![Definite Integral. Given a function f (x) f ( x) that is continuous on the interval [a,b] [ a, b] we divide the interval into n n subintervals of equal width, Δx Δ x, and from each interval choose a point, x∗ i x i ∗. Then the definite integral of f (x) f ( x) from a a to b b is. The definite integral is defined to be exactly the limit .... Linkin park chester](/img/512x512/332943866984.webp)
Nov 16, 2022 · First, when doing a substitution remember that when the substitution is done all the x x ’s in the integral (or whatever variable is being used for that particular integral) should all be substituted away. This includes the x x in the dx d x. After the substitution only u u ’s should be left in the integral. Nov 16, 2022 · We’ll start with a rational expression in the form, f(x) = P(x) Q(x) where both P(x) and Q(x) are polynomials and the degree of P(x) is smaller than the degree of Q(x). Recall that the degree of a polynomial is the largest exponent in the polynomial. Partial fractions can only be done if the degree of the numerator is strictly less than the ... In this section we solve linear first order differential equations, i.e. differential equations in the form y' + p(t) y = g(t). We give an in depth overview of the process used …My Students. If you are one of my students and you are here looking for homework assignments and/or due dates for homework assignments you won't find them here. For a variety of reasons I like to keep the site that contains my notes separate from the pages that are devoted to the classes that I'm actually teaching here at Lamar. Here …A linear equation is any equation that can be written in the form. ax +b = 0 a x + b = 0. where a a and b b are real numbers and x x is a variable. This form is sometimes called the standard form of a linear equation. Note that most linear equations will not start off in this form. Also, the variable may or may not be an x x so don’t get too ...Section 8.1 : Arc Length. In this section we are going to look at computing the arc length of a function. Because it’s easy enough to derive the formulas that we’ll use in this section we will derive one of them and leave the other to you to derive.In this section we are going to start talking about power series. A power series about a, or just power series, is any series that can be written in the form, ∞ ∑ n=0cn(x −a)n ∑ n = 0 ∞ c n ( x − a) n. where a a and cn c n are numbers. The cn c n ’s are often called the coefficients of the series.One of the equations is easy. The guess, \ (\eqref {eq:eq2}\), will need to satisfy the original differential equation, \ (\eqref {eq:eq1}\). So, let’s start taking some derivatives and as we did when we first looked at variation of parameters we’ll make some assumptions along the way that will simplify our work and in the process generate ...First, the always important, rate of change of the function. Although we now have multiple ‘directions’ in which the function can change (unlike in Calculus I). We will also see that partial derivatives give the slope of tangent lines to the traces of the function. Higher Order Partial Derivatives – In the section we will take a look at ...Calculus I. Here are a set of assignment problems for the Calculus I notes. Please note that these problems do not have any solutions available. These are intended mostly for instructors who might want a set of problems to assign for turning in. Having solutions available (or even just final answers) would defeat the purpose the problems.Paul's Online Math Notes. Good self-contained notes for Algebra, Calculus I/II/III, and Ordinary Differential Equations by Professor Dr. Paul Hawkins at Lamar University. The link address is: https://tutorial.math.lamar.edu/.Khan Academy: What is a differential equation? Video - 11:02, Introduction to differential equations and the terms order and linear/nonlinear · Paul's Notes: ...Spherical coordinates consist of the following three quantities. First there is ρ ρ. This is the distance from the origin to the point and we will require ρ ≥ 0 ρ ≥ 0. Next there is θ θ. This is the same angle that we saw in polar/cylindrical coordinates.Nov 16, 2022 · Section 2.4 : Bernoulli Differential Equations. In this section we are going to take a look at differential equations in the form, y′ +p(x)y = q(x)yn y ′ + p ( x) y = q ( x) y n. where p(x) p ( x) and q(x) q ( x) are continuous functions on the interval we’re working on and n n is a real number. Differential equations in this form are ... Nov 16, 2022 · The range of a function is simply the set of all possible values that a function can take. Let’s find the domain and range of a few functions. Example 4 Find the domain and range of each of the following functions. f (x) = 5x −3 f ( x) = 5 x − 3. g(t) = √4 −7t g ( t) = 4 − 7 t. h(x) = −2x2 +12x +5 h ( x) = − 2 x 2 + 12 x + 5. Sep 25, 2018 · Download pdf cheat sheets and tables for algebra, trig, calculus, and Laplace transforms from Pauls Online Notes. The cheat sheets come in full and reduced versions, with common facts, formulas, properties, and errors. Section 16.2 : Line Integrals - Part I. In this section we are now going to introduce a new kind of integral. However, before we do that it is important to note that you will need to remember how to parameterize equations, or put another way, you will need to be able to write down a set of parametric equations for a given curve.The curvature measures how fast a curve is changing direction at a given point. There are several formulas for determining the curvature for a curve. The formal definition of curvature is, κ = ∥∥ ∥d →T ds ∥∥ ∥ κ = ‖ d T → d s ‖. where →T T → is the unit tangent and s s is the arc length.Share. 1K views 3 years ago. Paul's Online Calculus 4-1 Rates of Change example 1 Thank you Professor Paul from http://tutorial.math.lamar.edu/ ...more. ...more.Sep 25, 2018 · Download pdf cheat sheets and tables for algebra, trig, calculus, and Laplace transforms from Pauls Online Notes. The cheat sheets come in full and reduced versions, with common facts, formulas, properties, and errors. Nov 16, 2022 · Table Notes. This list is not a complete listing of Laplace transforms and only contains some of the more commonly used Laplace transforms and formulas. Recall the definition of hyperbolic functions. cosh(t) = et +e−t 2 sinh(t) = et−e−t 2 cosh. . ( t) = e t + e − t 2 sinh. . ( t) = e t − e − t 2. Be careful when using ... Here is the definition of the logarithm function. If b is any number such that b > 0 and b ≠ 1 and x > 0 then, y = logbx is equivalent to by = x. We usually read this as “log base b of x ”. In this definition y = logbx is called the logarithm form and by = x is called the exponential form. Note that the requirement that x > 0 is really a ...Actually they are only tricky until you see how to do them, so don’t get too excited about them. The first one involves integrating a piecewise function. Example 4 Given, f (x) ={6 if x >1 3x2 if x ≤ 1 f ( x) = { 6 if x > 1 3 x 2 if x ≤ 1. Evaluate each of the following integrals. ∫ 22 10 f (x) dx ∫ 10 22 f ( x) d x.Biblical scholars do not agree on the number of epistles that Paul wrote; some think he wrote all 13 epistles that have his name on them, while others think he authored only a few ...Plug the product solution into the partial differential equation, separate variables and introduce a separation constant. This will produce two ordinary differential equations. Plug the product solution into the homogeneous boundary conditions. Note that often it will be better to do this prior to doing the differential equation so we can use ...Pauls Online Math Notes. Home. Welcome to my online math tutorials and notes. The intent of this site is to provide a complete set of free online (and downloadable) notes and/or tutorials for classes that I teach at Lamar University. I've tried to write the notes/tutorials in such a way that they should be accessible to anyone wanting to learn ...In order for something to be a vector space it simply must have an addition and scalar multiplication that meets the above axioms and it doesn't matter how ...The (implicit) solution to an exact differential equation is then. Ψ(x,y) = c (4) (4) Ψ ( x, y) = c. Well, it’s the solution provided we can find Ψ(x,y) Ψ ( x, y) anyway. Therefore, once we have the function we can always just jump straight to (4) (4) to get an implicit solution to our differential equation.Aug 22, 2012 ... https://www.patreon.com/ProfessorLeonard Calculus 1 Lecture 1.1: An Introduction to Limits.Nov 13, 2023 · Show Solution. We can also use the above formulas to convert equations from one coordinate system to the other. Example 2 Convert each of the following into an equation in the given coordinate system. Convert 2x−5x3 = 1 +xy 2 x − 5 x 3 = 1 + x y into polar coordinates. Convert r =−8cosθ r = − 8 cos. . Section 14.5 : Lagrange Multipliers. In the previous section we optimized (i.e. found the absolute extrema) a function on a region that contained its boundary.Finding potential optimal points in the interior of the region isn’t too bad in general, all that we needed to do was find the critical points and plug them into the function.One of the equations is easy. The guess, \ (\eqref {eq:eq2}\), will need to satisfy the original differential equation, \ (\eqref {eq:eq1}\). So, let’s start taking some derivatives and as we did when we first looked at variation of parameters we’ll make some assumptions along the way that will simplify our work and in the process generate ...Section 7.10 : Approximating Definite Integrals. In this chapter we’ve spent quite a bit of time on computing the values of integrals. However, not all integrals can be computed. A perfect example is the …Jul 29, 2016 ... Paul's Online Math Notes. Calculus III (Notes) / Surface Integrals / Stokes' Theorem [Notes] [Practice Problems] [Assignment. Problems].Paul Wahlberg is a chef, actor and brother of actor Mark and actor/musician Donnie Wahlberg. He stars in the reality TV show on A&E entitled “Wahlburgers,” which documents the fami...Oct 9, 2023 ... Welcome to my math notes site. Contained includes this site are the notes (free and downloadable) that I use to teach Algebra, Calculus (I, ...Pauls Online Math Notes Calc 1. We will discuss the interpretation/meaning of a limit, how to. Web calculus i here are the notes for my calculus i course that i ...Share. 1K views 3 years ago. Paul's Online Calculus 4-1 Rates of Change example 1 Thank you Professor Paul from http://tutorial.math.lamar.edu/ ...more. ...more.First, we need a little terminology/notation out of the way. We call the equations that define the change of variables a transformation. Also, we will typically start out with a region, R R, in xy x y -coordinates and transform it into a region in uv u v -coordinates. Example 1 Determine the new region that we get by applying the given ...Oct 9, 2023 · Pauls Online Math Notes. Home. Welcome to my online math tutorials and notes. The intent of this site is to provide a complete set of free online (and downloadable) notes and/or tutorials for classes that I teach at Lamar University. I've tried to write the notes/tutorials in such a way that they should be accessible to anyone wanting to learn ... Powers and Roots. In this section we’re going to take a look at a really nice way of quickly computing integer powers and roots of complex numbers. We’ll start with integer powers of z = reiθ z = r e i θ since they are easy enough. If n n is an integer then, zn =(reiθ)n = rnei nθ (1) (1) z n = ( r e i θ) n = r n e i n θ.Systems of differential equations can be converted to matrix form and this is the form that we usually use in solving systems. Example 3 Convert the following system to matrix form. x′ 1 =4x1 +7x2 x′ 2 =−2x1−5x2 x ′ 1 = 4 x 1 + 7 x 2 x ′ 2 = − 2 x 1 − 5 x 2. Show Solution. Example 4 Convert the systems from Examples 1 and 2 into ...My Students. If you are one of my students and you are here looking for homework assignments and/or due dates for homework assignments you won't find them here. For a variety of reasons I like to keep the site that contains my notes separate from the pages that are devoted to the classes that I'm actually teaching here at Lamar. Here …In this section we solve separable first order differential equations, i.e. differential equations in the form N(y) y' = M(x). We will give a derivation of the solution process to this type of differential equation. We’ll also start looking at finding the interval of validity for the solution to a differential equation.If you have furniture that you no longer need or want, donating it can be a great way to give back to your community and help those in need. St. Vincent de Paul is a well-known cha...Jun 6, 2018 · Chapter 9 : Partial Differential Equations. In this chapter we are going to take a very brief look at one of the more common methods for solving simple partial differential equations. The method we’ll be taking a look at is that of Separation of Variables. We need to make it very clear before we even start this chapter that we are going to be ... Section 15.1 : Double Integrals. Before starting on double integrals let’s do a quick review of the definition of definite integrals for functions of single variables. First, when working with the integral, ∫ b a f (x) dx ∫ a b f ( x) d x. we think of x x ’s as coming from the interval a ≤ x ≤ b a ≤ x ≤ b. For these integrals we ...Paul's Online Notes Home / Calculus I / Applications of Integrals / Area Between Curves. Prev. Section. Notes Practice Problems Assignment Problems. ... In this case it’s pretty easy to see that they will intersect at \(x = 0\) and \(x = 1\) so these are the limits of integration.End of Preview - Want to read all 11 pages? Access Now. Unformatted Attachment Preview. Calculus Cheat Sheet Limits Definitions Precise Definition : We say lim ...First, we need a little terminology/notation out of the way. We call the equations that define the change of variables a transformation. Also, we will typically start out with a region, R R, in xy x y -coordinates and transform it into a region in uv u v -coordinates. Example 1 Determine the new region that we get by applying the given ...This chapter will actually contain more than most text books tend to have when they discuss higher order differential equations. We will definitely cover the same material that most text books do here. However, in all the previous chapters all of our examples were 2 nd order differential equations or 2×2 2 × 2 systems of differential …Nov 2, 2022 · Paul's Online Notes View Quick Nav Download This menu is only active after you have chosen one of the main topics (Algebra, Calculus or Differential Equations) from the Quick Nav menu to the right or Main Menu in the upper left corner. We can also give a strict mathematical/formula definition for absolute value. It is, |p| = {p if p ≥ 0 −p if p < 0 | p | = { p if p ≥ 0 − p if p < 0. This tells us to look at the sign of p p and if it’s positive we just drop the absolute value bar. If p p is negative we drop the absolute value bars and then put in a negative in front ...Example 1 Use the definition of the limit to prove the following limit. lim x→0x2 =0 lim x → 0 x 2 = 0. Show Solution. These can be a little tricky the first couple times through. Especially when it seems like we’ve got to do the work twice. In the previous example we did some simplification on the left-hand inequality to get our guess ...The standard form of a complex number is. a +bi a + b i. where a a and b b are real numbers and they can be anything, positive, negative, zero, integers, fractions, decimals, it doesn’t matter. When in the standard form a a is called the real part of the complex number and b b is called the imaginary part of the complex number.It is also important to note that all we want are the critical points that are in the interval. Let’s do some examples. Example 1 Determine the absolute extrema for the following function and interval. g(t) = 2t3 +3t2 −12t+4 on [−4,2] g ( t) = 2 t 3 + 3 t 2 − 12 t + 4 on [ − 4, 2] Show Solution. In this example we saw that absolute ...First, we need to recall just how spherical coordinates are defined. The following sketch shows the relationship between the Cartesian and spherical coordinate systems. Here are the conversion formulas for spherical coordinates. x = ρsinφcosθ y = ρsinφsinθ z = ρcosφ x2+y2+z2 = ρ2 x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ ...In this section we give a general set of guidelines for determining which test to use in determining if an infinite series will converge or diverge. Note as well that there really isn’t one set of guidelines that will always work and so you always need to be flexible in following this set of guidelines. A summary of all the various tests, as well as …Here is the definition of the logarithm function. If b is any number such that b > 0 and b ≠ 1 and x > 0 then, y = logbx is equivalent to by = x. We usually read this as “log base b of x ”. In this definition y = logbx is called the logarithm form and by = x is called the exponential form. Note that the requirement that x > 0 is really a ...Oct 9, 2023 · Pauls Online Math Notes. Home. Welcome to my online math tutorials and notes. The intent of this site is to provide a complete set of free online (and downloadable) notes and/or tutorials for classes that I teach at Lamar University. I've tried to write the notes/tutorials in such a way that they should be accessible to anyone wanting to learn ... Nov 16, 2022 · A power series about a, or just power series, is any series that can be written in the form, ∞ ∑ n=0cn(x −a)n ∑ n = 0 ∞ c n ( x − a) n. where a a and cn c n are numbers. The cn c n ’s are often called the coefficients of the series. The first thing to notice about a power series is that it is a function of x x. Jul 11, 2023 · Here is a set of notes used by Paul Dawkins to teach his Calculus II course at Lamar University. Topics covered are Integration Techniques (Integration by Parts, Trig Substitutions, Partial Fractions, Improper Integrals), Applications (Arc Length, Surface Area, Center of Mass and Probability), Parametric Curves (inclulding various applications), Sequences, Series (Integral Test, Comparison ... Nov 16, 2022 · Method 1 : Use the method used in Finding Absolute Extrema. This is the method used in the first example above. Recall that in order to use this method the interval of possible values of the independent variable in the function we are optimizing, let’s call it I I, must have finite endpoints. Also, the function we’re optimizing (once it’s ... So with all of that out of the way here is a quick summary of the method of separation of variables for partial differential equations in two variables. Verify that the …Now that we’ve seen a couple of vector fields let’s notice that we’ve already seen a vector field function. In the second chapter we looked at the gradient vector. Recall that given a function f (x,y,z) f ( x, y, z) the gradient vector is defined by, ∇f = f x,f y,f z ∇ f = f x, f y, f z . This is a vector field and is often called a ...First, when doing a substitution remember that when the substitution is done all the x ’s in the integral (or whatever variable is being used for that particular integral) should all be substituted away. This includes the x in the dx. After the substitution only u ’s should be left in the integral.Nov 15, 2023 · Integration By Parts. ∫ udv = uv −∫ vdu ∫ u d v = u v − ∫ v d u. To use this formula, we will need to identify u u and dv d v, compute du d u and v v and then use the formula. Note as well that computing v v is very easy. All we need to do is integrate dv d v. v = ∫ dv v = ∫ d v. When working with a line integral in which the path satisfies the condition of Green’s Theorem we will often denote the line integral as, ∮CP dx+Qdy or ∫↺ C P dx +Qdy ∮ C P d x + Q d y or ∫ ↺ C P d x + Q d y. Both of these notations do assume that C C satisfies the conditions of Green’s Theorem so be careful in using them.To determine a condition that must be true in order for a Taylor series to exist for a function let’s first define the nth degree Taylor polynomial of f(x) as, Tn(x) = n ∑ i = 0f ( i) (a) i! (x − a)i. Note that this really is a polynomial of degree at most n.Section 9.1 : Parametric Equations and Curves. To this point (in both Calculus I and Calculus II) we’ve looked almost exclusively at functions in the form y = f (x) y = f ( x) or x = h(y) x = h ( y) and almost all …Jul 11, 2023 · So, the sequence converges for r = 1 and in this case its limit is 1. Case 3 : 0 < r < 1. We know from Calculus I that lim x → ∞rx = 0 if 0 < r < 1 and so by Theorem 1 above we also know that lim n → ∞rn = 0 and so the sequence converges if 0 < r < 1 and in this case its limit is zero. Case 4 : r = 0. Trig Cheat Sheet - Here is a set of common trig facts, properties and formulas. A unit circle (completely filled out) is also included. Currently this cheat sheet is 4 pages long. Complete Calculus Cheat Sheet - This contains common facts, definitions, properties of limits, derivatives and integrals.Jun 6, 2018 · Trig Cheat Sheet - Here is a set of common trig facts, properties and formulas. A unit circle (completely filled out) is also included. Currently this cheat sheet is 4 pages long. Complete Calculus Cheat Sheet - This contains common facts, definitions, properties of limits, derivatives and integrals. In this section we give a general set of guidelines for determining which test to use in determining if an infinite series will converge or diverge. Note as well that there really isn’t one set of guidelines that will always work and so you always need to be flexible in following this set of guidelines. A summary of all the various tests, as well as …Nov 16, 2022 · In this section we are going to be looking at quadric surfaces. Quadric surfaces are the graphs of any equation that can be put into the general form. Ax2+By2 +Cz2 +Dxy +Exz+F yz+Gx+H y +I z +J = 0 A x 2 + B y 2 + C z 2 + D x y + E x z + F y z + G x + H y + I z + J = 0. where A A, … , J J are constants. There is no way that we can possibly ... These methods allow us to at least get an approximate value which may be enough in a lot of cases. In this chapter we will look at several integration techniques including Integration by Parts, Integrals Involving Trig Functions, Trig Substitutions and Partial Fractions. We will also look at Improper Integrals including using the Comparison ...Section 4.1 : Rates of Change. The purpose of this section is to remind us of one of the more important applications of derivatives. That is the fact that f ′(x) f ′ ( x) represents the rate of change of f (x) f ( x). This is an application that we repeatedly saw in the previous chapter. Almost every section in the previous chapter ...Jun 6, 2018 · Trig Cheat Sheet - Here is a set of common trig facts, properties and formulas. A unit circle (completely filled out) is also included. Currently this cheat sheet is 4 pages long. Complete Calculus Cheat Sheet - This contains common facts, definitions, properties of limits, derivatives and integrals. In this section we will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section. Collectively, they are called improper integrals and as we will see they may or may not have a finite (i.e. not infinite) value. Determining if they have finite values will, in fact, be one of the major ...Sep 25, 2018 · Download pdf cheat sheets and tables for algebra, trig, calculus, and Laplace transforms from Pauls Online Notes. The cheat sheets come in full and reduced versions, with common facts, formulas, properties, and errors. Paul's Online Notes Home / Download pdf File. Show Mobile Notice Show All Notes Hide All Notes. Mobile Notice. You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode.Here is the definition of the logarithm function. If b is any number such that b > 0 and b ≠ 1 and x > 0 then, y = logbx is equivalent to by = x. We usually read this as “log base b of x ”. In this definition y = logbx is called the logarithm form and by = x is called the exponential form. Note that the requirement that x > 0 is really a ...In this section we discuss how the formula for a convergent Geometric Series can be used to represent some functions as power series. To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible. However, use of this formula does quickly illustrate how functions can be represented as …Powers and Roots. In this section we’re going to take a look at a really nice way of quickly computing integer powers and roots of complex numbers. We’ll start with integer powers of z = reiθ z = r e i θ since they are easy enough. If n n is an integer then, zn =(reiθ)n = rnei nθ (1) (1) z n = ( r e i θ) n = r n e i n θ.In terms of our new function the surface is then given by the equation f (x,y,z) = 0 f ( x, y, z) = 0. Now, recall that ∇f ∇ f will be orthogonal (or normal) to the surface given by f (x,y,z) = 0 f ( x, y, z) = 0. This means that we have a normal vector to the surface. The only potential problem is that it might not be a unit normal vector.Online Math Lab. Khan Academy: What is a differential equation? Introduction to differential equations and the terms order and linear/nonlinear. Definitions for order, linear, initial …
First, we need a little terminology/notation out of the way. We call the equations that define the change of variables a transformation. Also, we will typically start out with a region, R R, in xy x y -coordinates and transform it into a region in uv u v -coordinates. Example 1 Determine the new region that we get by applying the given .... The project's
Complex Numbers Primer. Before I get started on this let me first make it clear that this document is not intended to teach you everything there is to know about complex numbers. That is a subject that can (and does) take a whole course to cover. The purpose of this document is to give you a brief overview of complex numbers, notation ...An alternating series is any series, ∑an ∑ a n, for which the series terms can be written in one of the following two forms. an = (−1)nbn bn ≥ 0 an = (−1)n+1bn bn ≥ 0 a n = ( − 1) n b n b n ≥ 0 a n = ( − 1) n + 1 b n b n ≥ 0. There are many other ways to deal with the alternating sign, but they can all be written as one of ...Nov 16, 2022 · For example, the hyperbolic paraboloid y = 2x2 −5z2 y = 2 x 2 − 5 z 2 can be written as the following vector function. →r (x,z) = x→i +(2x2−5z2) →j +z→k r → ( x, z) = x i → + ( 2 x 2 − 5 z 2) j → + z k →. This is a fairly important idea and we will be doing quite a bit of this kind of thing in Calculus III. Oct 9, 2023 · Pauls Online Math Notes. Home. Welcome to my online math tutorials and notes. The intent of this site is to provide a complete set of free online (and downloadable) notes and/or tutorials for classes that I teach at Lamar University. I've tried to write the notes/tutorials in such a way that they should be accessible to anyone wanting to learn ... Apr 12, 2020 ... ... online instructor. Over the past 15 years, I've taught 1000s of students. eBassGuitar is the only video bass guitar lessons website ...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. This will include deriving a second linearly independent solution that we will need to form the general solution to the system. We will also show how to sketch phase ...From the Quadric Surfaces section notes we can see that this is a cone that opens along the \(x\)-axis. We are much more likely to need to be able to write down the parametric equations of a surface than identify the surface from the parametric representation so let’s take a look at some examples of this.Note that all these properties also hold for the two one-sided limits as well we just didn’t write them down with one sided limits to save on space. Let’s compute a limit or two using these properties. The next couple of examples will lead us to some truly useful facts about limits that we will use on a continual basis.ax + by = p cx + dy = q. We first write down the augmented matrix for this system, [a b p c d q] and use elementary row operations to convert it into the following augmented matrix. [1 0 h 0 1 k] Once we have the augmented matrix in this form we are done. The solution to the system will be x = h and y = k.We can also give a strict mathematical/formula definition for absolute value. It is, |p| = {p if p ≥ 0 −p if p < 0 | p | = { p if p ≥ 0 − p if p < 0. This tells us to look at the sign of p p and if it’s positive we just drop the absolute value bar. If p p is negative we drop the absolute value bars and then put in a negative in front ...Linear Algebra equivalent of Paul's Online Math Notes (Lamar) I'm not sure if this is question is a better fit for the main site or the meta site, but I was wondering if ….