Spring Moon Essay, Research Paper

Jamie Burton

Period 1

Calculus

To Use Calculus, or Not to Use Calculus?

That is the Question

In the past, if you have studied Algebra and Trigonometry, then your knowledge has prepared you to master the next step: calculus. Calculus is complicated, but not quite as bad as everyone thinks. It is the study of changing quantities. Take for example, the curve as a path of a rocket. A tangent line at any point on the orbit displays the direction that the rocket is flying at that point. If gravity disappears, the rocket would move off on the tangent line.

Within changing quantities, calculus is based on two fundamentals: derivatives and integrals. ( and ) Computing the derivative is easy, but has a lot of algormathically. The derivative finds the slope of a function. Integration is the same as the anti-derivative in that it finds the original equation of the function and computes the area under the function. These are two concepts that calculus is molded.

Derivatives and integrals are all to do with limits. The concept of limits is essential to all calculus. The limit is the basis for all calculus problems. A knowledgeable understanding of limits will help explain many theories in calculus. In limits, the only thing that matters is how a function is defined near the point a ( point a is the quantity that the limit gets close to but never reaches).

In taking the derivative of a height function, the out come is a velocity function. In taking the derivative of a velocity function ( the rate of change of a position), you get a acceleration function ( the rate of change of velocity). Or if given an acceleration function you can integrate to get a velocity function. Newton realized that the pattern of velocities is simpler, while the pattern of accelerations is simpler still. The two known operations of calculus, differentiation (taking the derivative) and integration, let us pass form any of these patterns to any other. So in other words, we can work from the easiest, acceleration, and find the one we really want…..height.

Calculus finds speeds if you know the position of an object at all times. It finds the position if you know the speeds at all times. If you want to know the total change of position over a period of time, you just need to know how fast the quantity is changing. And if you know how much there is at anytime, it will tell you how rapidly the quantity is changing. The quantity could be population in a lake, city, country , or bacterial.

Analytic Geometry is a big factor of calculus, so its good to familiar with Cartesian coordinates, the slopes of lines, the equations of figures such as circles, ellipses, and parabolas, and function notation such as . Knowing these and learning derivatives and integrals and their separate ideas, then your most likely to move on to bigger and better things.

Big and better wasn?t always easy in the 17th century. Some of our integration techniques date back to Ancient Greece, but calculus today was developed by Isaac Newton in 1666 and G.W. Leibniz in 1675. Fluxions is what Newton called his development.

Newton was born during the Cromwell Rebellion and was the son of a farmer in Lincolnshire. He gave little hope to prosper in the battle of life. At first he was extremely inattentive to his school work and ranked very poorly in his school. He liked carpentering, mechanics, the writing of verses, and drawing.

Newton showed no achievement in life, not even after he went to Trinity College. It wasn?t until he earned his B.A. in 1665 that he was on the right track. When he achieved his M.A. he was already considered the most promising mathematics and physicist in England.

In 1665 he started his work with calculus, described by Newton as the theory of fluxions and he used it to find the tangent and radius of curvature at any point on a curve. So at age 27 he had begun his intelligent work on the calculus and mathematical physics.

Besides Newton, there was another great mathematician that discovered calculus. His name is Gottfried Wilhelm Leibniz. He studied mathematics in Germany during the 17th century. He showed a great future in math considering he read the most influential treatises on the subject before the age of 20. He tried the law thing, but finally moved to Hannover and became the librarian to the duke.

At his palatial house, he had the opportunity to develop the differential and integral calculus. He started to think and contemplate the idea in 1673, some seven years after Newton. Two years later he had his theory well put together, but didn?t publish anything until 1684 when he published Acta Eruditorum, a explanation of the method and its possibilities.

It is not doubtful that Leibniz thought of calculus independently, and that Leibniz and Newton are both permitted credit for their unseen inventions. The two sides were dramatically different although both lead to the same conclusions. Leibniz is said to have made Cavalieri scientific. He also provided the first step for the theory of envelopes and explained the osculating circle and displayed its significance in the study of curves. Today we use Leibniz?s version.

The calculus we know today was developed by these two brilliant mathematicians in a time when technology was almost unheard of. Limits, integrals, and differentiation was all thought of and proved by them. So when you go to sit down and do calculus and want to know what is calculus? You can thank them for a fun and integrating experience.