The Calculus-I topic I enjoyed most
Unit Two Derivatives
Ayanna Wilson
Calculus 151- F1
Dr. Verma
12/5/2017
The calculus topic I would like to discuss comes from unit two, derivatives. Derivatives are enjoyable because in most cases, they are simple to solve. Also, derivatives make other classes involving calculus and derivatives easier to understand. Within this paper, I will be elaborating on differentiation, the derivative, rate of change, the rules and purpose of derivatives and how to understand them.
As a chemistry major, math altogether is important in understanding my major and its curriculum. Calculus is a subject that many have difficulty understanding, including myself. Understanding derivatives is very necessary because it is the basics of calculus and modern mathematics. What is differentiation? Differentiation is the algebraic method of finding the derivative for a function at any point. (Wyzant)
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For example, the slope of a line like 2x is 2, or 3x is 3 and so on. Also, the constant rule, the power rule, the constant multiple rule, the sum rule, and the difference rule are all common rules one should be familiar with when involved in calculus. A constant function is the simplest function and the differentiation rule for a constant function is: This rule is derived from the power rule. The power rule is the most basic rule of differentiation. This rule is: Next, the constant multiple rule is the derivative of a constant times a function, is just the constant times the derivative and It states: The Sum Rule says the derivative of f + g = f’ + g’. So, we can work out each derivative separately and then add them with the use of the power rule. Lastly, the difference rule, along with the sum rule, gives us rules for finding the derivatives of the sums or differences of any of these basic functions and their constant
This equation shifts from the parent function based on the equation f(x) = k+a(x-h) . In this equation, k shifts the parent function vertically, up or down, depending on the value of k. The h value shifts the parent function to the left or right. If h equals 1, it goes to the right 1 unit, if it is negative 1, it goes to the left 1 unit. If a is negative, the parent function is reflected on the x-axis. If x is negative, the parent function is reflected on the y-axis.
Ross, Danice, Re-Ann Sabubu, and Era Manitas. "Applications - Bernoulli's Principle." Bernoulli's Principle. N.p., n.d. Web. 26 Jan. 2014.
When explaining the topic, I was completely lost and had trouble catching up but as soon as there was a demonstration, I soon caught on and was able to complete each equation with confidence.
Kinematics unlike Newton’s three laws is the study of the motion of objects. The “Kinematic Equations” all have four variables.These equations can help us understand and predict an object’s motion. The four equations use the following variables; displacement of the object, the time the object was moving, the acceleration of the object, the initial velocity of the object and the final velocity of the object. While Newton’s three laws have co-operated to help create and improve the study of
The derivative of a function is the rate of change of that function. It shows how fast or how slow the function is changing. This can be useful in determining things such as instantaneous rates of change, velocity, acceleration and maximum profits. A good way to explain the concept of a derivative is to do it graphically. To illustrate, think of a drag car race. The track is only ¼ of a mile long, or 1320 feet. The dragster crosses the finish line in six seconds. How fast was the dragster going when it crossed the finish line? The dragster traveled 1320 feet in 6 seconds, so the average speed of the dragster is 1320 divided by 6 which equals 220 feet per second, or 150 miles per hour. The following graph represents the dragster’s position function as the red curve. The position function for the dragster is 36 2/3 x^2. The green line is the secant line connecting the dragster’s starting point and end point. The slope of this secant line is the average speed of the dragster, 220 feet per second, or 150 miles per hour.
Watkins, James. An Introduction to Mechanics of Human Movement. MTP Press Limited. Lancaster, England. 1983.
In chapter 14, we will analyse motion of a particle using the concepts of work and energy. The resulting equation will be useful for solving problems that involve force, velocity, and displacement. Before we do this, we must first define the work of a force. Force, F will do work on a particle only when the particle undergoes a displacement in the direction of the force.
Finite difference scheme can be categorized and formulated in Taylor series expansions. When a function and its derivatives are single valued, finite and the continuous functions then the Taylor series expansion for function may be written at particular point as:
In doing this, let us consider that freely falling objects moves in a vertical direction that is, along the y-axis. instead of using Δx, we will use Δy.
There are many reasons why Algebra matters in life. One reason that comes to mind is from an early age, your understanding and success in algebra can help build math confidence, notable achievements in high school coursework and college readiness, and more importantly help predict one’s salary earnings on so many levels. As one would know that nearly all sports statistics are produced using algebraic equations. Average points per game are used to determine the Most Valuable Player. Winning percentages are used to determine top rankings. These are calculated with concepts learned in Algebra. Formulas are a part of our lives, whether we drive a car and need to calculate the distance, or need to work out our food volume intake for dieting; algebraic formulas are used every day without us even realizing it.
The argument in this paper that even though the onus of the discovery of calculus lies with Isaac Newton, the credit goes to Leibniz for the simple fact that he was the one who published his works first. Appending to this is the fact that the calculus wars that ensue was merely and egotistic battle between humans succumbing to their bare primal instincts. To commence, a brief historical explanation must be given about both individuals prior to stating their cases.
I am currently taking Maths, Chemistry and Biology to Advanced Higher, all of which are challenging and stimulating subjects. They have all influenced me in different ways and were integral in my decision to pursue a degree in Chemical Engineering. They have also helped me obtain a foundation of core skills and extended knowledge to hopefully prepare me undertaking my desired degree. There is no doubt that Maths and chemistry have helped fine tune and advance my problem solving skills and think in a more logical manner, all of which I believe to be essential for this degree.
In my previous studies, I have covered all the four branches of mathematics syllabus and this has made me to develop a strong interest in pure mathematics and most importantly, a very strong interest in calculus.
Ever wonder how scientists figure out how long it takes for the radiation from a nuclear weapon to decay? This dilemma can be solved by calculus, which helps determine the rate of decay of the radioactive material. Calculus can aid people in many everyday situations, such as deciding how much fencing is needed to encompass a designated area. Finding how gravity affects certain objects is how calculus aids people who study Physics. Mechanics find calculus useful to determine rates of flow of fluids in a car. Numerous developments in mathematics by Ancient Greeks to Europeans led to the discovery of integral calculus, which is still expanding. The first mathematicians came from Egypt, where they discovered the rule for the volume of a pyramid and approximation of the area of a circle. Later, Greeks made tremendous discoveries. Archimedes extended the method of inscribed and circumscribed figures by means of heuristic, which are rules that are specific to a given problem and can therefore help guide the search. These arguments involved parallel slices of figures and the laws of the lever, the idea of a surface as made up of lines. Finding areas and volumes of figures by using conic section (a circle, point, hyperbola, etc.) and weighing infinitely thin slices of figures, an idea used in integral calculus today was also a discovery of Archimedes. One of Archimedes's major crucial discoveries for integral calculus was a limit that allows the "slices" of a figure to be infinitely thin. Another Greek, Euclid, developed ideas supporting the theory of calculus, but the logic basis was not sustained since infinity and continuity weren't established yet (Boyer 47). His one mistake in finding a definite integral was that it is not found by the sums of an infinite number of points, lines, or surfaces but by the limit of an infinite sequence (Boyer 47). These early discoveries aided Newton and Leibniz in the development of calculus. In the 17th century, people from all over Europe made numerous mathematics discoveries in the integral calculus field. Johannes Kepler "anticipat(ed) results found… in the integral calculus" (Boyer 109) with his summations. For instance, in his Astronomia nova, he formed a summation similar to integral calculus dealing with sine and cosine. F. B. Cavalieri expanded on Johannes Kepler's work on measuring volumes. Also, he "investigate[d] areas under the curve" ("Calculus (mathematics)") with what he called "indivisible magnitudes.
...d a better understanding of differentiation, I have had several of my students tell me that I am the best math teacher they have ever had. They express their happiness by telling me that I teach math in a way they understand. They state, “You do not stand in front of the classroom and explain how to do the problem, give us homework, and move on to the next topic”. I take pride in this. I try very hard to help each of my students understand the necessary standards so when they leave my room, they are able to take a real-world problem and find solutions to them.