4. Asymptotes

If a function f gets close to a certain number L when x gets larger and larger, then we say that the limit as x goes to infinity is L and we write: . Likewise, if f gets close to L when x gets smaller and smaller, then the limit as x goes to negative infinity is L and we write: . In both cases, the line y = L is a horizontal asymptote of f.

If a function gets larger and larger as x gets close to a number a, then it “goes to infinity” and we write: . The line x = a is a vertical asymptote of f. Similarly, if gets smaller and smaller as x gets close to a, then it “goes to negative infinity” and we write: . Again, the line x = a is a vertical asymptote of f.

An important result:

If then . This is because one over a very large positive number and one over a huge negative number are both close to 0.

Slant asymptotes have an equation y = kx + q and their position is arbitrary except vertical. In order a straight line y = kx + q be an asymptote, the coefficients k and q must satisfy at least one pair of the following conditions

and (k and b are numbers).

Naturally these limits must be finite real numbers. A certain function can have maximally two inclined asymptotes.

5. General Scheme for the Investigation of the Graph of a Function

To analyze the function and graph it, we must:

1. Determine the domain;

2. Determine the parity (evenness);

3. Determine the periodicity or non-periodicity of a function;

4. Find the points of intersection of a function and the axes of coordinates;

5. Treat a function for the continuity. Find the points of discontinuity and define the type of discontinuity;

6. Treat a function for the min and max;

7. Find the intervals of concavity and convexity, the inflection;

8. Find the asymptotes of a curve;

9. Construct the graph based on the below-mentioned data.

LECTURE 7

Functions of several variables. Range of definition. Quotient derivatives. Full differential. Extremum of several variables. Necessary and sufficient conditions for the existence of an extremum

1. Concept of functions of several variables

Definition of functions of several variables

We will be studying functions of several variables, say . It is often convenient to organize this list of input variables into a vector x . When n is two or three, we usually dispense with the subscripts and write x = or x = .

For example, consider the function f from to defined by

.

With x = , we can write this as

.

As we shall see, sometimes it is very helpful to think of the input variables as united into a single vector variable x, while other times it is helpful to think of them individually and separately.

We will also be considering functions from to . These take vector variables as input, and return vector variables as output. For example, consider the function F from to given by

.

Introducing the functions and , and with f(x; y) defined as in , we can rewrite as

.

Often, the questions we ask about F(x) can be answered by considering the functions f, g and h one at a time.

Graph of functions of two variables

The graph of a function of two variables, f(x, y), is the set of all points (x, y, z) such that z = f(x, y). In general, the graph of a function of two variables is a surface in 3-space.

Example 1: Describe and graph the function f(x, y) = xy.

Solution:

We begin by making a table of values. Then we attempt to plot those points in 3-space and connect the points. As we include more and more points, our graph becomes the desired surface. We see that if x or y is 0, the function is 0. As x and y both increase, the values of the function get larger. For a fixed value of either x or y, the function looks like a line. Using this information, we can attempt to make a sketch of the function.

Figure 1: Graph of f(x, y) = xy

We remark that as x and y are both positive or both negative, the function values are getting larger and positive. When one is positive and the other is negative, the function values are getting larger and negative. This is illustrated by the slants of the surface.