Functions of several variables

Introduction:

Functions of Two Variable:

We will be working with the functions of two variables. These functions will help us in understanding how a function can be used with two or more variables. For example, if we wish to take the distance from one point to another point, then it is quite simple that we just need to use two or more variables as given below:

distance = 0;

dist(p1,p2)

This function can be applied on any two points as long as there are no restrictions in the position of these points and also no restrictions in the direction in which these points are pointing. This type of equation is called a “formal” equation, which means that this type of equation can be solved using a method such as graphing or solving.

Functions of More than two variables:

The functions of more than two variables are quite similar to the functions of two variables but there are a couple of differences. The difference lies in the syntax and how it is used. Here, instead of a function as we had with two variables, we have to use sqrt(x2 + y2). For example:

distance = 0;

dist(p1,p2)

This type of equation is called an “informal” equation because it can be solved using different methods such as graphing or solving. This type of equation cannot be solved using the method that we have learned previously because this method only works when there is a restriction on one or more variables in the formula. This equation is not a definite one because of the “sqrt” function.

Functions of more than three variables:

The functions of more than three variables are quite similar to the functions of more than two variables but there are a couple of differences that we need to keep in mind here. In addition, the equation should obey some restrictions such as if x is greater than 0 then y should be less than 1 and so forth. Again, the equation should be an informal one. These equations cannot always be solved using the method that we have learned previously, because this method only works when there is a restriction on one or more variables in the formula. This type of equation is thus not a definite one.

Level Curves:

 A level curve is a curve that passes through all the points, which are at the same level. This type of level curve is called a “plane”, which means that all these planes are describing a region or an area.

These types of level curves are quite helpful in reducing the number of calculations that we have to do. For example, if there is an area which has a rectangular shape and we want to find all the points that have the same height (all points that are at the same level), then this type of calculation can be done easily. This is because when we draw a plane at a certain point, it will cut that entire rectangular area but if we do not use the level curve, then this would not be the case. This type of calculation, especially when it comes to areas with rectangles, is called finding “all intersections” or finding “all meeting points”.

Curves of Constant Level:

A curve of constant level is a curve that passes through all the points, which are at the same level. This type of constant level is called a “plane”, which means that all these planes are describing a region or an area.

To be able to solve quadratic equations, we must first identify the degrees of the largest and smallest variable.

Degrees of Variables:

A degree is a number associated with a variable that relates to the highest power to which that variable can be raised. For example, if we were to take two variables, then we would say that it has two degrees or values. Alternatively, if we were to take one variable as given below:

x = 2

then it would have a degree of 1 because this expression can only be raised by multiplying it by itself.

Degrees of Quadratic Equations:

A degree is a number associated with a variable that relates to the highest power to which that variable can be raised. For example, if we were to take two variables, then we would say that it has two degrees or values. Alternatively, if we were to take one variable as given below:

x = 21/4

then it would have a degree of 2 because this expression can only be raised by multiplying it by itself. We will start solving problems with quadratic equations by taking the first step in solving such an equation. This step includes identifying the variable degrees and finding the value of these degrees.

Key Equations:

Vertical trace: f(a,y)=z  for  x=a  or  f(x,b)=z  for  y=b 

Level surface of a function of three variables: f(x,y,z)=c

Conclusion:

Functions are the most important concepts that we need to know especially when it comes to high school mathematics, because they greatly help us in coming up with formulas and equations that we can use in solving different types of problems. Here, we have learned how to draw a function and how to graph functions. We were also introduced to some key concepts such as domain, range, graph of the function and finding intersections. On top of all this, we were also given a short introductory lecture on how functions can be used in solving different types of problems.