A linear regression fits the lineor as Excel prefers to call itto the existing data set. It does so through a technique known as minimizing the sum of the squares of the error terms.
Due to the nature of the mathematics on this site it is best views in landscape mode.
If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. Differential Equations Here are my notes for my differential equations course that I teach here at Lamar University.
Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
Because I wanted to make this a fairly complete set of notes for anyone wanting to learn differential equations have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here.
In general, I try to work problems in class that are different from my notes. However, with Differential Equation many of the problems are difficult to make up on the spur of the moment and so in this class my class work will follow these notes fairly close as far as worked problems go.
With that being said I will, on occasion, work problems off the top of my head when I can to provide more examples than just those in my notes. Sometimes questions in class will lead down paths that are not covered here. You should always talk to someone who was in class on the day you missed and compare these notes to their notes and see what the differences are.
This is somewhat related to the previous three items, but is important enough to merit its own item. Using these notes as a substitute for class is liable to get you in trouble.
As already noted not everything in these notes is covered in class and often material or insights not in these notes is covered in class. Here is a listing and brief description of the material that is in this set of notes.
Basic Concepts - In this chapter we introduce many of the basic concepts and definitions that are encountered in a typical differential equations course. We will also take a look at direction fields and how they can be used to determine some of the behavior of solutions to differential equations.
Definitions — In this section some of the common definitions and concepts in a differential equations course are introduced including order, linear vs.
Direction Fields — In this section we discuss direction fields and how to sketch them. We also investigate how direction fields can be used to determine some information about the solution to a differential equation without actually having the solution.
Final Thoughts — In this section we give a couple of final thoughts on what we will be looking at throughout this course. First Order Differential Equations - In this chapter we will look at several of the standard solution methods for first order differential equations including linear, separable, exact and Bernoulli differential equations.
In addition we model some physical situations with first order differential equations. Linear Equations — In this section we solve linear first order differential equations, i.
We give an in depth overview of the process used to solve this type of differential equation as well as a derivation of the formula needed for the integrating factor used in the solution process. Separable Equations — In this section we solve separable first order differential equations, i.
We will give a derivation of the solution process to this type of differential equation. Exact Equations — In this section we will discuss identifying and solving exact differential equations.
We will develop of a test that can be used to identify exact differential equations and give a detailed explanation of the solution process. We will also do a few more interval of validity problems here as well.
Bernoulli Differential Equations — In this section we solve Bernoulli differential equations, i. This section will also introduce the idea of using a substitution to help us solve differential equations.
Intervals of Validity — In this section we will give an in depth look at intervals of validity as well as an answer to the existence and uniqueness question for first order differential equations.Introduction.
A trendline shows the trend in a data set and is typically associated with regression analysis. Creating a trendline and calculating its coefficients allows for the quantitative analysis of the underlying data and the ability to both interpolate and extrapolate the data for forecast purposes.
Given a few terms of a sequence, we are often asked to find the expression for the nth term of this sequence. While there is a multitude of ways to do this, In this article, we discuss an algorithmic approach which will give the correct answer for any polynomial expression.
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate, x, is x 2 − 4x + torosgazete.com example in three variables is x 3 + 2xyz 2 − yz + 1.
Section The Heat Equation.
Before we get into actually solving partial differential equations and before we even start discussing the method of separation of variables we want to spend a little bit of time talking about the two main partial differential equations that we’ll be solving later on in the chapter.
Polynomial Roots. A root of a polynomial is a number such torosgazete.com fundamental theorem of algebra states that a polynomial of degree has roots, some of which may be degenerate. For example, the roots of the polynomial. Polynomial Graphs and Roots. We learned that a Quadratic Function is a special type of polynomial with degree 2; these have either a cup-up or cup-down shape, depending on whether the leading term (one with the biggest exponent) is positive or negative, respectively.
Think of a polynomial graph of higher degrees (degree at least 3) as quadratic graphs, but with more twists and turns.