If this thing over here is greater than 0, then the absolute value sign is irrelevant. What are these two values? Step Functions — In this section we introduce the step or Heaviside function. Remember, up here we said, what are the x's that are exactly 10 away from positive 5?

Mechanical Vibrations — In this section we will examine mechanical vibrations. This is essentially how much of an error did we make, right? Negative 18 minus 1 is negative So these are the two solutions to the equation.

Sometimes questions in class will lead down paths that are not covered here. Welcome to She Loves Math! Admittedly, it would take a calculator to determine just what those numbers are, but they are numbers and so we can do the same thing here.

We also give a nice relationship between Heaviside and Dirac Delta functions. Both with be solutions provided we solved the two equations correctly. Check the answers — they work! The Heat Equation — In this section we will do a partial derivation of the heat equation that can be solved to give the temperature in a one dimensional bar of length L.

Ask the student to consider these two solutions in the context of the problem to see if each fits the condition given in the problem i.

Systems of Differential Equations - In this chapter we will look at solving systems of differential equations. We will solve differential equations that involve Heaviside and Dirac Delta functions. So both of these x values satisfy the equation. In addition, we will see that the main difficulty in the higher order cases is simply finding all the roots of the characteristic polynomial.

What value can we substitute for x to get an answer of 4? Repeated Eigenvalues — In this section we will solve systems of two linear differential equations in which the eigenvalues are real repeated double in this case numbers.

We will also define the Wronskian and show how it can be used to determine if a pair of solutions are a fundamental set of solutions.

Consider the following number line. Nonhomogeneous Differential Equations — In this section we will discuss the basics of solving nonhomogeneous differential equations.

Now we don't care if it's above or below, we just care about the absolute distance fromor the absolute value of that difference, so we took the absolute value. Plugging in the corresponding value for each variable and then evaluating the expression we get: I like to then make the expression on the right hand side without the variables both positive and negative and split the equation that way.

Once we have the eigenvalues for a matrix we also show how to find the corresponding eigenvalues for the matrix. This is commonly referred to as taking the logarithm of both sides.

Both sides of the equation contain absolute values and so the only way the two sides are equal will be if the two quantities inside the absolute value bars are equal or equal but with opposite signs.

If the answer to an absolute value equation is negative, then the answer is the empty set. The second portion of the book introduces trigonometry. If, say, w wasminus is going to be negative 1.

If x minus 5 evaluated to negative 10, when you take the absolute value of it, you would get 10 again. We will also work several examples finding the Fourier Series for a function. Here is a listing and brief description of the material that is in this set of notes. Any errors reported will be cataloged here.

This will include deriving a second linearly independent solution that we will need to form the general solution to the system. So you kind of have this v-shaped function, this v-shaped graph, which is indicative of an absolute value function.

In this case, we have to separate in four cases, just to be sure we cover all the possibilities. Exact Equations — In this section we will discuss identifying and solving exact differential equations.When you take the absolute value of a number, the result is always positive, even if the number itself is negative.

For a random number x, both the following equations are true: |-x| = x and |x| = x. This means that any equation that has an absolute value in it has two possible solutions. Section Solving Exponential Equations.

Now that we’ve seen the definitions of exponential and logarithm functions we need to start thinking about how to solve equations involving them. A more dubious reason is that the author is writing about Bat Durston, that is, they are being lazy by writing a space dfaduke.comns are set in the wild west, the corresponding location in science fiction is an interstellar colony.

Algebra 1 Here is a list of all of the skills students learn in Algebra 1! These skills are organized into categories, and you can move your mouse over any skill name to preview the skill. If you're behind a web filter, please make sure that the domains *dfaduke.com and *dfaduke.com are unblocked.

Precalculus: An Investigation of Functions (2nd Ed) David Lippman and Melonie Rasmussen. IMPORTANT NOTE: This page contains details on the current, second edition of the dfaduke.com you are looking for the original first edition (black cover), please go here.

Precalculus: An Investigation of Functions is a free, open textbook covering a two-quarter pre-calculus sequence including trigonometry.

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