Calculus, originally called infinitesimal calculus or “the calculus of infinitesimals”, is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while integral calculus concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.
Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, calculus has widespread uses in science, engineering, and economics.
In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. The word calculus (plural calculi) is a Latin word, meaning originally “small pebble” (this meaning is kept in medicine – see Calculus (medicine)). Because such pebbles were used for counting (or measuring) a distance travelled by transportation devices in use in ancient Rome, the meaning of the word has evolved and today usually means a method of computation. It is therefore used for naming specific methods of calculation and related theories, such as propositional calculus, Ricci calculus, calculus of variations, lambda calculus, and process calculus.
Binomial Theorem: where and are real numbers and are binomial cofficient. and are exponents and is called index. The exponent of decreases from index to zero. The exponent of increases from zero to index. The degree of each terms is equal to its index. The numbers of terms in the expansion is one greater than […]
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What is Homogeneous Function Definition: A function defined by of any number of variables are said to be homogeneous of degree in these variables if multiplication of these variables by any number result in the multiplication of the function by . provide that is in the domain of . Taking the equation becomes . .
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Statement of Mean Value Theorem (MVT) : Let a function be then there exist a point such that MVT Proof: Define a new function by where A is a constant to determined such that The functiom is contineous on and differentiable on . Hence is contineous and differentiable respecctively on and . The function satisfies
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Euler’s Theorem Statement: If is a homogeneous function of of degree , then . Proof: we have ,Therefore . . and . . Thus from (1) and (2), we have . . . Hence proved. Example: If , show that . Solution: we have . . . . . Thus is homogeneous function of of
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What is hyperbola:We have already stated that a conic section is a hyperbola if you can check. Let and be a fixed point and be a line not containing . Also let be a point in the plane and be the perpendicular distance of from .The set of all points such that is called a
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In this article, we will discuss the equation of the circle also the standard equation of a circle in an easy way. The equation of a circle comes in two forms. If the equation of a circle is in standard form, we can easily identify the center of the circle and radius The set of
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What is Parabola: If the intersecting plane is parallel to the generator of the cone but cut only one nappe is called parabola. We have already stated that a conic section is a parabola if . Here e is the eccentricity. We shall first derive an equation of a parabola in standard form and study
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Negation: If p is any proposition its negation is denoted by ~p, read‘not p‘. It follows from this definition that if p is true, ~p is false and ifp is false, ~p is true. The adjoining table, called the truth table, gives thepossible truth- values of p and ~p. Conjunction: The conjunction of two statements
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A compound statement of the form if p then q , also written p implies q , is called aconditional or an implication, p is called the antecedent or hypothesis and q is called theconsequent or the conclusion.A conditional is regarded as false only when the antecedent is true and consequent isfalse. In all other
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Just as operations of addition, subtraction etc., are performed on numbers, theoperations of unions, intersection etc., are performed on sets. We are already familiar withthem. A review of the main rules is given below: –Union of two sets: The Union of two sets and , denoted by , is the set of all elements,which belong
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