Category: Calculus

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.…

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.…

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…

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…

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…

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…

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…

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…

The next step is to consider powers of the independent variable x. A term of the form axm is called a monomial of degree m. Summing monomials, we get a polynomial function: Here n is the degree of the polynomial. A few polynomial functions are shown in Fig. 2.8. A quick glance at the three plots suggests a few observations: We define concepts like local or global minimum…

A linear affine function has the following general form: Figure 2.7 shows a few linear functions. Strictly speaking, only the first function is linear. A function f is linear if the following condition holds: Fig. 2.7 Graphs of linear (affine) functions. for arbitrary numbers αi and xi, i = 1, 2. However, this holds only when the coefficient q in (2.4) is zero. To see…