"Calculus Essays"

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Derivatives and Definite Integral Essay

Words: 688 Length: 2 Pages Document Type: Essay Paper #: 34926082

Derivatives and Definite Integrals

Word Count (excluding title and works cited page): 628

Calculus pioneers of the seventeenth century such as Leibniz, Newton, Barrow, Fermat, Pascal, Cavelieri, and Wallis sought to find solutions to puzzling mathematical problems. Specifically, they expressed the functions for derivatives and definite integrals. Their areas of interest involved discussions on tangents, velocity and acceleration, maximums and minimums, and area. This introductory paper shall briefly introduce four specific questions related to these problems and the solutions that were sought.

In calculus, how a function changes in response to input is measured using a derivative. The derivative of a function is the result of mathematical differentiation. It measures the instantaneous rate of change of one certain quantity in relationship to another and is expressed as df (x)/dx. It can be interpreted geometrically as the slope of the curve of a mathematical function f (x) plotted as a function of x. The integral of a function can be geometrically interpreted as the area under the curve of the mathematical function f (x) plotted as a function of x (Nave).

How does the tangent line evolve from the secant line and how does the derivative relate to the tangent line? A primary problem in calculus is to discover the slope of a curve. To do this, mathematicians use tangent and secant lines. Segments drawn in tangent to or through a curved line create angles which help define and measure the slope of a curve. The slope of a curve is found at a point. The method is to draw a secant line through that point and to first find the slope of the secant line. The formula is ?y/?x. The next step is to slide the second point along the curve toward the first. The secant approaches the limit and, as the two points meet, it becomes a tangent line. The slope of the tangent becomes the slope of the curve and is called the derivative. It is written dy/dx .

How are derivatives used to solve velocity and acceleration problems? Another common use…… [Read More]

Nave, R. Derivatives and integrals. Hyper Physics, Retrieved from http://hyperphysics.phy-astr.gsu.edu

Kouba, D.A. The Calculus Page, U.C. Davis Department of Mathematics. http://www.math.ucdavis.edu, Retrieved January 25, 2011
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Jeremy Bentham Tried to Establish Essay

Words: 1029 Length: 3 Pages Document Type: Essay Paper #: 95472163

Smoking rates do seem to be down, as a result of sin taxes and smoking bans in public areas like restaurants.

However, although this might be an example of when Bentham's moral science might seem to work (although it is controversial how helpful mandatory sentences may be) it is hard not to think of a familiar phrase: "one man's meat is another man's poison" -- in short, what gives pleasure to some might not give pleasure to all. A good example of this might be a child who is starving for parental attention. The child begins to 'act out' and is punished. The parents think they are acting to deter the behavior, but in reality, rather than experiencing an intense and swift punishment, the child experiences the punishment as a kind of reward, because it is at least feels like some kind of attention. Another example might be that of an individual who enjoys flouting the law or ethics. Some people might like to take advantage of a corporate expense account, not so much for the pleasure given by the stole items, but the pleasure of getting away with a petty crime, which should not rationally exceed the possible painful consequences of getting caught.

Although it might seem as if incarceration is always a terrible punishment, it could be noted that the 'calculus,' that the punishment must seem worse than the possible pleasure accrued from the action, will not be the same for all individuals. A wealthy individual might think twice about committing a crime because of all he has to lose in terms of financial and social stability. A poor individual who believes he has no way out of his circumstances other than to commit a crime will likely see his situation far differently, and not view the threat of prison with as much fear. Also it could be added that in some instances, like a person addicted to drugs, their moral calculus may not function 'normally' as the punishment-setter. The wealthy individual might be addicted to heroin and feel that it was worth risking everything to get the drug for their 'fix' although logically most rational people would believe that no drug would be worth the loss of one's liberty, position, wealth, and power.

There may also be a certain injustice in Bentham's calculus because it refuses to take into consideration the individual circumstances of the perpetrator,…… [Read More]

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Utilitarianism the Philosophy of Utilitarianism Essay

Words: 1787 Length: 5 Pages Document Type: Essay Paper #: 73817

Lastly, it runs counter to the view that morality is essentially related to the concept of justice. Many critics of this theory argue that, "morality is not based on consequences of actions. Instead, it is based on the fundamental concept of justice" (Lee). In the final analysis, the dilemma in utilitarianism is that it unable to deal with a wide range of moral issue and actins and, as such, tends to subvert a more comprehensive and wide ranging understanding of morality and ethics.… [Read More]

Consequentialism. Retrieved June 26, 2009, from http://plato.stanford.edu/entries/consequentialism

Hedonic Calculus. Retrieved June 26, 2009, from ttp://www.utilitarianism.com/felicalc.htm
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Frequency Distribution Below Shows the Essay

Words: 870 Length: 3 Pages Document Type: Essay Paper #: 38176982

The probability that both a and B. will occur is different from the probabilities that a will occur and that B. will occur.

Refer to the following data to answer questions 7 and 8. Just the answer

A random sample of song playing times in seconds is as follows:

7. Find the standard deviation. 28.65

8. Are any of these playing times considered unusual in the sense of our textbook? Explain. Does this differ with your intuition? Explain. The longest playing time, 293 seconds, could be considered somewhat unusual in that it is more than 1.5 standard deviations away from the mean, meaning more than 93% of data in the population (if normally distributed) would fall below this level -- 93% of songs would be shorter. Intuition says that this is not the case, and that though the song is longer this is simply part of the variation that exists in music.

Refer to the following situation for Questions 9, 10, and 11.

The boxplots below show the real estate values of single family homes in two neighboring cities, in thousands of dollars.

For each question, give your answer as one of the following: (a) Tinytown; (b) BigBurg; (c) Both cities have the same value requested; (d) it is impossible to tell using only the given information. Please explain answer in each case.

9. Which city has greater variability in real estate values? (b) BigBurg

10. Which city has the greater percentage of households with values $85,000 and over? (c) Both cities have the same value requested

11. Which city has a greater percentage of homes with real estate values between $55,000 and $85,000? (a) Tinytown

12. A random sample of the lifetime of 49 UltraIllum light bulbs has a mean of 3,960 hours and a standard deviation of 200 hours. Construct a 95% confidence…… [Read More]

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Personal Challenges Early on in Essay

Words: 874 Length: 3 Pages Document Type: Essay Paper #: 69239900

During my junior year, I decided to undergo a significant change in my educational career, which is to transfer from Los Angeles to San Francisco. My decision to transfer to another school was anchored on two objectives. My first objective was to transfer to a school where I will develop my knowledge and know my limits in studying, particularly in the field of natural science. My second objective, meanwhile, was to attend a school that has a curriculum structure and format that complements my needs -- that is, to attend classes centering on natural science and mathematics.

It was noticeable during my junior year that I experienced difficulty in excelling in my humanities class. This was due to the difference in the format of teaching humanities between my old and new school. In the new school I enrolled in, humanities class was handled differently, requiring greater classroom (teacher-student) interaction, which my humanities class in my old school did not engage in. However, this difficulty was just a transitional phase; soon after I became acquainted with my new school's curriculum format and getting used to my new schedule and classes, I started improving my grades, and successfully excelled in my classes.

My adjustment in my new school also enabled me to cultivate new relationships with other people. I met new people who share with me the same passion I have for the sciences and socio-civic activities. I have become a familiar face to teachers and mentors who I constantly inquire about not only academic concerns, but mundane issues relevant to us as well. I have come full circle in my new school, and the great period of adjustment had passed.

Looking back, my decision to change schools and take on greater academic responsibility and workload is still the hardest and most stressful yet bravest decision I ever did in my life. I knew that transferring during my junior year would have serious repercussions in my academic record; my difficult adjustment in my humanities class was an example of these repercussions. However, I considered these consequences as minor setbacks that had ultimately aided me…… [Read More]

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Art and Mathematics Are Related Essay

Words: 2688 Length: 10 Pages Document Type: Essay Paper #: 96643501

Note the distinct similarities.

An examination of Escher's Circle Limit III can thus tell us much about distance in hyperbolic geometry. In both Escher's woodcut and the Poincare disk, the images showcased appear smaller as one's eye moves toward the edge of the circle. However, this is an illusion created by our traditional, Euclidean perceptions. Because of the way that distance is measured in a hyperbolic space, all of the objects shown in the circle are actually the same size. As we follow the backbones of the fish in Escher's representation, we can see, then, that the lines separating one fish from the next are actually all the same distance even though they appear to grow shorter. This is because, as already noted, the hyperbolic space stretches to infinity at its edges. There is no end. Therefore, the perception that the lines are getting smaller toward the edges is, in fact, a result of two-dimensional perspective drawing attempting to illustrate the nature of an infinite hyperbolic space.

Put another way, hyperbolic lines are represented by circular arcs perpendicular to the bounding circle of the disk, shown by the spines on the fish in Circle Limit III. Ever-decreasing Euclidean distances represent equal distances in this hyperbolic space as the eye approaches the edge of the disk (Dunham 23). The objects along the edge of the circle are the same size, thus, as those on the interior and the distances are equal. In theory, the fish should continue to exist ad infinitum, although there were no doubt physical limitations on what Escher's hand could manage.

Developing an Appropriate Class Project simple, yet appropriate, classroom project that would synthesize the various elements of art and geometry that have been discussed up to this point is easy to devise at this point. To begin with, the students must…… [Read More]

Corbitt, Mary Kay. "Geometry." World Book Multimedia Encyclopedia. World Book, Inc., 2003.

Dunham, Douglas. "A Tale Both Shocking and Hyperbolic." Math Horizons Apr. 2003: 22-26.
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Ethics the Ford Pinto Case Offers an Essay

Words: 1587 Length: 5 Pages Document Type: Essay Paper #: 11956081


The Ford Pinto case offers an ideal opportunity to apply utilitarian ethics to a real world situation. First, it is important to list the actors and stakeholders in this case. Lee Iacocca was the leader of the Ford Motor Company. He is credited with creating the inflexible parameters for the Pinto automobile as weighing no more than 2000 pounds and costing no more than $2,000. Therefore, the utilitarian analysis can and should apply primarily to Iacocca and his corporate brethren at the helm of Ford. It was their decision that led to the consequences associated with the poor design of the automobile, causing deaths.

However, the Ford Pinto case also highlights the ethical responsibilities of all members of the Ford Motor Company. In particular, the case showcases the role that engineers play in carrying out their jobs. It can easily be said that any engineer who felt that Iacocca's decision was unsound or unethical could have left his or her post with Ford Motor Company, but also that another engineer would have seamlessly replaced the other. The engineers are therefore more like passive actors, versus the more active decision makers in corporate headquarters. It is important to determine which entities are responsible for the ethical decision-making, and in this case, those entities are Iacocca and his corporate comrades.

The stakeholders in the Ford Pinto case include all consumers of the automobile. Ancillary stakeholders include all members of the general public who might come into contact with the automobile in their community. Because of the tendency of the Pinto to explode on impact, bystanders who had not purchased the car might also be injured in its wake.

Before completing a utilitarian analysis of the Ford Pinto case, it is important to note the historical context in which the situation arose. Iacocca was reacting to specific market forces. In particular, foreign automobile manufacturers were creating and selling cheap cars that were cutting into Ford's market share. Iacocca had to act fast in order to retain the competitive advantage of his American company. His stakeholders are the shareholders of Ford, who depend on Iacocca to make decisions that maximize Ford's profitability. For Iacocca and the shareholders in the Ford Motor Company, profit is the primary (and even arguably ethical) objective. If Iacocca were to shirk his responsibility as president of Ford…… [Read More]

DeGeorge, Richard T. "Ethical Responsibilities in Large Corporations."

Horas, Matthew R. "The Ford Pinto." Retrieved online: http://thelittlecarefreecar.webs.com/
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Newton Sir Isaac Newton Isaac Newton Bio Essay

Words: 703 Length: 3 Pages Document Type: Essay Paper #: 18754207


Sir Isaac Newton

Isaac Newton (Bio, N.d.)

Sir Isaac Newton is one of the most recognizable names in all of science. He was a mathematician, a natural philosopher, an inventor, an English physicist, and pretty much an all around genius. His work included the study of how light reacts to reflection, formulating laws of universal gravitation and motion, and building the first ever reflecting telescope. Newton arguably contributed more to the science than any single person in the entire history of science. Newton's book, Principia, is considered to be among the most influential science books in the history of science, possibly of mankind. In this book he provided the foundation for classical mechanics. Newton described universal gravitation and the three laws of motion which have been the background of classical physics for over three centuries. Since Sir Isaac Newton was such an influential mind, I thought it would be fun to read about his life and his education. This report consists of some of the interesting tidbits I found about Newton.

Newton's Life

Isaac Newton was born in 1642 after his father passed away. Although his father was fairly prosperous, his family mostly consisted of a poor farming family in Woolsthorpe, England. Not only was Newton born without his father, he was born three-month early and most people didn't think he would make it. After Isaac survived his own birth, his grandma took him and worked to raise him. Not only did he not have a father, his mother remarried another man and left Isaac. However, when he was 12 he reunited with his mother and got the chance to attend school.

Newton had been enrolled at the King's School in Grantham, a town in Lincolnshire, where he lodged with a local apothecary and was introduced to the fascinating world of chemistry (Bio, N.d.). His mother eventually pulled him out of school, because she needed (or thought she need him) to be farmer. His mother had Isaac take care of the…… [Read More]

Bio. (N.d.). Isaac Newton Biography. Retrieved from Bio True Story: http://www.biography.com/people/isaac-newton-9422656
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Dedicate Myself to an Interest Essay

Words: 675 Length: 2 Pages Document Type: Essay Paper #: 90537785

I intend to pursue a career in medicine, a career that I regard as a calling as well as an aspiration. My sports activities have disciplined me, toughened me, and honed my competitive instincts in a way that I believe is necessary for pre-medical studies, and later, perhaps, for medical school. I intend to enter the health care field, either a practitioner, or perhaps from a business angle, as I have grown convinced, after seeing my own family's struggle with the bureaucratic aspects of the American medical system, that there is a need for an infusion of compassion and reform into the system from all areas, on the part of administrators as well as doctors.

At present, to give me a strong academic founding for my rigorous college studies in science and business, I am currently enrolled in three AP classes: Calculus AB AP, Psychology AP, and Spanish V AP. I hope these classes will give me a foundation in the sciences, and also sharpen my ability to understand and communicate better with those whom I serve. Additionally, after speaking with people through my charity work for whom Spanish is their first language, I have grown to appreciate how much speaking in someone's first language can mean to them, and help to establish a sense of comfort between two people.

My desire to enter the field of health care is inspired primarily by my sister, who has Downs Syndrome. She has inspired me to help others through her own personal accomplishments and her determination to overcome every obstacle. I know that it is incumbent upon every person to make the most of his or her life. I once asked my sister what mattered most to her in this world. I expected her to name a favorite toy or food, but instead she just smiled and said, family and friends, making a difference in the world and changing the world from the better. I could not have said it better…… [Read More]

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Attend the University of Toronto Essay

Words: 306 Length: 1 Pages Document Type: Essay Paper #: 22223292

During my two years at the Sias Marketing Group, I have put the theories I learn every day into practice through various company projects. In these projects, I am often in charge of bringing together statistical data on marketing projects for analysis and further examination. While analyzing this data with the senior management of the corporation, I have gained valuable experience in the practical use of mathematics in combination with economic theories. I enjoy this type of work very much, and hope to continue to gain more experience and knowledge for my continued career success.

I feel my education and the experience in finance qualifies me for a chance to capture the crown jewel of the world of mathematical finance, and you will accept my application to attend the University of Toronto. I look forward to hearing from…… [Read More]

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Personal Statement for Pharmacy School Essay

Words: 713 Length: 2 Pages Document Type: Essay Paper #: 50622349

Personal Statement for Pharmacy School

Pharmacy School is highly important to me as part of my career path, and I believe I am both personally and professionally well-qualified to attend and be successful. I have work experience in the pharmaceutical field, having worked at my aunt's pharmacy in the past. This work taught me a great deal about pharmacy as a profession, but also gave me a lot of insight about how to properly treat patients and interact with others. I consider myself more well-rounded because of the interaction I had with so many people during my time at the pharmacy. I smile frequently, consider myself a nice person, and have developed a high level of patience with people from all walks of life. In 2000, when I was 17, I came to the United States and earned my diploma from Valencia High School. Just a week later I began attending the summer semester at College of the Canyons. I was a pre-pharmacy major, in the Honors program, and a member of Educational Opportunity Programs and Services (EOPS).

I also performed volunteer work in the pharmacy at Henry Mayo Newhall Memorial Hospital in Valencia and wrote a diabetes research paper for the Honors program. Additionally, I performed volunteer work such as walking and gift wrapping for the cancer society. I was a calculus tutor, and was offered my own workshop and a teaching assistant position. I transferred to Saddleback College in Orange County in 2003, and received my IGETC certificate to transfer to the University of California. I was rewarded for my high GPA and my involvement in the EOPS program. I became a member of the pharmacy club at Irvine Valley College, and also worked at Bank of America for four years. Since I enjoyed helping people so much, I was very good at customer service. In 2004 I transferred to University. I had applied to 10 different universities, and was accepted to all of them including UC Berkley, UCLA, UCSB, and UCI, among others.

I was concerned that my educational downfall may have come in 2005, when I experienced trouble in my classes and…… [Read More]

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Recommendation Letters Essay

Words: 573 Length: 2 Pages Document Type: Essay Paper #: 66657850

Pharmacy Personal Statement

Over the course of my life, I have been confronted with many obstacles, yet one core passion has kept me focused: that of my desire to enter the medical profession as a pharmacist. I came to the United States when I was seventeen, relatively late to enter the U.S. school system, but I was still able to become a part of the Honors Program at the College of the Canyons as a pre-pharmacy major, with a focus in chemistry. I had already worked at my aunt's pharmacy, and seeing how the work of a pharmacist could make such a difference in people's daily lives cemented my decision to pursue the profession. While at College of the Canyons I was a participant in the EOPS (Educational Opportunity Programs and Services) and volunteered at the pharmacy of Henry Mayo Newhall Memorial Hospital in Valencia. The latter enabled me to continue to learn about the field from a hands-on perspective. I also volunteered for the Cancer Society and tutored calculus. As part of the honors program I submitted a research paper on the treatment of diabetes.

I then transferred to Saddleback College where I became a member of pharmacy club at IVC (Irvine Valley College) and worked at Bank of America to support my studies. However, after losing my brother in a car accident in 2005, I was forced to interrupt my education to help my parents through this difficult time. I attended several universities afterwards including University of California-Irvine (UCI), California State University-Fullerton (CSUF), followed by DeVry University and some online classes.

Finally, now that my family is stable once again and moving on from my brother's death, I feel that I am able to…… [Read More]

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Philosophy of Education Math Field Essay

Words: 1152 Length: 3 Pages Document Type: Essay Paper #: 86411019

Philosophy of Education

Norma Cunningham

I am a nontraditional student and I am returning back to college due to a job loss. I have been given a second chance at obtaining an education. Since I have been attending college, I was accepted into the nursing program, but I turned it down. I did this because I remember my dream has always been to be a math teacher. Everyone knows teachers are not in the profession for the money, and that nurses make more money, so people may ask, why a teacher? Well, I remember when I was growing up, every time someone would ask me what I wanted to be, I always answered, a math teacher. Certain teachers that I have had in the past, and present, have helped me decide that I want to spend the rest of my life teaching math. Helping any age student to learn math gives me chills, because, like those teachers, I love solving problems and trying to figure out the right answers. Like John Travolta sang in Grease, "I've got chills, they're multiplying." Ever since I graduated in 1976, math has been my passion. I helped my children with their math throughout their education and also helped my older brother complete his college math while I was still in high school. Today, I still want to be a teacher -- but not just any teacher. I want to be one who reaches all of her students in a memorable way by following excellent teaching practices.

All children need to learn the basics of math in order to lead successful lives, not necessarily algebra or calculus, but definitely how to add, multiply, divide, subtract and yes, even problem solving. Problem solving which is observed in Maslow's hierarchy of needs involves various types of needs which are required to reach an individual's full potential, described as self-actualization (%% reference). Otherwise, to become everything that one is capable of becoming. This…… [Read More]

Bloom, B. (1956). Taxonomy of Educational Objectives, the classification of educational goals. New York: McKay.

Brown, J., & Duguid, P. (2002). Knowledge and organization: a social-practice perspective. Organization Science, 12(2), 198-213.
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Jaime Escalante Hero Teaching Hope Essay

Words: 1767 Length: 5 Pages Document Type: Essay Paper #: 4316610

movie Stand and Deliver (Menendez & Musca, 1988), which is based on the true story of Jamie Escalante, an individual who overcame ethnic, cultural, and socioeconomic issues to become a highly successful mathematics teacher. Discuss the beliefs he held and the strategies he employed in his classroom that contributed to high achievement levels in his students.

The final report of the National Mathematics Advisory Panel (2008) presents a three-pronged argument for an effective math curricula: 1) It must foster the successful mathematical performance of students in algebra and beyond; 2) it must be taught by experienced teachers of mathematics who instructional strategies that are research-based; and, 3) the instruction of the math curriculum must accomplish the "mutually reinforcing benefits of conceptual understanding, procedural fluency, and automatic recall of facts" (National Mathematics Advisory Panel, 2008, p. xiv). Jamie Escalante began teaching before this report was released, but he knew from experience -- and instinct -- that students who do not achieve mastery of foundational concepts of mathematics will face unforgiving -- perhaps harsh -- consequences in their lives (Won, 2010).

Frontrunner Excellence. A report published in 2006 by the National Council of Teachers of Mathematics made recommendations for math curriculum that are currently being implemented in a majority of U.S. states as the Common Core State Standards. One recommendation has been particularly influential and is pivotal to curriculum, instruction, and assessment aligned with the Common Core: Math curriculum and instruction should cover fewer topics at greater depth. Best practices in mathematics instruction have established the need to ensure students have sufficient time to learn concepts deeply so that they can build on the learning in subsequent grades and so that redundancy does not need to be built into the grade level curriculum and instruction. Rather, math curricula are designed to ensure continuity across the grade-level instruction, with each deep instruction provided at each grade. This successful approach has been adopted by a majority of nations where students are top-performers in mathematics.

An important outcome of these comprehensive reports on mathematics instruction in the U.S. is the recommendation that the issue of identifying which of two primary methods of math instruction…… [Read More]

____. (2004, April 13). "Hero'" Teacher Escalante Addresses Students At Wittenberg Commencement May 9. Wittenberg University. Retrieved http://www4.wittenberg.edu/news/1998/commspeaker.shtml

____. (2008). National Mathematics Advisory Panel, Foundations for Success. The Final Report of the National Mathematics Advisory Panel, U.S. Department of Education. Washington, D.C. Retrieved http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf