Month: February 2023

The direct application of the definition to find the derivative of a function is typically a rather difficult and cumbersome procedure, possibly requiring some intuition. Example 2.19 (Derivative of logarithm and exponential function) One of the most useful results concerning derivatives is that the derivative of the exponential is just the exponential itself: As a first… Continue reading RULES FOR CALCULATING DERIVATIVES

If the derivative of function f at point Xo exists, then we say that the function is differentiable at point x0; if this holds for all points on an interval or domain, the function is differentiable on that interval or domain. If the derivative f′(x) exists at all points x on an interval and the derivative is a continuous function, we say that… Continue reading Continuity and differentiability

Consider a point x0 and the increment ratio of function f at that point: Fig. 2.17 The derivative is the limit of an increment ratio. For a nonlinear function, keeping x0 fixed, this ratio is a function of h. Now consider smaller and smaller steps h, as illustrated in Fig. 2.17. If we let h → 0, we get the “tangent” line to the graph of f at point x0.… Continue reading Definition of the derivative

We have seen that a linear (affine) function f(x) = mx + q has a well-defined slope. Whatever value of the independent variable we consider, the slope of the function is always the same. If we are at point x and we move to point x + h, by any displacement h, the increment ratio14 is: Fig. 2.16 A nonlinear function does not have constant increment ratios.… Continue reading DERIVATIVES

The logarithm arises as the inverse of an exponential function. To further motivate this, let us consider again continuous compounding of interest rates. As we have pointed out, continuous compounding leads to an exponential function that streamlines financial calculations considerably. However, in practice, interest rates are not quoted like this. Typically, interest rates are quoted… Continue reading The logarithm

A function maps an input value x into an output value y = f(x). There are cases in which we want to go the other way around; i.e., given y, we would like to find a value x such that y = f(x). Actually, this is what we do whenever we want to solve an equation. For instance, given a function that evaluates the NPV of… Continue reading INVERSE FUNCTIONS

So far, we have considered linear, polynomial, rational, and exponential functions. From our high school math, we might recall something about trigonometric functions; since we will not use them in the following, we leave them aside. A natural way to build quite complicated, but hopefully useful, functions is function composition. Given functions g and h, we may build the… Continue reading COMPOSITE FUNCTIONS

Before we proceed in our treatment of functions, we should pause a little and discuss a fundamental feature of functions: continuity, or lack thereof. Compare the graphs of polynomial functions in Fig. 2.8 against the graph of the rational function in Fig. 2.9. There is a striking qualitative difference between the two figures; in the first case, if… Continue reading CONTINUOUS FUNCTIONS

Polynomial functions involve powers like xk, where the exponent k is an integer number. We recall some fundamental rules that are quite handy when dealing with powers and should be familiar from high school mathematics: In a monomial function f(x) = αxk, the basis x is the independent variable and the exponent k is a fixed parameter. In exponential functions we reverse their roles and… Continue reading Exponential functions

If P(x) and Q(x) are polynomial functions, the function is a rational function. In other words, a rational function is just a ratio of two polynomials. Unlike linear and polynomial functions, the domain of a rational function need not be the whole real line. We are in trouble when the denominator polynomial is zero, i.e., when Q(x) = 0. Loosely… Continue reading Rational functions