Author Archives: hangtime

Alternating Series, and Absolute & Conditional Convergence

Just when they finally give you a type of series that almost always converges -- alternating series -- they have to muck it up by giving you sub-categories of convergence: absolute and conditional. Mean calculus! Oh well, they're still only as bad as the other series, so you just end up using all the tests you've learned up to this point, just with a couple of new vocab words to slap on.

This video appears on the page: Series Convergence & Divergence

Nth Root Test

Not the most useful test, but what the heck, most schools and books cover it, so I guess we should mention it here. It basically only applies when there's an exponent of n surrounding the whole problem, which let's face it, is kind of rare.

This video appears on the page: Series Convergence & Divergence

Ratio Test

The easiest and most useful test when checking the convergence of nastier problems, in this video I'll show you the key "tells" to watch out for to help you decide when to use the relatively straightforward ratio test rather than the more difficult comparison test.

This video appears on the page: Series Convergence & Divergence

Limit Comparison Test

Like the regular comparison test, this one tests for divergence or convergence by comparing two series. On the bright side, this method is a lot more plug-and-chug: once you pick the series to compare, you just throw them into a limit problem and execute. The devil is the details, though, since with this it's a bit trickier to pick the series you're comparing.

This video appears on the page: Series Convergence & Divergence

Comparison Test

One of the more difficult tests for convergence, this one is annoying because it works for some cases and not others, and it's not plug-and-chug: you have to be a bit clever with picking what to compare your series to. This video will sort most of that out.

This video appears on the page: Series Convergence & Divergence

Integral Test For Convergence

Just when you thought there wouldn't be any more calculus in calculus, you get to do some integration! It's pretty wacky integration, though, requiring Improper Integrals, so if you're not rock-solid on those, you might want to review this previous chapter.

This video appears on the page: Series Convergence & Divergence

P-Series Test

This video introduces the first of the new series types we'll be working with (and expected to memorize) in this chapter: the P-Series. Though it sounds like it might be an abbreviation for Power Series, it most definitely is not, so be careful. P-Series are more like geometric series, the famous Harmonic Series being the P-series where p=1.

This video appears on the page: Series Convergence & Divergence

Geometric Series Convergence

Don't blow this video off because it's just geometric series and you've seen these before. Because you haven't seen them like this. These ones are infinite. These ones have sigma notation. And this being calculus, naturally there's a special rule just for geometric series that you have to memorize and apply. You'll be glad you did, though, because it turns out these are some of the easiest problems in this chapter if you know how to spot them.

This video appears on the page: Series Convergence & Divergence

Nth Term Test For Convergence

This test basically tells you that if your terms aren't approaching zero, there's no way the series converges. Even if the terms are approaching zero the series could still diverge, though, hence the Harmonic Series, so this test isn't super-useful on its own. But hey, gotta start somewhere when delving into the exciting world of infinite series!

This video appears on the page: Series Convergence & Divergence

Intro to Convergence & Divergence (free)

As you probably noticed, there are a LOT of videos in this chapter, most covering some bizarre-sounding "test" for "convergence" or "divergence". Naturally, you're wondering what the heck any of it means, so that's what this video is all about. Getting the overview and concepts before the mind-numbing nitty gritty of the videos to follow!

This video appears on the page: Series Convergence & Divergence

Convergence & Divergence:
Nth Term Test
Geometric Series Test
P-Series Test
Integral Test
Comparison Test
Limit Test
Ratio Test
Alternating Series

Everything you could possibly need to know about convergence and divergence of infinite series is all in this one chapter because otherwise it would be really confusing. In addition to the topics listed to the left, there's a free overview of series convergence, as well as more obscure topics like the remainder (error) of an alternating series approximation.

Part of the course(s): ,Test Image Problem ,Calculus

Two-Dimensional Projectile Problems

You'll definitely want to watch the one-dimensional video before this first, if you haven't already. This video expands on that one, showing you how to break a velocity vector into its horizontal and vertical components. Once the problem is split into X and Y, you basically work the two halves of the problem separately, solving one half for time (t) and then plugging that back into the other half to get the final answer.

This video appears on the page: Projectile Motion ,Free Sample Videos ,physics free videos

One-Dimensional Gravity Problems

In this first video we'll cover problems where an object is either getting dropped, or thrown straight up or straight down. Thus the "one dimension" we're talking about is vertical, and the problems will ask things like "How far does the object fall in 5 seconds" and "What is the max height the ball reaches" and "What if the ball is dropped on the moon where gravity is less." The next video covers two-dimensional projectile problems, where the object follows a parabolic arc.

This video appears on the page: Projectile Motion

Projectile Motion
(Pre-Calc Version)
One-Dimensional Gravity Problems
Two-Dimensional Projectile Problems

This chapter covers kinematics projectile motion problems as you would see in Pre-Calculus or Algebra 2 math classes. This topic is covered in more depth on the physics page. One-dimensional and two-dimensional gravity problems, range, vector components of velocity, etc.

Part of the course(s): ,Physics ,Trigonometry ,Test Image Problem ,College Algebra ,Pre-Calculus ,Calculus

Parametric Equations of Conic Sections: Circles, Ellipses & Hyperbolas

Many teachers make you know circles. A few do ellipses. Almost none do hyperbolas. But the three equations are so similar that I thought I'd throw them all into the same video so you can get them easier by seeing how they're all the same. Gonna have to memorize these!

This video appears on the page: Parametric Equations

Converting Rectangular Equations to Parametric

Sort of a reverse of the problems in the previous video, this is actually easier than you'd think. When all else fails, just set x=t at the start of the problem and you'll have smooth sailing.

This video appears on the page: Parametric Equations

Converting Parametric Equations to Rectangular

In this video we'll go through the surprisingly simple process of switching between the two types of equations by "eliminating the parameter". We can do this in basic polynomial situations, but also when complex trig functions and identities are involved.

This video appears on the page: Parametric Equations

Graphing Parametric Equations

In this video we'll talk about what the heck parametric equations are, how to graph them, and do a quick projectile problem since projectile equations are a "classic" parametric equation situation.

This video appears on the page: Parametric Equations

Parametric Equations
Graphing Parametric Equations
Converting Parametric Equations to Rectangular
Converting Rectangular Equations to Parametric
Parametric Equations of Conic Sections: Circles, Ellipses & Hyperbolas

This chapter covers converting parametric equations to rectangular and back again, eliminating the parameter, parametric forms of circles and ellipses, and graphing them.

Part of the course(s): ,Trigonometry ,Test Image Problem ,College Algebra ,Pre-Calculus ,Calculus

Solving Differential Equations Using Separation of Variables

This video covers how to rearrange a differential equation so that you can solve it by integrating both sides. General solutions, specific solutions, initial conditions: everything is better once the variables have been separated!

This video appears on the page: Differential Equations