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calculus, branch of mathematics concerned with the calculation of instantaneous rates of change (differential calculus) and the summation of infinitely many small factors to determine some whole (integral calculus). Two mathematicians, Isaac Newton of England and Gottfried Wilhelm Leibniz of Germany, share credit for having independently developed the calculus in the 17th century. Calculus is now the basic entry point for anyone wishing to study physics, chemistry, biology, economics, finance, or actuarial science. Calculus makes it possible to solve problems as diverse as tracking the position of a space shuttle or predicting the pressure building up behind a dam as the water rises. Computers have become a valuable tool for solving calculus problems that were once considered impossibly difficult.
Calculating curves and areas under curves
The roots of calculus lie in some of the oldest geometry problems on record. The Egyptian Rhind papyrus (c. 1650 bce) gives rules for finding the area of a circle and the volume of a truncated pyramid. Ancient Greek geometers investigated finding tangents to curves, the centre of gravity of plane and solid figures, and the volumes of objects formed by revolving various curves about a fixed axis.
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Numbers and Mathematics
By 1635 the Italian mathematician Bonaventura Cavalieri had supplemented the rigorous tools of Greek geometry with heuristic methods that used the idea of infinitely small segments of lines, areas, and volumes. In 1637 the French mathematician-philosopher René Descartes published his invention of analytic geometry for giving algebraic descriptions of geometric figures. Descartes’s method, in combination with an ancient idea of curves being generated by a moving point, allowed mathematicians such as Newton to describe motion algebraically. Suddenly geometers could go beyond the single cases and ad hoc methods of previous times. They could see patterns of results, and so conjecture new results, that the older geometric language had obscured.
For example, the Greek geometer Archimedes (287–212/211 bce) discovered as an isolated result that the area of a segment of a parabola is equal to a certain triangle. But with algebraic notation, in which a parabola is written as y=x2, Cavalieri and other geometers soon noted that the area between this curve and the x-axis from 0 to a is a3/3 and that a similar rule holds for the curve y=x3—namely, that the corresponding area is a4/4. From here it was not difficult for them to guess that the general formula for the area under a curve y=xn is an+1/(n+1).
Calculating velocities and slopes
The problem of finding tangents to curves was closely related to an important problem that arose from the Italian scientist Galileo Galilei’s investigations of motion, that of finding the velocity at any instant of a particle moving according to some law. Galileo established that in t seconds a freely falling body falls a distance gt2/2, where g is a constant (later interpreted by Newton as the gravitational constant). With the definition of average velocity as the distance per time, the body’s average velocity over an interval from t to t+h is given by the expression [g(t+h)2/2−gt2/2]/h. This simplifies to gt+gh/2 and is called the difference quotient of the function gt2/2. As h approaches 0, this formula approaches gt, which is interpreted as the instantaneous velocity of a falling body at time t.
This expression for motion is identical to that obtained for the slope of the tangent to the parabola f(t)= y=gt2/2 at the point t. In this geometric context, the expression gt+gh/2 (or its equivalent [f(t+h)−f(t)]/h) denotes the slope of a secant line connecting the point (t,f(t)) to the nearby point (t+h,f(t+h)) (see figure). In the limit, with smaller and smaller intervals h, the secant line approaches the tangent line and its slope at the point t.
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Thus, the difference quotient can be interpreted as instantaneous velocity or as the slope of a tangent to a curve. It was the calculus that established this deep connection between geometry and physics—in the process transforming physics and giving a new impetus to the study of geometry.
Differentiation and integration
Independently, Newton and Leibniz established simple rules for finding the formula for the slope of the tangent to a curve at any point on it, given only a formula for the curve. The rate of change of a function f (denoted by f′) is known as its derivative. Finding the formula of the derivative function is called differentiation, and the rules for doing so form the basis of differential calculus. Depending on the context, derivatives may be interpreted as slopes of tangent lines, velocities of moving particles, or other quantities, and therein lies the great power of the differential calculus.
An important application of differential calculus is graphing a curve given its equation y=f(x). This involves, in particular, finding local maximum and minimum points on the graph, as well as changes in inflection (convex to concave, or vice versa). When examining a function used in a mathematical model, such geometric notions have physical interpretations that allow a scientist or engineer to quickly gain a feeling for the behaviour of a physical system.
The other great discovery of Newton and Leibniz was that finding the derivatives of functions was, in a precise sense, the inverse of the problem of finding areas under curves—a principle now known as the fundamental theorem of calculus. Specifically, Newton discovered that if there exists a function F(t) that denotes the area under the curve y=f(x) from, say, 0 to t, then this function’s derivative will equal the original curve over that interval, F′(t)=f(t). Hence, to find the area under the curve y=x2 from 0 to t, it is enough to find a function F so that F′(t)=t2. The differential calculus shows that the most general such function is x3/3+C, where C is an arbitrary constant. This is called the (indefinite) integral of the function y=x2, and it is written as ∫x2dx. The initial symbol ∫ is an elongated S, which stands for sum, and dx indicates an infinitely small increment of the variable, or axis, over which the function is being summed. Leibniz introduced this because he thought of integration as finding the area under a curve by a summation of the areas of infinitely many infinitesimally thin rectangles between the x-axis and the curve. Newton and Leibniz discovered that integrating f(x) is equivalent to solving a differential equation—i.e., finding a function F(t) so that F′(t)=f(t). In physical terms, solving this equation can be interpreted as finding the distance F(t) traveled by an object whose velocity has a given expression f(t).
The branch of the calculus concerned with calculating integrals is the integral calculus, and among its many applications are finding work done by physical systems and calculating pressure behind a dam at a given depth.
John L. BerggrenI am an expert in mathematics, particularly in the field of calculus. My depth of knowledge extends to the historical development of calculus, its key concepts, and its applications across various scientific disciplines. Allow me to demonstrate my expertise by delving into the concepts presented in the article.
Calculus Overview: Calculus is a branch of mathematics focused on two main pillars—differential calculus and integral calculus. These branches were independently developed by Isaac Newton of England and Gottfried Wilhelm Leibniz of Germany in the 17th century. The discipline is essential for understanding instantaneous rates of change (differential calculus) and the summation of infinitely small factors to determine a whole (integral calculus).
Key Figures:
- Isaac Newton: Developed calculus independently, contributed to both differential and integral calculus.
- Gottfried Wilhelm Leibniz: Independently developed calculus, contributing to notation and foundational principles.
- John Wallis: A mathematician not explicitly mentioned in the provided text but made significant contributions to the development of calculus in the 17th century.
- Augustin-Louis Cauchy: A prominent mathematician whose work, though post-17th century, greatly influenced calculus.
Fundamental Theorem of Calculus: The fundamental theorem of calculus, a key discovery by Newton and Leibniz, establishes a deep connection between differentiation and integration. If there exists a function F(t) denoting the area under the curve y=f(x) from 0 to t, then the derivative of this function (F'(t)) equals the original curve over that interval (F'(t) = f(t)).
Differential Calculus: Differential calculus involves finding the rate of change of a function, known as its derivative (denoted by f′). This process, called differentiation, has various interpretations, such as slopes of tangent lines or velocities of moving particles. It plays a crucial role in graphing curves and understanding the behavior of physical systems.
Integral Calculus: Integral calculus focuses on finding the integral or antiderivative of a function. The indefinite integral (∫) is represented by an elongated S and involves summing infinitesimally small increments of the variable over which the function is integrated. Newton and Leibniz discovered that integrating a function is equivalent to solving a differential equation, with applications ranging from finding areas under curves to calculating work done by physical systems.
Applications of Calculus: Calculus finds applications in diverse fields such as physics, chemistry, biology, economics, finance, and actuarial science. It enables the solution of problems like tracking the position of a space shuttle, predicting dam pressure, and understanding the behavior of physical systems.
Evolution of Calculus: The roots of calculus trace back to ancient geometry problems, with significant advancements by mathematicians like Cavalieri and Descartes. The use of algebraic notation, coupled with the concept of infinitely small segments, allowed mathematicians to generalize formulas and establish patterns, paving the way for the development of calculus.
In summary, calculus is a foundational and versatile mathematical discipline that has played a transformative role in understanding the physical world and solving complex problems across various domains.