Detailed notes and examples are provided and there are extension activities for students to complete, together with learning outcomes and assessment criteria. In this resource, students explore how to build a mathematical model of liquid draining through a tank and how to use the model to determine the time required for a tank to completely drain. In this resource, students explore how calculations for displacement, velocity and acceleration, caused during loading, are used to ensure that they are not so large as to adversely affect the performance of a dump truck.
Many machines compress or expand gas or fluid as part of their working design. Examples include a simple bicycle pump, a refrigerator and an internal combustion engine. To compress gas energy needs to be expended to reduce its volume. When gas is allowed to expand energy is released. A Japanese mathematician, Kowa Seki, lived at the same time as Leibniz and Newton and independently elaborated some of the fundamental principles of integral calculus.
The derivative measures the sensitivity of one variable to small changes in another variable. Consider the formula:. The speed of a car, as measured by the speedometer, is the derivative of the car's displacement as a function of time. Calculus is a mathematical tool for dealing with this complex but natural and familiar situation. Differential calculus determines the instantaneous speed at any given specific instant in time, not just average speed during an interval of time.
In mathematical language, this is an example of taking a limit. More formally, differential calculus defines the instantaneous rate of change the derivative of a mathematical function's value, with respect to changes of the variable.
The derivative is defined as a limit of a difference quotient. The derivative of a function, if it exists, gives information about small pieces of its graph. It is useful for finding the maxima and minima of a function — because at those points the graph is flat i. Another application of differential calculus is Newton's method, an algorithm to find zeroes of a function by approximating the graph of the function by tangent lines. Differential calculus has been applied to many questions that are not first formulated in the language of calculus.
The derivative lies at the heart of the physical sciences. Maxwell's theory of electromagnetism and Einstein's theory of gravity general relativity are also expressed in the language of differential calculus, as is the basic theory of electrical circuits and much of engineering. It is also applied to problems in biology, economics, and many other areas. The definite integral evaluates the cumulative effect of many small changes in a quantity. The simplest instance is the formula. The distance moved is the cumulative effect of the small distances moved in each of the many seconds the car is on the road.
The calculus is able to deal with the natural situation in which the car moves with changing speed. Integral calculus determines the exact distance traveled during an interval of time by creating a series of better and better approximations, called Riemann sums , that approach the exact distance. More formally, we say that the definite integral of a function on an interval is a limit of Riemann sum approximations.
Applications of integral calculus arise whenever the problem is to compute a number that is in principle approximately equal to the sum of the solutions of many, many smaller problems.
The classic geometric application is to area computations. Then I learned that studying calculus and mathematics I studied, five courses of mathematics and four physical allows you to prepare your attitude for the resolution of complex problems.
In my view, it is a kind of prior training, as the athlete begins training with jogging or the singer performs exercises to prepare her voice. Secondly, there is sometimes specific knowledge not offered at the university that you will need when you're at work.
You may need to get a book and learn it yourself. Sometimes that knowledge is based on physics or math, and if you don't have a base it will prove difficult to handle and understand. Finally, engineering has many branches. With these applets, or a spreadsheet, you can apply the tools of calculus with greater ease and flexibility than has been possible before. There are more advanced programs that are often available, such as MAPLE and Mathematica, which allow you to do much more with similar ease.
With them you can deduce the consequences of models of various kinds in a wide variety of contexts. Once you understand calculus they can make its use much easier, but they provide answers given inputs, which does not provide understanding of how they do it. Also, we will put much greater emphasis on modeling systems.
With ideas on modeling and methods for solving the differential equations they lead to, you can achieve the empowerment we have claimed. Okay, probably not. But you might. And also you might be provoked to learn more about the systems you want to study or about mathematics, to improve your chances to do so. Also you might be able to understand the probable consequences of models a little better than you do now.
Also you may get to love the concepts and ideas of calculus. We also describe decimal expansions which describe "real numbers" and examine the notion of countability. We mutter about complex numbers as well. We start with an abstract definition of a function as a set of argument-value pairs and then describe the standard functions.
But what is the exponential function, and what are substitution and inversion? The exponential function is mysteriously defined using calculus: it is the function that is its own derivative, defined to have the value 1 at argument 0.
It turns out, however, to be something you have seen before. And it turns out to bear a close relation to the sine function of trigonometry. Substitution of one function f into another g produces a new function, the function defined to have, at argument x, the value of f at an argument which is the value of g at argument x.
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