea as plants do. Students, being the recipient of knowledge, must learn it with remarkable efficiency. Teachers always find the most efficient means to impart the subject to the students the best that they can deliver. Teachers feed them different ways how to solve mathematical problems by breaking these into its components slowly

Engineering Mathematics 1 students batch 2013 First Semester

and slowly. Then finally problems are transformed to its basic operations. This is basically the features of the classroom lectures!
Still classroom lectures are the best means for students to learn fast in mathematics. Teachers are there to instruct, discuss, assist, and show different approaches in mathematics by working out examples and illustrations. Classroom lecture offers opportunities for students to ask questions, offers suggestions and presents ideas, and shares knowledge with others.
In order to help students, the following are tips on how to maximize learning on classroom lectures;

1. Take an advanced reading of the topic for discussion before going to the class. Just read and picture out the new subject matter even you do not have clear understanding. Just a glimpse of the new topic! The idea is to let you feel as if you are in a review mode when you are in the classroom.

2. Listen carefully the discussion done by the teachers. Make notes on important points! This keeps you alert for the whole session. This avoids feeling sleepy or daydreaming. Sometimes teachers could share their own knowledge on things that may not be found in your textbook. Be an active participant.

3. Ask question if you fail to grasp some important points during class lectures. This keeps you from guessing and spending time later on your review. By asking questions, you learn and solidify important points. Understand the why behind every steps of the solution.

High-achievers undoubtedly do more efficient way to maximize their learning in classroom. Though not all students are endowed with wisdom to learn quickly like King Solomon, above suggested tips will help the slow learners in Mathematics. Remember there is no day in our life without mathematics!

# Mount Hermon; A Mysterious Mountain

By Vic Odarve

Let us discover Mount Hermon. Known for its majestic beauty, this mountain is approximately 2,814 above the sea level. It borders Israel, Syria, and Lebanon. It is the main source of the Jordan River. Mount Hermon is a very large triple peaked mountain. One of its three summits is the highest point on the east coast of the Mediterranean Sea. This mountain is believed to be the site of the transfiguration of Jesus Christ. In the book of Enoch, this mountain is also the site that the 200 fallen angels descended from heaven. These events have profound impacts among the religious faithful; Christians, Muslims, and Jews. Why these events happen on this mountain remain a mystery.

THE DANCING SISTERS

TRANSFIGURATION OF JESUS

Transfiguration is a dramatic change in appearance. In mathematical language, this is similar to transformations! A change of forms in expressions or formulas that makes our mathematical calculations simpler and easier! That’s why we have logarithms, Laplace, Fourier, Mellin and Z transformations!

Jesus also does transformation. A dramatic change of appearance or transfiguration! It was witnessed by the apostles. He transforms to other dimensions – from earthly appearance to spiritual or heavenly dimension of existence. Among the twelve disciples, only Peter, James, and John had witnessed their Master dramatically transfigured. This transfiguration reveals his glory as the son of God. A true nature of his divine existence!

By transfiguration, Jesus gives his disciples a glimpse of what he looks like when he will be on his kingdom with his Father. Just a glimpse! See what happened! Jesus face shone like the sun, and his garments became white as light! Below are the verses taken from Matthew 17: 1-2;

1 Six days later, Jesus took with him Peter and James and his brother John and led them up a high mountain where they were alone.
2 Jesus’ appearance was changed before them: his face shone like the sun and his clothes became bright as light.

This thing happened at Mount Hermon….a mysterious mountain!

THE FALLEN ANGELS

There were two hundred fallen angels descended to Earth at the summit of Mount Hermon. These watchers as they are called in the book of Enoch mated with humans. This class of spiritual beings was in charged with the observance and study of creation. And what happened? They transgressed by intermarrying with humanity and introducing forbidden technologies to earth.
They saw that daughters of Adam were beautiful and good looking. All of them took unto themselves wives. This was against God’s law as angels did not need to procreate like humans. As a punishment they could not go back to heaven anymore. These angels violated both their own nature and their office. God cannot forgive their sins. Having families, these angels taught their human wives charms, enchantments, astrology, cutting of roots and acquainted with plants, sorcery, and others.

Since then, these fallen angels were living on and around the area of Mount Hermon. Their hybrid children were monstrous giants called Nephilim. The aftermath of these giants was awful. Upon their death, they would release evil or demonic spirits that will remain bound to the earth. Since then, mankind won’t have any peace. That’s why we have demons and evil spirits.
It was believed that the tribe of Dan which the Antichrist will come from moved to this area during the days of the Judges. They adopted the Canaanite worship of these angels. Caanites Gods, Baal and Ashtaroth, have the origin at Mount Hermon! Following are taken from the book of Enoch 6: 1-3;

And the angels, the children of the heaven, saw and lusted after them, and said to one another: ‘Come, let us choose us wives from among the children of men and beget us children.’ And they were in all two hundred; who descended in the days of Jared on the summit of Mount Hermon, and they called it Mount Hermon, because they had sworn and bound themselves by mutual imprecations upon it.

Again, this happened at Mount Hermon…. A mysterious mountain!

Today this mysterious mountain is one of the hot spots in Middle East. Christians, Muslims, and Jews have contested this mountain for thousands of years. After the 1967 Six-Day war, both Syria and Israel administered military posts on Mount Hermon’s slopes.

Today, thousands of visitors come and visit this mysterious mountain! Some folks believe that this mysterious mountain is one of the Earth’s portals to other universes. A gateway to another world! Nobody knows!