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Episode 5 of Netflix’s ‘Stranger Things’ Season 3 left us with the chilling image of the gooey monster in the hospital hallway with Nancy and Jonathan. Episode 6 picks up from right there and if you were expecting narrative respite, there is none. The protagonists are hard on the trail of the aliens now, be it from the Upside Down or from mother Russia. With only two more episodes to go, it does not look like the Duffer brothers are going to ease up.

Five things to know about the Braves’ final homestand of the regular season, which begins Tuesday night with the opener of a three-game series against the Phillies: 1.

However, there are some new character dynamics like the clique including Hopper, Joyce, Alexei, and Murray. Their scenes include some sharp and funny scriptwriting, which comes as a relief when compared to the looming threat of the monster. I believe they could give the Starcourt Squad a run for their money when it comes to humor and rapport.

  1. Sequences - Finding a Rule. To find a missing number in a Sequence, first we must have a Rule. A Sequence is a set of things (usually numbers) that are in order. Each number in the sequence is called a term (or sometimes 'element' or 'member'), read Sequences and Series for a more in-depth discussion.
  2. Things: Directed by Andrew Jordan. Gillis, Amber Lynn, Bruce Roach, Doug Bunston. An impotent husband, driven by a fanatical desire to father children, forces his wife to undergo a dangerous experiment.

Titled ‘E Pluribus Unum’ which means ‘out of many, one’, and which happens to be the motto of the USA, this episode has set the stage for the final fireworks to take place in the season. The title literally ties in with the theme of this episode, and since it happens during the Fourth of July celebrations, we can expect the final action to unfurl at the fair that Mayor Klines has organized for the people of Hawkins.

3.3.6 average

Episode Recap

As I said, the biggest worry was whether Nancy and Jonathan would make it alive out of the hospital after being trapped with the creature. However, the creature seemed to be more interested in Nancy who got a good thrashing from this monster, I might add. Just when it looked like Nancy was running out of time and Jonathan could not do anything, El and her friends intervened. In one of the most spectacular sequences yet, El gave the monster a good flinging with one hand, before using her powers to literally fling the monster out of the hospital.

Meanwhile, it seems like our Russian Terminator is not such a good tracker after all, because he leans on Kline to help in finding Hopper, and subsequently Alexei. Meanwhile, we see Alexei helping Hopper and Joyce understand what the Russians are doing. Their motivations are now clear. They have a key to open a gate to another world (Upside Down), and while they have many keys in Russia, Hawkins is the ideal location because the gate has been opened here once, already. Acting on this information, Hopper calls in the cavalry to try and deal with the Russians.

Speaking of dealing with Russians, our Starcourt Squad runs into trouble as they are discovered. Steve and Robin play the heroes, allowing themselves to be captured while Erica and Dustin escape. Though it is sad to see the Steve-Dustin duo break up, the Duffer brothers do a good job exploring Steve’s relation with Robin, and Dustin’s with Erica, in the meanwhile. Steve and Robin get the harsher deal though, as Steve is brutally beaten up in interrogation, and later they are both injected with what seems to be truth serum. However, Dustin and Erica double back to rescue their friends, and we will see in the upcoming episode if the squad makes it out of the facility safe and sound.

This brings me to my favorite part of the episode, and certainly the most stunning. El, in trying to track down the monster’s location, accesses Billy’s mind and memories. As I have already mentioned, Dacre Montogomery has done a stellar job this season, as far as acting is concerned. It is nice to see that the showrunners have hashed out Billy’s character beautifully by showing a montage of his troubled childhood and his parents’ marital problems, which probably made Billy the way he is. Anyhow, El does find the monster’s hideout, but in the process, she gives herself away to the Mind Flayer and the flayed humans under his control. There is also a hint that El might share a deeper bond with the Upside Down than what we are aware of, but that will surely be made clear in the last two episodes.

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In the meanwhile, another of my theories comes to fruition in the last few scenes of Episode 6. As posited, the Duffer brothers were using the narrative strategy of keeping the monster lurking underneath, till it was powerful enough to rise to the surface in all its might. It seems, the Mind Flayer is now powerful, having absorbed the gooey substance from all the flayed, and out of many, one powerful, monstrous creature has risen to the surface. We will be sure to see how Hawkins fights back against this monster in the future episodes.

Episode Review

Okay, first things first. Of all the ‘Stranger Things’ Season 3 episodes till now, this has been the most visually stunning. Using El’s powers to access Billy’s memories, allows for some excellent scenes to play out, like the tranquility at the beach during Billy’s childhood being juxtaposed against the monstrous storm when things begin to go wrong in his family. At the center of the storm is the core of the monstrosity, the Mind Flayer, which resides in the abandoned factory, as El glimpses.

We also loved how Steve and Robin got a chance to show their emotional range, from scared to nostalgic, to laugh out stoned. This episode also speeds things up a lot and sets up for two very action-packed final episodes. One big revelation in the Russian facility was the discovery of the cage by Erica, which seems big enough to hold a Demodog. Do the Russians have an infernal creature from the Upside Down locked away? Only time will tell.

Meanwhile, Hopper and his gang are on their way to beat the Ruskies and perhaps we will see another brilliant confrontation between ‘fat Rambo‘ as Alexei calls Hopper, and the Russian Terminator. The storylines are beginning to converge, and we will probably see the Starcourt Squad intersect with Hopper’s clique soon enough.

Overall, this has been an episode that has maintained its brutality and hilarity in equal measures, with the most impactful moments coming from El and Billy’s interactions, which helps us gain a greater insight into the mind of Billy, and only makes us more excited to see what the Mind Flayer does in the episodes to come.

Rating: 3/5

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Functions are what we use to describe things we want to talk about mathematically. I find, though, that I get a bit tongue tied when I try to define them.

The simplest definition is: a function is a bunch of ordered pairs of things (in our case the things will be numbers, but they can be otherwise), with the property that the first members of the pairs are all different from one another.

Thus, here is an example of a function:

[[{1, 1}, {2, 1}, {3, 2}]]

This function consists of three pairs, whose first members are (1, 2) and (3).
It is customary to give functions names, like (f, g) or (h), and if we call this function (f), we generally use the following notation to describe it:

[f(1) = 1, f(2) = 1, f(3) = 2]

The first members of the pairs are called arguments and the whole set of them is called the domain of the function. Thus the arguments of (f) here are (1, 2) and (3), and the set consisting of these three numbers is its domain.

The second members of the pairs are called the values of the functions, and the set of these is called the range of the function.

The standard terminology for describing this function f is:

The value of (f) at argument (1) is (1), its value at argument (2) is (1), and its value at argument (3) is (2), which we write as (f(1) = 1, f(2) = 1, f(3) = 2).

We generally think of a function as a set of assignments of values (second members of our pairs) to arguments (their first members).

The condition that the first members of the pairs are all different is the condition that each argument in the domain of (f) is assigned a unique value in its range by any function.

Exercise 3.1 Consider the function (g), defined by the pairs ((1, 1), (2, 5), (3, 1)) and ((4, 2)). What is its domain? What is the value of (g) at argument (3)? What is (g(4))?

If you stick a thermometer in your mouth, you can measure your temperature, at some particular time. You can define a function (T) or temperature, which assigns the temperature you measure to the time at which you remove the thermometer from your mouth. This is a typical function. Its arguments are times of measurement and its values are temperatures.

Of course your mouth has a temperature even when you don't measure it, and it has one at every instant of time and there are an infinite number of such instants.

This means that if you want to describe a function (T) whose value at any time t is the temperatures in your mouth at that time, you cannot really list all its pairs. There are an infinite number of possible arguments (t) and it would take you forever to list them.

Instead, we employ a trick to describe a function (f): we generally provide a rule which allows you, the reader, to choose any argument you like in (f)'s domain, and, by using the rule, to compute the value of your function at that argument. This rule is often called a formula for the function. The symbol (x) is often used to denote the argument you will select, and the formula tells you how to compute the function at that argument.

The simplest function of all, sometimes called the identity function, is the one that assigns as value the argument itself. If we denote this function as (f), it obeys

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[f(x) = x]

for (x) in whatever domain we choose for it. In other words, both members of its pairs are the same wherever you choose to define it.

We can get more complicated functions by giving more complicated rules, (These rules are often called formulae as we have noted already). Thus we can define functions by giving any of the following formulae among an infinity of possibilities:

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[3x, x^2, x^2-1, frac{3}{x}, x^3, frac{x}{x^2 + 1}, 3x + 5, x^2 + 7x - 1]

These represent, respectively, (3) times (x), (x) squared, (x) squared minus (1), (3) divided by (x), (x) cubed, (x) divided by the sum of the square of (x) and (1), and so on.

We can construct functions by applying the operations of addition, subtraction, multiplication and division to copies of (x) and numbers in any way we see fit to do so.

There are two very nice features of functions that we construct in this way, and the first applies to all functions.

We can draw a picture of a function, called its graph, on a piece of graph paper, or on a spreadsheet, chart, or with a graphing calculator. We can do it by taking argument-value pairs of the function and describing each by a point in the plane, with (x) coordinate given by the argument and y coordinate given by the value for its pair.

Of course it is impossible to plot all the pairs of a function that has an infinite domain, but we can get a pretty good idea of what its graph looks like by taking perhaps a hundred evenly spaced points in any interval of interest to us. This sounds like an impossibly tedious thing to do and it used to be so, but now it is not. On a spreadsheet, the main job is to enter the function once (with its argument given by the address of some other location). That and some copying is all you have to do, and with practice it can be done in (30) seconds for a very wide variety of functions.

The second nice feature is that we can enter any function formed by adding, subtracting, multiplying, dividing and performing still another operation, on the contents of some address very easily on a spreadsheet or graphing calculator. Not only that, these devices have some other built in functions that we can use as well.

The two of these facts mean that we can actually look at any function formed by adding subtracting multiplying or dividing copies of the identity function (x) and other built in functions, and any number we want, and see how they behave, with very limited effort.

We will soon see that we can use the same procedure used for graphing functions to graph their derivatives (we have not defined these yet) as well, but that is getting ahead of the story. You should realize though that we can compute derivatives for most functions numerically with only a small amount of effort as well.

Exercise 3.2 What is the value of the function (x^2 + 3) at (x = 5)? At argument (10)?

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Would you please give some examples?