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We offer various automated calculus tutorials. We cover the following topics: limits, continuity, derivatives and antiderivatives, applications of integrals. Our tutorials are automated and interactive, feel free to use our application when preparing for exams or doing your homework assignments.
Limits
The concept of a limit or limiting process, essential to the understanding of calculus, has been around for thousands of years. In fact, early mathematicians used a limiting process to obtain better and better approximations of areas of circles. Yet, the formal definition of a limit—as we know and understand it today—did not appear until the late 19th century. We therefore begin our quest to understand limits, as our mathematical ancestors did, by using an intuitive approach. At the end of this chapter, armed with a conceptual understanding of limits, we examine the formal definition of a limit.
Definition of Limit
The limit of a function at a point is the value that the function approaches as the input approaches that point.
Suppose f(x) is defined when x is near the number a. (This means that f is defined on some open interval that contains a, except possibly at a itself.) Then we write
and say "the limit of f(x), as x approaches a, equals L" if we can make the values of f(x) arbitrarily close to L (as close to L as we like) x to be sufficiently close to a (on either side of a) but not equal to a.
Derivatives
When we credit Newton and Leibniz with developing calculus, we are really referring to the fact that Newton and Leibniz were the first to understand the relationship between the derivative and the integral. Both mathematicians benefited from the work of predecessors, such as Barrow, Fermat, and Cavalieri. The initial relationship between the two mathematicians appears to have been amicable; however, in later years a bitter controversy erupted over whose work took precedence. Although it seems likely that Newton did, indeed, arrive at the ideas behind calculus first, we are indebted to Leibniz for the notation that we commonly use today.
Derivatives
Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time is varied. The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time is varied. The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).