The process ofdifferentiation can be applied several times in succession, leading in particular to thesecond derivativef″ of thefunctionf, which is just thederivative of the derivativef′. The second derivative often has a useful physical interpretation. For example, iff(t) is the position of an object at timet, thenf′(t) is its speed at timet andf″(t) is its acceleration at timet.Newton’s laws of motion state that the acceleration of an object is proportional to the total force acting on it; so second derivatives are of central importance indynamics. The second derivative is also useful for graphing functions, because it can quickly determine whether eachcritical point,c, corresponds to a local maximum (f″(c) < 0), a local minimum (f″(c) > 0), or a change in concavity (f″(c) = 0). Third derivatives occur in such concepts as curvature; and even fourth derivatives have their uses, notably in elasticity. Thenth derivative off(x) is denoted byf⁽ⁿ⁾(x) ordⁿf/dxⁿ and has important applications inpower series.

Aninfinite series of the forma₀ +a₁x +a₂x² +⋯, wherex and thea_j are real numbers, is called a power series. Thea_j are the coefficients. The series has alegitimate meaning, provided the seriesconverges. In general, there exists areal numberR such that the series converges when −R <x <R but diverges ifx < −R orx >R. The range of values −R <x <R is called the interval of convergence. The behaviour of the series atx =R orx = −R is more delicate and depends on the coefficients. IfR = 0 the series has little utility, but whenR > 0 the sum of theinfinite series defines a functionf(x). Any functionf that can be defined by a convergent power series is said to be real-analytic.

The coefficients of the power series of a real-analytic function can be expressed in terms of derivatives of that function. For values ofx inside the interval of convergence, the series can bedifferentiated term by term; that is,f′(x) =a₁ + 2a₂x + 3a₃x² +⋯, and this series also converges. Repeating this procedure and then settingx = 0 in the resulting expressions shows thata₀ =f(0),a₁ =f′(0),a₂ =f″(0)/2,a₃ =f′′′(0)/6, and, in general,a_j =f^(j)(0)/j!. That is, within the interval of convergence off,

This expression is the Maclaurin series off, otherwise known as theTaylor series off about 0. A slight generalization leads to the Taylor series off about a general valuex: All these series are meaningful only if they converge.

For example, it can be shown thate^x = 1 +x +x²/2! +x³/3! +⋯,sin (x) =x −x³/3! +x⁵/5! − ⋯,cos (x) = 1 −x²/2! +x⁴/4! − ⋯, and these series converge for allx.

The fundamental theorem of calculus

The process of calculatingintegrals is called integration. Integration is related to differentiation by thefundamental theorem of calculus, which states that (subject to the mild technical condition that the function be continuous) the derivative of the integral is the original function. In symbols, the fundamentaltheorem is stated asd/dt(Integral on the interval [a,t ] of∫atf(u)du) =f(t).

The reasoning behind this theorem (seefigure) can be demonstrated in a logical progression, as follows: LetA(t) be the integral off froma tot. Then the derivative ofA(t) is very closely approximated by the quotient (A(t +h) −A(t))/h. This is 1/h times the area under the graph off betweent andt +h. For continuous functionsf the value off(t), fort in the interval, changes only slightly, so it must be very close tof(t). The area is therefore close tohf(t), so the quotient is close tohf(t)/h =f(t). Taking the limit ash tends to zero, the result follows.

Antidifferentiation

Strictmathematical logic aside, the importance of the fundamental theorem of calculus is that it allows one to find areas by antidifferentiation—the reverse process to differentiation. Tointegrate a given functionf, just find a functionF whose derivativeF′ is equal tof. Then the value of the integral is the differenceF(b) −F(a) between the value ofF at the two limits. For example, since the derivative oft³ is 3t², take the antiderivative of 3t² to bet³. The area of the region enclosed by the graph of the functiony = 3t², the horizontal axis, and the vertical linest = 1 andt = 2, for example, is given by the integralIntegral on the interval [1,2 ] of∫12 3t²dt. By the fundamental theorem of calculus, this is the difference between the values oft³ whent = 2 andt = 1; that is, 2³ − 1³ = 7.

All the basic techniques ofcalculus for finding integrals work in this manner. They provide arepertoire of tricks for finding a function whose derivative is a given function. Most of what is taught in schools and colleges under the namecalculus consists of rules for calculating the derivatives and integrals of functions of various forms and of particular applications of those techniques, such as finding thelength of a curve or thesurface area of a solid of revolution.

Table 2 lists the integrals of a small number of elementary functions. In the table, the symbolc denotes an arbitraryconstant. (Because the derivative of a constant is zero, the antiderivative of a function is not unique: adding a constant makes no difference. When anintegral is evaluated between two specific limits, this constant is subtracted from itself and thus cancels out. In the indefinite integral, another name for the antiderivative, the constant must be included.)

TheRiemann integral

The task of analysis is to provide not a computational method but a sound logical foundation for limiting processes. Oddly enough, when it comes to formalizing the integral, the most difficult part is to define the termarea. It is easy to define the area of a shape whose edges are straight; for example, the area of a rectangle is just the product of the lengths of two adjoining sides. But the area of a shape with curved edges can be moreelusive. The answer, again, is toset up a suitable limiting process that approximates the desired area with simpler regions whose areas can be calculated.

The first successful general method for accomplishing this is usually credited to the German mathematicianBernhard Riemann in 1853, although it has manyprecursors (both in ancient Greece and in China). Given some functionf(t), consider the area of the region enclosed by the graph off, the horizontal axis, and the vertical linest =a andt =b. Riemann’s approach is to slice this region into thin vertical strips (see part A of thefigure) and to approximate its area by sums of areas of rectangles, both from the inside (part B of the figure) and from the outside (part C of the figure). If both of these sumsconverge to the same limiting value as the thickness of the slices tends to zero, then their common value is defined to be the Riemann integral off between the limitsa andb. If this limit exists for alla,b, thenf is said to be (Riemann) integrable. Every continuous function is integrable.