The problem of finding tangents to curves was closely related to an important problem that arose from the Italian scientist Galileo Galilei’s investigation of motion, that of finding the velocity at any instant of a particle moving according to a law. Galileo established that in t seconds a freely falling body falls a distance gt2/2, where g is a constant (later defined by Newton as the gravitational constant). With the definition of average velocity as distance per time, the average velocity of the body over the 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 quant of the function gt2/2. As h approaches 0, this formula approaches gt, which is interpreted as the instantaneous velocity of the 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. In this geometric context, the expression gt + gh/2 (or equivalently [f(t + h) − f(t)]/h) is the slope of the secant line joining the point (t, f(t)). shows the to the nearest 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.
Thus, the difference quantile can be expressed as the instantaneous velocity or as the slope of the tangent to a curve. It was 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 a formula for the slope of a tangent at any point on it, given only one formula for the curve. The rate of change of a function f (denoted by f′) is known as its derivative. Finding the formula for a derivative function is called differentiation, and the principles of doing so form the basis of differential calculus. Depending on the context, derivatives can be interpreted as the slope of tangent lines, the speed of moving particles, or other quantities, and therein lies the great power of differential calculus.
An important application of differential calculus is to graph a curve given its equation y = f(x). This includes, in particular, the detection of local maximum and minimum points on the graph, as well as changes in inflection (concave to concave, or vice versa). When examining a function used in a mathematical model, such geometric concepts have physical interpretations that allow the scientist or engineer to gain an immediate sense of the behavior of the physical system.
Newton and Leibniz’s other great discovery was that finding the derivatives of functions was, in a sense, the inverse of the problem of finding the areas under a curve—a principle now known as the Fundamental Theorem of Calculus. In particular, Newton discovered that if there exists a function F(t) that represents the area under the curve y = f(x), say, from 0 to t, then the derivative of the function is the origin on the interval will be equal to the curve, F′ (t) = f(t). Therefore, to find the area under the curve y = x2 from 0 to t, it is sufficient to find a function F such that F′(t) = t2. Differential calculus shows that the most common such function is x3/3 + C, where C is an arbitrary constant. This is called the (indefinite) integral of the function y = x2, and is written ∫x2dx. The initial symbol ∫ is a long S, which stands for sum, and dx denotes the infinitesimal increment of the variable, or axis, over which the function is being summed. Leibniz introduced it because he thought of integration as finding the area under a curve by summing the areas of infinitely many thin rectangles between the x-axis and the curve. Newton and Leibniz discovered that integrating f(x) is equivalent to solving a differential equation—that is, finding a function F(t) such 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 is expressed as f(t).
Integral calculus is the branch of calculus concerned with calculating integrals, and its many applications include finding the work done by physical systems and calculating the pressure behind a dam at a given depth.
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