Nov 16, 2022 · A geometric series is any series that can be written in the form, ∞ ∑ n = 1arn − 1. or, with an index shift the geometric series will often be written as, ∞ ∑ n = 0arn. These are identical series and will have identical values, provided they converge of course. If we start with the first form it can be shown that the partial sums are ... Nov 16, 2022 ... In this chapter proofs of many of the facts/properties/theorems given through out the material are given. Also included are a brief review ...The apostle Paul likely died of decapitation due to a beheading by the Romans. While there are no definitive records of Paul’s death, decapitation is the commonly accepted reasonin...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 ...This method is often called the method of disks or the method of rings. Let’s do an example. Example 1 Determine the volume of the solid obtained by rotating the region bounded by y = x2 −4x+5 y = x 2 − 4 x + 5, x = 1 x = 1, x = 4 x = 4, and the x x -axis about the x x -axis. Show Solution. In the above example the object was a solid ...Note as well that in order to use this formula \(n\) must be a number, it can’t be a variable. Also note that the base, the \(x\), must be a variable, it can’t be a number. It will be tempting in some later sections to misuse the Power Rule when we run in some functions where the exponent isn’t a number and/or the base isn’t a variable.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.This is called the scalar equation of plane. Often this will be written as, ax+by +cz = d a x + b y + c z = d. where d = ax0 +by0 +cz0 d = a x 0 + b y 0 + c z 0. This second form is often how we are given equations of planes. Notice that if we are given the equation of a plane in this form we can quickly get a normal vector for the plane.Nov 16, 2022 · a ∫e2t + sec(2t)tan(2t)dt Show Solution. b ∫sin(t)(4cos3(t) + 6cos2(t) − 8)dt Show Solution. c ∫xcos(x2 + 1) + x x2 + 1 dx Show Solution. So, as we’ve seen in the previous set of examples sometimes we can use the same substitution more than once in an integral and doing so will simplify the work. Example 2 Evaluate each of the ... First, the standard form of a quadratic equation is. ax2 +bx +c = 0 a ≠ 0 a x 2 + b x + c = 0 a ≠ 0. The only requirement here is that we have an x2 x 2 in the equation. We guarantee that this term will be present in the equation by requiring a ≠ 0 a ≠ 0. Note however, that it is okay if b b and/or c c are zero.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.In the previous section we saw that there is a large class of functions that allows us to use. lim x → af(x) = f(a) to compute limits. However, there are also many limits for which this won’t work easily. The …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. Apr 5, 2019 · As we will see in later sections we can use Laplace transforms to reduce a differential equation to an algebra problem. The algebra can be messy on occasion, but it will be simpler than actually solving the differential equation directly in many cases. Laplace transforms can also be used to solve IVP’s that we can’t use any previous method on. Are you dreaming of a luxurious vacation filled with adventure and breathtaking beauty? Look no further than Paul Gauguin Cruises in Tahiti. With their exceptional service, stunnin...At present I've gotten the notes/tutorials for my Algebra (Math 1314), Calculus I (Math 2413), Calculus II (Math 2414), Calculus III (Math 2415), Linear Algebra (Math 2318) and …Let’s work a couple of examples now. Find a general solution to the following differential equation. \ [2y'' + 18y = 6\tan \left ( {3t} \right)\] First, since the formula for variation of parameters requires a coefficient of a one in front of the second derivative let’s take care of that before we forget.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...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 ...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 …Donating to Saint Vincent de Paul is a wonderful way to give back to the community and help those in need. Whether you have clothing, furniture, or household items that you no long...Binomial Series. So, similar to the binomial theorem except that it’s an infinite series and we must have |x| < 1 | x | < 1 in order to get convergence. Let’s check out an example of this. In this section we will give the Binomial Theorem and illustrate how it can be used to quickly expand terms in the form (a+b)^n when n is an integer.2.2 Triple Integrals in Cylindrical and Spherical Coordinates. In this lesson, you will learn to evaluate triple integrals in cylindrical and spherical coordiates. Set up and evaluate …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 …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 ...Section 6.4 : Volume With Cylinders. In the previous section we started looking at finding volumes of solids of revolution. In that section we took cross sections that were rings or disks, found the cross-sectional area and then used the following formulas to find the volume of the solid. V = ∫ b a A(x) dx V = ∫ d c A(y) dy V = ∫ a b A ...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 ...Work to Understand the Principles. You can pass a history class by simply memorizing a set of dates, names and events. You will find, however, that in order to pass a math class you will need to do more than just memorize a set of formulas. While there is certainly a fair amount of memorization of formulas in a math class you need to do more.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 ...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 ...Fill Paul's Online Notes Integration By Parts, Edit online. Sign, fax and printable from PC, iPad, tablet or mobile with pdfFiller ✓ Instantly. Try Now!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 ...St. Vincent de Paul is a well-known charitable organization that has been making a significant impact in communities around the world for centuries. With a mission to serve those i...Nov 16, 2022 · W =F d W = F d. However, most forces are not constant and will depend upon where exactly the force is acting. So, let’s suppose that the force at any x x is given by F (x) F ( x). Then the work done by the force in moving an object from x = a x = a to x = b x = b is given by, W =∫ b a F (x) dx W = ∫ a b F ( x) d x. Determine the dimensions of the box that will maximize the enclosed volume. Solution. We want to build a box whose base length is 6 times the base width and the box will enclose 20 in 3. The cost of the material of the sides is $3/in 2 and the cost of the top and bottom is $15/in 2. Determine the dimensions of the box that will minimize the cost.This is called the scalar equation of plane. Often this will be written as, ax+by +cz = d a x + b y + c z = d. where d = ax0 +by0 +cz0 d = a x 0 + b y 0 + c z 0. This second form is often how we are given equations of planes. Notice that if we are given the equation of a plane in this form we can quickly get a normal vector for the plane.Nov 16, 2022 · Quotient Rule. If the two functions f (x) f ( x) and g(x) g ( x) are differentiable ( i.e. the derivative exist) then the quotient is differentiable and, ( f g)′ = f ′g −f g′ g2 ( f g) ′ = f ′ g − f g ′ g 2. Note that the numerator of the quotient rule is very similar to the product rule so be careful to not mix the two up! The ... 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.Welcome to my math notes site. Contained in this site are the notes (free and downloadable) that I use up teach Algebra, Calculus (I, II furthermore III) as well as Diff Equations during Lamar Your. The notes contains of usual topics that were taught in the courses as well-being when a few extra topics that I decided to include just because I …In this section we want to revisit tangent planes only this time we’ll look at them in light of the gradient vector. In the process we will also take a look at a normal line to a surface. Let’s first recall the equation of a plane that contains the point (x0,y0,z0) ( x 0, y 0, z 0) with normal vector →n = a,b,c n → = a, b, c is given by ...Nov 16, 2022 · With a Fourier series we are going to try to write a series representation for f (x) f ( x) on −L ≤ x ≤ L − L ≤ x ≤ L in the form, f (x) = ∞ ∑ n=0Ancos( nπx L)+ ∞ ∑ n=1Bnsin( nπx L) f ( x) = ∑ n = 0 ∞ A n cos ( n π x L) + ∑ n = 1 ∞ B n sin ( n π x L) So, a Fourier series is, in some way a combination of the ... In this section we will give a quick review of trig functions. We will cover the basic notation, relationship between the trig functions, the right triangle definition of the trig functions. We will also cover evaluation of trig functions as well as the unit circle (one of the most important ideas from a trig class!) and how it can be used to evaluate trig …Learn Algebra, Trig, Calculus, Differential Equations and more with free online notes and tutorials from Pauls Online Math Notes. The notes are written for students …Determining asymptotes is actually a fairly simple process. First, let’s start with the rational function, f (x) = axn +⋯ bxm +⋯ f ( x) = a x n + ⋯ b x m + ⋯. where n n is the largest exponent in the numerator and m m is the largest exponent in the denominator. We then have the following facts about asymptotes.A web page with online notes on surface integrals, parametric surfaces, vector fields, Stokes' theorems and divergence theorem. The notes include colour graphics, external …Section 4.3 : Minimum and Maximum Values. Many of our applications in this chapter will revolve around minimum and maximum values of a function. While we can all visualize the minimum and maximum values of a function we want to be a little more specific in our work here. In particular, we want to differentiate between two types …Apr 6, 2020 ... Paul's Online Calculus 4-1 Rates of Change example 3 Thank you Professor Paul from http://tutorial.math.lamar.edu/Here are my online notes for my Linear Algebra course that I teach here at Lamar University. Despite the fact that these are my “class notes”, they should be accessible to anyone wanting to learn Linear Algebra or needing a refresher. These notes do assume that the reader has a good working knowledge of basic Algebra. This setDetermine the dimensions of the box that will maximize the enclosed volume. Solution. We want to build a box whose base length is 6 times the base width and the box will enclose 20 in 3. The cost of the material of the sides is $3/in 2 and the cost of the top and bottom is $15/in 2. Determine the dimensions of the box that will minimize the cost.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.Basic Concepts – In this section we will introduce some common notation for vectors as well as some of the basic concepts about vectors such as the magnitude of a vector and unit vectors. We also illustrate how to find a vector from its starting and end points. Vector Arithmetic – In this section we will discuss the mathematical and ...When disaster strikes, homeowners and business owners often find themselves overwhelmed and in need of immediate assistance. That’s where Paul Davis Restoration Services comes in. ...Mar 2, 2020 ... In the example given in Paul's Online Math Notes ( see image below) , I do not understand why the author seems to give up his initial goal, ...When disaster strikes, homeowners and business owners often find themselves overwhelmed and in need of immediate assistance. That’s where Paul Davis Restoration Services comes in. ...Nov 16, 2022 · The notation for the general triple integrals is, Note that when using this notation we list the \ (x\)’s first, the \ (y\)’s second and the \ (z\)’s third. Note that we integrated with respect to \ (x\) first, then \ (y\), and finally \ (z\) here, but in fact there is no reason to the integrals in this order. MthSc 208, Summer Session II, 2012. Distinction Equations (Math 3301) [ Bills] - Topics built stylish this set of notes/tutorial are : First Order Differentials Equations - Linear Equations, Separable Equations, Exact Equations, Equilibrium Solutions, Modeling Problems. Second Order Differential Mathematische - Homogeneous and …Note that it is okay to have the constant sitting outside of the series like this. In fact, there is no way to bring it into the series so don’t get excited about it. Finally, because the power series representation from Example 3 had a radius of convergence of \(R=5\) this series will also have a radius of convergence of \(R=5\).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. Nov 16, 2022 · This can be written in several ways. Here are a couple of the more standard notations. lim x→a y→b f (x,y) lim (x,y)→(a,b)f (x,y) lim x → a y → b f ( x, y) lim ( x, y) → ( a, b) f ( x, y) We will use the second notation more often than not in this course. The second notation is also a little more helpful in illustrating what we are ... Properties of the Indefinite Integral. ∫ kf (x) dx =k∫ f (x) dx ∫ k f ( x) d x = k ∫ f ( x) d x where k k is any number. So, we can factor multiplicative constants out of indefinite integrals. See the Proof of Various Integral Formulas section of the Extras chapter to see the proof of this property. ∫ −f (x) dx = −∫ f (x) dx ∫ ...Using summation notation the area estimation is, A ≈ n ∑ i=1f (x∗ i)Δx A ≈ ∑ i = 1 n f ( x i ∗) Δ x. The summation in the above equation is called a Riemann Sum. To get a better estimation we will take n n larger and larger. In fact, if we let n n go out to infinity we will get the exact area. In other words,Paul's Online Math Notes provides class notes and practice problems for Algebra, Calculus I, Calculus II, Calculus III, and Differential Equations. This is an ...Section 6.4 : Volume With Cylinders. In the previous section we started looking at finding volumes of solids of revolution. In that section we took cross sections that were rings or disks, found the cross-sectional area and then used the following formulas to find the volume of the solid. V = ∫ b a A(x) dx V = ∫ d c A(y) dy V = ∫ a b A ...W =F d W = F d. However, most forces are not constant and will depend upon where exactly the force is acting. So, let’s suppose that the force at any x x is given by F (x) F ( x). Then the work done by the force in moving an object from x = a x = a to x = b x = b is given by, W =∫ b a F (x) dx W = ∫ a b F ( x) d x.zn =(reiθ)n = rnei nθ (1) (1) z n = ( r e i θ) n = r n e i n θ. There really isn’t too much to do with powers other than working a quick example. Example 1 Compute (3 +3i)5 ( 3 + 3 i) 5 . Show Solution. So, there really isn’t too much to integer powers of a complex number. Note that if r = 1 r = 1 then we have,So, L’Hospital’s Rule tells us that if we have an indeterminate form 0/0 or ∞/∞ ∞ / ∞ all we need to do is differentiate the numerator and differentiate the denominator and then take the limit. Before proceeding with examples let me address the spelling of “L’Hospital”. The more modern spelling is “L’Hôpital”.Surface Integrals – In this section we introduce the idea of a surface integral. With surface integrals we will be integrating over the surface of a solid. In other words, the variables will always be on the surface of the solid and will never come from inside the solid itself. Also, in this section we will be working with the first kind of ...Paul McCartney is a name that needs no introduction in the world of music. As a member of the iconic band, The Beatles, he has earned his place in history as one of the greatest mu...When disaster strikes, whether it’s a fire, flood, or mold infestation, it can leave homeowners feeling overwhelmed and unsure of where to turn. That’s where Paul Davis Restoration...Proof of Integral Test. First, for the sake of the proof we’ll be working with the series ∞ ∑ n=1an ∑ n = 1 ∞ a n. The original test statement was for a series that started at a general n =k n = k and while the proof can be done for that it will be easier if we assume that the series starts at n =1 n = 1.Find the best master's in math education online degrees with our list of top rated online programs. Updated October 3, 2022 thebestschools.org is an advertising-supported site. Fea...Apr 5, 2019 · As we will see in later sections we can use Laplace transforms to reduce a differential equation to an algebra problem. The algebra can be messy on occasion, but it will be simpler than actually solving the differential equation directly in many cases. Laplace transforms can also be used to solve IVP’s that we can’t use any previous method on. Nov 16, 2022 · Green’s Theorem. Let C C be a positively oriented, piecewise smooth, simple, closed curve and let D D be the region enclosed by the curve. If P P and Q Q have continuous first order partial derivatives on D D then, ∫ C P dx +Qdy =∬ D ( ∂Q ∂x − ∂P ∂y) dA ∫ C P d x + Q d y = ∬ D ( ∂ Q ∂ x − ∂ P ∂ y) d A. Before ... 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 ...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 ...Repeated Roots – In this section we discuss the solution to homogeneous, linear, second order differential equations, ay′′ +by′ +cy = 0 a y ″ + b y ′ + c y = 0, in which the roots of the characteristic polynomial, ar2 +br+c = 0 a r 2 + b r + c = 0, are repeated, i.e. double, roots. We will use reduction of order to derive the second ...In this section we look at integrals that involve trig functions. In particular we concentrate integrating products of sines and cosines as well as products of secants and tangents. We will also briefly look at how to modify the work for products of these trig functions for some quotients of trig functions.A geometric series is any series that can be written in the form, ∞ ∑ n = 1arn − 1. or, with an index shift the geometric series will often be written as, ∞ ∑ n = 0arn. These are identical series and will have identical values, provided they converge of course. If we start with the first form it can be shown that the partial sums are ...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 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 theorem note that the surface S S can actually be any surface so long as its boundary curve is given by C C. This is something that can be used to our advantage to simplify the surface integral on occasion. Let’s take a look at a couple of examples. Example 1 Use Stokes’ Theorem to evaluate ∬ S curl →F ⋅ d →S ∬ S curl F ...
Use either 1. or 2.4. n and m both even. Use double angleand/or half angle formulas to reduce theintegral into a form that can be integrated.1. n odd. Strip 1 tangent …. I left my heart in san francisco
f (x) = P (x) Q(x) f ( x) = P ( x) Q ( x) where both P (x) P ( x) and Q(x) Q ( x) are polynomials and the degree of P (x) P ( x) is smaller than the degree of Q(x) 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 case we call h′(b) h ′ ( b) the partial derivative of f (x,y) f ( x, y) with respect to y y at (a,b) ( a, b) and we denote it as follows, f y(a,b) = 6a2b2 f y ( a, b) = 6 a 2 b 2. Note that these two partial derivatives are sometimes called the first order partial derivatives. Just as with functions of one variable we can have ...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 ... This website was inspired by Paul's Online Math Notes when a student I was tutoring in calculus asked me if there was a resource with explanations and ...Work to Understand the Principles. You can pass a history class by simply memorizing a set of dates, names and events. You will find, however, that in order to pass a math class you will need to do more than just memorize a set of formulas. While there is certainly a fair amount of memorization of formulas in a math class you need to do more.Highly recommend every part of this website, including the tips, the study guides, the notes and the practice problems. 100% yes, Paul's Online Notes saved my ass in university. While all of the professors I had for calculus were brilliant mathematicians, all but one were actually good at teaching (in a way that made sense to me that is).Basic Concepts – In this section we will introduce some common notation for vectors as well as some of the basic concepts about vectors such as the magnitude of a vector and unit vectors. We also illustrate how to find a vector from its starting and end points. Vector Arithmetic – In this section we will discuss the mathematical and ...Paul's Online Math Notes. Paul's Online notes are a collection of short course notes, examples, and problem sets designed to assist first-year college students or advanced …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. f (x) = P (x) Q(x) f ( x) = P ( x) Q ( x) where both P (x) P ( x) and Q(x) Q ( x) are polynomials and the degree of P (x) P ( x) is smaller than the degree of Q(x) 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 ...Nov 16, 2022 · Example 1 Find and classify all the critical points of f (x,y) = 4+x3 +y3 −3xy f ( x, y) = 4 + x 3 + y 3 − 3 x y . Let’s do one more example that is a little different from the first two. Example 3 Determine the point on the plane 4x−2y +z = 1 4 x − 2 y + z = 1 that is closest to the point (−2,−1,5) ( − 2, − 1, 5) . Sep 25, 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. Welcome to my math notes site. Contained in this site are the notes (free and downloadable) that I use to teach Algebra, Calculus (I, II and III) as well as Differential Equations at Lamar University. ... Calculus II (Math 2414), Calculus III (Math 2415) and Differential Equations (Math 3301) class online. I've also got a couple of Review ...Nov 5, 2020 · The Calculus II notes/tutorial assume that you've got a working knowledge Calculus I, including Limits, Derivatives, and Integration (up to basic substitution). It is also assumed that you have a fairly good knowledge of Trig. Several topics rely heavily on trig and knowledge of trig functions. Example 1 Determine the linear approximation for f (x) = 3√x f ( x) = x 3 at x = 8 x = 8. Use the linear approximation to approximate the value of 3√8.05 8.05 3 and 3√25 25 3 . Linear approximations do a very good job of approximating values of f (x) f ( x) as long as we stay “near” x = a x = a. However, the farther away from x = a x ...Nov 16, 2022 · Green’s Theorem. Let C C be a positively oriented, piecewise smooth, simple, closed curve and let D D be the region enclosed by the curve. If P P and Q Q have continuous first order partial derivatives on D D then, ∫ C P dx +Qdy =∬ D ( ∂Q ∂x − ∂P ∂y) dA ∫ C P d x + Q d y = ∬ D ( ∂ Q ∂ x − ∂ P ∂ y) d A. Before ... Apr 6, 2020 ... Paul's Online Calculus 4-1 Rates of Change example 3 Thank you Professor Paul from http://tutorial.math.lamar.edu/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.Divergence Theorem. Let E E be a simple solid region and S S is the boundary surface of E E with positive orientation. Let →F F → be a vector field whose components have continuous first order partial derivatives. Then, ∬ S →F ⋅ d→S = ∭ E div →F dV ∬ S F → ⋅ d S → = ∭ E div F → d V. Let’s see an example of how to ....