The Mathematics of Financial Modelingand Investment Management

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140 The Mathematics of Financial Modeling and Investment Management

in general, integrals can be used to represent cumulated quantities such

as cumulated gains.

■ Integrals and derivatives. The fundamental theorem of calculus proves

that integrals and derivatives are inverse operations, insofar as the

derivative of the integral of a function returns the function.

■ The derivative of the product of a constant and a function is the prod-

uct of the constant and the derivative of the function.

■ The integral of the product of a constant and a function is the product

of the constant and the integral of the function.

■ The derivative and the integral of a sum of functions is the sum of

derivatives or integrals.

■ Derivation and integration are linear operations.

■ The derivative of a product of functions is the derivative of the first

function times the second plus the first function times the derivative of

the second.

■ The derivative of a function of function is the product of outer function

with respect to the inner function times the derivative of the inner func-

tion.

■ A derivative of order n of a function is defined as the function that

results from applying the operation of derivation n times.

■ A function that is differentiable to any order at a given point a can be

represented as a series of the powers of (x – a) times the n-th derivative

at a times the reciprocal of n!; this expansion is called a Taylor series

expansion.

■ Taylor series truncated to the first or second terms are called first and

second order approximations, respectively.

■ Laplace and Fourier transforms of a function are the integral of that

function times an exponential.

■ Laplace and Fourier transforms are useful because they transform dif-

ferentiation and integration into algebraic operations, thereby provid-

ing a method for solving linear differential equations.

■ Differentiation and integration can be extended to functions of more

than one variable.

■ A function of n variables has n first derivatives, n-square second deriv-

atives and so on.