Calculus I - Lecture 01 - A Review of Pre-Calculus

thing will do before we get into calculus proper is to do a little review of precalculus so that we're all on the same page as we continue so that's why this first unit is called functions a review of precalculus and first we'll look at the beginning idea of a function and start with the definition of a function so the idea before I write down the definition is the following and I'm hoping you've seen this before so that this will be a reminder remember how functions can be schematically drawn if this X is a value that goes into a function we often write functions like this we say this F represents the function it takes a value X and produces a value we call f of X this f of X is referred to as the image of X under F and sometimes the X is referred to as the argument that results in this image there are other visualizations we can use and I'll put a few of them here you can think of X going into a function machine and the function machine might look something like this it's got an input in an output it's called F and produces the value f of X so you can think of a function as a machine that takes in a value and produces a value that's fine too you can also use the words input and output where function here in the middle might be if you like a computer program and it produces an output finally we often say X goes into the function f and produces a letter Y if we don't want to remind yourself that this came from the function f we sometimes use the letter Y in that case we'll say that X is the independent variable that's the variable that is unaffected yet by the function it's the one that starts the process and then Y which is the result of the function acting on X we call the dependent variable because it depends on what X is and that seems to make good sense so with that in mind as a reminder of the idea let us go ahead and write down the proper definition of a function and use that from here on so a function f is a rule of some kind that has to satisfy a certain property is a rule that associates with each input using that notation with each input a unique and by unique of course we mean exactly one so we associate with each input a unique or exactly one output this is the crucial definition of a function input to output each input gives you exactly one output now if the input is written as it often is if it is written say X then the output is written as you saw on the other page as f of X and we write it this way so that we remember that it's F acting upon X this is not a multiplication of course and let me write that down so we're clear on this this is not a multiplication this is not a multiplication of F times X this is rather the function f acting upon X in this symbol representing what comes out of that process the output another way to say that is so the function itself is represented perhaps by something like this F of and a box goes here and this is where the input goes the input goes into here so F of this empty box is a way of representing the function also now functions can be given to you in several different ways a function can be represented and represented can be described or visualized in several different forms and here are some of them that you'll see often in practice one is it can be represented as a table of values secondly can be represented as a graph and we'll talk more about this a little later it can also be represented as a formula an algebraic formula which is a very familiar way for many students to remember functions but it's not the only one as you can see here also it can be represented in words we can describe what the function does without using any symbols these are all equally legitimate ways of representing a function although formulas is what you see most often please be aware that the other ways are legitimate and will be used in different circumstances in each case just to reiterate in each case a function must be absolutely predictable absolutely predictable and what I mean by that is just another way of saying the definition an input say X always has exactly the same or how about this exactly one in the same output f of X it doesn't matter when you do it the function will always produce the same output for this given input X it will never change that is the key idea to what makes a function different from other sorts of relations finally let me say a word about notation here just to see it in operation a function might be given for example here as this polynomial 3x squared minus 4x plus 2 this is an example of a function given by a formula we might also write y equals 3x squared minus 4x plus 2 using Y and f of X is not quite as informative doesn't tell us what the name of the function is but very often we use this when we want to graph functions which we'll talk about later when we want to evaluate a function at a point say evaluating F at minus 1 this is the north' notation we use for that we put the minus 1 in where the X was and then that means everywhere there is an X we substitute minus 1 so this becomes 3 x minus 1 squared minus 4 x minus 1 plus 2 now all of that will add up to 9 what we then have is this input which is minus 1 leading to this output which is 9 by the action of the function and sometimes we put these two together in the form of an ordered pair and that will lead us directly into our next topic which is the graphs of functions now that we've looked at the notion of a function let's look at how to visualize a function in the form of what you know as graphs let me first begin by giving you some examples here of a number of graphs of functions that you may be familiar with here's the function y equals x squared here is another function which is the y equals x function the line through the origin here is another function that will become very important for us it has the vertical line in the center is what we call an asymptote and this is the function y equal 1 over X and here perhaps is another example where it starts at the origin goes off to the right like this this is the function y equal the square root of x which we can also write as X to the one half these are each examples of the graphs of functions the way we visualize functions most commonly and they're very very handy notice that in all of these pictures each x value was what we previously called an input each x value is associated with exactly one y value and that was what we previously called an output for example if we have an x value here in the x squared picture there is exactly one y value here associated with it and that's of course because when you graph you have X x squared as the coordinates of that point so you need an x value and the y value x squared of course is y here the same thing for all these others if we pick an x value notice that above it there's only one y value above this previous x value there's only a single Y value anywhere along this function there's exactly one y value above it here there's exactly one or sometimes if we pick an X over here there is no y value this is a feature of all functions and leads to a graphical test for whether or not a picture is the graph of a function this is called the vertical line test and it's a test for the graph of a function to see whether a curve in the plane is in fact the graph of a function so this is the vertical line test for the graph of a function and it is an infinitely if statement goes like this a curve in the XY plane is the graph of a function of some function say we call it f that is true if and only if we use a double arrow here I'll come back and Mark that in a second if and only if no vertical line intersects no vertical line that is intersects the curve more than once so that means a vertical line could intersect the curve either not at all which is zero times or once but not more than once and that's the key to the pictures we saw on the previous page here are some examples of curves in the plane that are very familiar but which are not the graphs of functions this is a circle unit circle say centered at the origin this is not the graph of a function y because vertical lines in many places cross the curve in this one this case crosses the curve twice so that's not allowed a vertical line must only intersect the curve once so these are not the graphs of functions and let me give you a second one if you just take something like this you have something that is a perfectly good curve in the plane and may be useful at some later date but does not represent a function as we have defined it now let us look at some examples of functions that are going to be useful to us first of all we have a handy function this is called the absolute value function and we will be using this many times for many purposes the function is defined this way f of X we'll call it which is the absolute value function there's the absolute value notation and we will define this in three parts it is equal to X itself if X is a positive number X is greater than 0 it is equal to zero if X is equal to 0 and is equal to minus X if X is less than 0 that is to say when X is negative the graph of this function is 1 that's very well known it's a simply a 45-degree V with its nose down here at the origin so this would be a line where the slope is equal to 1 and this line here on the left would be one whose slope is equal to minus 1 this is the point origin the origin point here and the features of this curve are that if you have a point two here and a point minus 2 their absolute values are identically the same the absolute value of two and minus two is exactly equal to two and you can see this in this graph a couple of other notes about the absolute value function it is helpful to note that algebraically the absolute value function allows you to write the square root of x squared as a single object the absolute value of x because when you square a number remember the minus sign is removed it is gone it's destroyed and then when you take the square root of that positive number you only recapture the positive square root which means this number whatever it is will always be positive and that will be the absolute value of X that's very very handy in calculations lastly let me point out something that students need to be aware of here I have written minus X be careful - X is not negative here because X is less than zero X is negative so minus X is in fact positive which is what the absolute values should produce in general when you see a negative in front of a variable expression you have no idea whether that expression is positive or whether it's negative you have to know something about the variable in this case we do but in most cases you should find out before you assume that this was a negative number in addition to the absolute value function functions of other sorts can also be defined in pieces now these come up often enough and are called piecewise defined functions that we ought to take a look at one so here's an example f of X equals and we'll define this in three parts so three pieces first part it will be zero if X is less than or equal to minus one we'll define it to be one minus x squared if X is caught between minus one and one and we'll define it to be X if X is greater than or equal to one writing it in that fashion this part of this expression here defines the domain of F and that's how these piecewise functions are organized the domain is here this is the functional value on those domain strips and I will explain what I mean by strips in a moment but you see the function is divided up into parts that are less than minus one between minus 1 and 1 and greater than minus 1 that means when we look at a picture of the function if we Mark minus one here and one here and draw vertical dotted lines what we see is that the plane is divided up into 1 2 3 big strips and on each strip the function is different on the strip for X less than or equal to minus 1 the function is 0 so it's 0 here going off to the left forever we'll skip the middle one for a moment and look at the one at the other end if X is greater than or equal to one the function is X so let's see at one X would be a height of one and go off at this 45-degree direction now what about the part in the middle well to remind you of something let's look at what this is derived from recall that this is from the unit circle and here's how the original unit circle is x squared plus y squared equal one that's the equation of the full unit circle if we solve this for y you'll find that you get Y is equal to plus or minus the square root of 1 minus x squared the plus or minus indicate either the upper semicircle or the lower semicircle so by choosing the plus version which is what we have up here we have the upper semicircle and the reason that we need to distinguish the two is that the semicircles individually are graphs of functions but the circle itself as we saw is not the graph of a function so that graph will be a graph that looks like this with an open hold at this end because the function is not at at one is defined to be X up here which is one which is this point is no hole there's a hole there so there's no point there and there's also a hole here but you can't see it because it's filled by the zero from the previous part of the function now this function is fully defined and what we have here are as I said three vertical strips and that's the way to think about drawing piecewise defined functions next we'll look at what are called the domain and range of functions remember the image I had of a function before that X goes to f of X via the function f if you look at the set of all possible X's so we write that out as the set of all allowable and that will be determined by the function the set of all allowable in and you put them all together that is called the domain of F so all the numbers that can be allowed into the function is called the domain of the function and likewise the set of all these f of X is the set of all resulting outputs that is to say outputs that result from these allowable inputs the set of all resulting outputs is what we call the range of and this is the domain of F this is the range of F there we go now most of the time we will be concentrating on the domain because that determines what values the function is allowed to operate on and the range is just what comes out we sometimes will look at the range but the domain will be more important for us and very often we will not even mention the domain of a function so we need to address that issue if no domain is mentioned which often happens we will make the following assumption we will always assume the largest possible domain that's what we always do so that we don't have to keep writing down domains for every function will always assume the largest possible domain called the natural domain of the function f so the natural domain of a function is simply the largest possible set of allowable values that the function will act upon let's look at some examples find in all of these the natural domain now this is a good exercise and something you want to keep in mind as you work with functions don't start thinking of functions especially when they're given as formulas as just result as just algebraic expressions to be manipulated remember there is a domain involved and you have to be clear on what that is f of x equals x cubed in this case the domain of F and here I'll abbreviate domain of by writing do em nice convenient abbreviation the domain of F is all real numbers and sometimes that's written with a boldface R because any number can be cubed there's no restriction whatsoever here if we look however at this function f of X equals say 1 over X minus 1 times X minus 3 in this case the domain of this function is all real numbers except two numbers except 1 and 3 and the reason is if you put in 1 or 3 here you'd get division by 0 so those numbers have to be left out they are not allowable in this function therefore the domain of this function can be written as the union of these intervals from minus infinity to 1 open Union 1 2 3 open and then Union 3 to infinity open that's the way we would illustrate this with the symbols of intervals if you wanted to illustrate this with a number line you could you might have a picture that looks something like this here's one here's 3 they're open holes so they're not included but everything else on the real number line is included finally for examples let's take this one a little bit more complicated looking x squared minus 5x plus 6 under the square root symbol now you may already notice that that can be factored so I'm going to go ahead and do that X minus 3 times X minus 2 and let's examine the domain of this function now here in the previous one what could go wrong what could go wrong is division by zero in the square root situation what could go wrong is if there were a negative number under the square root the square root of a negative number would leave us to lead us to a complex number which we won't be examining in calculus so this has to be non-negative underneath the square root symbol so domain of F will be all real numbers except for some that will eliminate here all real numbers where the underneath part which is originally x squared minus 5x plus 6 wherever that is greater than or equal to zero now what you should do here and this is a good algebraic exercises verify this calculation which leads to the domain being minus infinity to two including two Union three including three to infinity and you can draw a number line picture but it's going to be everything here from two to the left and everything here from three to the right if you like so verify this calculation for practice but these are examples of domains natural domains for very familiar functions so let's summarize the usual reasons to restrict a domain these are things you want to keep in mind the usual regions to restrict a domain the first one we've already seen is avoid division by zero that is not allowed so we must make sure that that can't happen we also want to avoid little more generally even roots of negative numbers now our example was only the square root but any even root would have the same difficulty finally here is another reason to restrict a domain we might be working in an applied problem in which there are restrictions physical restrictions that the problem imposes so we might need to meet the conditions of an applied problem for example in an applied problem you might be working with a length or an area those numbers have to be zero or positive they can never be negative also in a problem you might have minimum or maximum values that are determined by the physical situation you can't go larger or smaller than a certain number because of something happening in the problem that would be a restriction that we can't mathematically define but you'll have to get from reading the problem the other two are very easy to observe these mathematical restrictions division by zero you just look in the quotient avoiding even roots of negative numbers is also straightforward so keep these in mind as you look at domains now for a couple of exercises the first one is the following compare the natural domains of the following two functions f of X equals x squared plus x over X plus 1 and G of X the other function which is just the function X give those a try something to be aware of in finding the natural domains of these functions and then comparing them let's see what we have first thing is let us observe in the first function f of X certainly you see that there's an X in both of the terms at the top and so you would be tempted to factor that out and you should so if I write this as x squared plus x over X plus 1 I can factor an X out of the top and get x times X plus 1 over X plus 1 now this is where you have to be careful the naive student might say well X plus 1 over X plus 1 is 1 and I will remove that so that f of X becomes X which is the same as G of X and then conclude that the two functions have the same natural domain but that would not be correct the reason is X plus 1 has a variable in it because it has a variable it might possible zero sometime and if it's ever zero dividing zero over zero is not going to give you one and of course it is zero one case it is zero when X is equal to minus one so what happens in fact the function x has two parts it's equal to X if X is not equal to minus one if you don't have minus one for X then this cancellation is indeed correct however if X does equal minus one the function f is undefined it has no value whatsoever which means the domain of the function f is equal to everything in the real numbers except minus one so you could write it this way as two open intervals or you could draw it this way in the number line is this is minus one it is everything except that number that is different from the domain of G which is all real numbers no gaps because you could put minus one in here and get back minus one no difficulty so this is a warning to be careful when you have quotients and not to cancel without thinking for a second problem let's look at one that's a bit more elaborate and takes a bit of time to write out and it is a nice long word problem you might say that will allow you to think about several things so here we go a company has a joined a 1000 foot squared rectangular enclosure so think of a yard or a pen enclosure to its building okay we are told that three sides of the enclosure are fenced in all right then there is the side of the building that butts up against this enclosure so the side of the building say adjacent to the enclosure this pen that we are creating is a 100 feet long and the enclosure may not be a hundred feet long itself so we'll write down that a portion of this side of the building is used as the fourth side of the enclosure okay let X and y be the dimensions of the enclosure and since there are two possibilities let's just set this where X is parallel to the building and finally let's let the letter L capital L be the length of the fencing we need of the fencing required now I haven't even gotten to the question yet this is just writing down the situation accompanies adjoined a 1000 foot squared rectangular can closure to its building three sides of the enclosure or fenced in the side of the building adjacent to the enclosure is 100 feet long and a portion of the site is used as the fourth side of the enclosure let's call the dimensions of the enclosure x and y where X is measured parallel to the building let L be the length of the fencing required now here are your questions question a find a formula for L in terms of x and y second question B find a formula for L in terms of X only not x and y just X final question what is the domain of the function that you write down in Part B so there are your three questions it's time for you to do a little work on this problem here are the three questions you're asked to answer find a formula for L in terms of x and y find a formula for L in terms of X and then what is the domain of the function in Part B which will depend on the circumstances of this problem so let's take a look and what kind of an answer we can come up with well the first thing you should do is draw a picture now I know that the side of the building is 100 feet long so I'll write this down this way 100 feet and I'll write side of building I don't know anything else about the building so I probably shouldn't draw anything else then I know I have this rectangular enclosure here of some kind butting up against the building so here's the enclosure I'm told to label the part that is parallel to the building by X and then the other part by Y and notice that both pieces here are the same length so I'll mark Y on both sides I also have this piece of information that the area of the enclosure is 1,000 feet squared okay well let's see if we can answer our questions L is the length of the fencing around this enclosure and we want to write this in terms of x and y what we just need to look here we see we have x + 2 Y's so X plus 2y does the trick Part B we want to write the same L the length of the fencing but we don't want to use Y so we'll have X plus 2 and now I need to find some other expression for Y which means I need to go back here and say is there some information here that would allow me to write x and y in terms of one another well here's a piece of information we have a news death the fact that the area is 1000 foot squared now the area of a rectangle is defined to be the product of the two sides which would be X Y in this case X Y of course is given to be 1,000 so there we are we can write Y as 1,000 over X and then substitute that down here and get 1,000 over X so now we have L written in terms tautly of X finally we want to find out what the domain of L is well before I write anything down here let's make a few remarks first of all X cannot be 0 why would that be well look up here x times y has to be 1,000 if X is 0 that would make this side 0 and that wouldn't happen so X certainly cannot be 0 X can't be negative why is that well that's from the circumstances of the problem there's no mathematical reason why X can't be negative but in the problem notice that X represents a length and a length is either 0 or positive but never negative so here's a restriction that's determined by the problem not by the mathematics finally this picture also suggests that X is the length of the enclosure in this direction the side of the building is only 100 feet and this has to butt up against it so the largest X could be would be 100 so X has to be less than or equal to 100 also determined by the problem putting these all together tells us that the domain is going to be 0 to 100 100 here where 100 is included so it can be 100 feet or it can be anything bigger than 0 but not 0 so that is the domain for this function those are the X values that are allowed so these are the allowable X values and I hope that you found this problem intriguing as I did viewing windows which is most appropriate for graphing calculators but does have bearing on computer software also viewing windows means that whenever you produce a graph on a piece of technology you can only look through a given window the window that the software or the graphing calculator allows you to look through what we do in the way of marking this is we call this say the a B by C D window now what is the definition here this is considered to be the closed interval a B horizontally and this is the closed interval C D vertically then the four corners are labeled by whatever points are necessary here if this is a B and this is C D that makes this upper point a D the one to the right will be B D the one to the lower left will be AC the one to the lower right will be B C this is horizontal first vertical second that's the way it's always done now this is general on calculators what you'll find is that the a and B and the C and D that define a window actually have different terminology so you might find for example this is fairly common although every calculator is slightly different this is common on the ti calculators you might write x-min and x-max for instead of a and B by Y min and Y max so this is the X minimum which is this point here this is the X maximum this point Y minimum this height and Y maximum this height so these seem to be very natural choices and are used fairly commonly now there is the problem with a window of choosing a window and it is important to do it well and sometimes you can use the calculus to decide what points ought to be in the window and sometimes you have to just use what knowledge you have of algebra but in every case you need to be careful what window you choose and not always choose the window that's oriented around the origin that may not be where the function is most interesting let me give you an example of what I'm looking at suppose we have a function like this so it might looks like something like this and suppose we have a window like this window that I bring up here now if I look right there the function is fairly nice and smooth it has a nice concave up looking figure if I look around the origin I see nothing at all so I would see no graph of the function if I look over here to the right I find out that the function has many ups and downs that I didn't see in other windows so depending on where you look you are determined what you see is determined by the window you choose so you want to make sure that you choose a window that reflects as much of the graph as possible and sometimes you can't do that you'll have to take separate windows with separate scales but that is something you want to check every time you look at choosing a window you're probably familiar with the very useful feature of graphing calculators and software that you can zoom in and out well when you zoom in and out you have to be careful and here are a couple of notes on that let's take an example here suppose we have the function y equals x to the fifth times X minus 2 and you want to look at this on your own graphing device let me draw two pictures now here is picture number one and here will be picture number two and both of them will imagine are centered on the origin so these are windows that are centered on the origin and both of them go from -5 to 5 in the x-direction and here is the graph of the first one and here is the graph of the second one now my question is which is the graph of y now they look very different this one seems to be flat and just goes back up this one comes down as flat then it has a little dip below so there's certainly different graphs which one would be the graph of the function y do you think well it turns out that the answer is both they are both the graph of this function they are simply zoomed in different ways this first one actually the window is -5 5 by minus 1000 1000 the next one is minus 5 5 of course by minus 10 10 in the first one you've zoomed out so far that you've lost all the structure in here you're just too far away to see what's going on in this one you're close enough that you can see this little dip which is completely gone here that is the advantage and the disadvantage of zooming you have to know enough about your function so that you don't miss out on detail like this that's where the calculus can come in to help you find and locate this detail so you can find a window good enough to show what you want to see and in this case this is a better window than this one is but sometimes you'd actually need both because there might be features here that would disappear at this scale so knowing that keep in mind that zooming is something to be careful about finally there's some common errors you should be aware of when you use graphing technology errors in resolution one of them is if you look at the aspect ratio you might have a distortion and mathematically that doesn't really matter but psychologically it does matter for example suppose you're interested in graphing a circle which is centered at the origin and has radius say 5 just to make it large enough if you have two pictures like this here's the first one centered at the origin here's another one also centered at the origin then this is what you'd like your picture to look like an actual circle here you might have this picture also which is exactly the same thing as it turns out except that the window is different - 10 - 10 by - 10 - 10 this is a perfectly standard window where you have you're going ten units in each of the four directions both x and y here in order to get this to look like a circle we have to change the orientation the actual window has to be from minus 17 - 17 by - 10 - 10 so the X distance had to be spread out a little bit in order to make this look like a circle that's going to depend on your own calculator or the software that you're using so sometimes although you'll have mathematically the correct curve it may not look like the curve you're used to because the scaling will not have the right appearance here's another thing that can go wrong pixel problems is one way to put this nice and simply and the problem is that pixels especially on graphing calculators where they're so large can leave gaps apparent gaps in curves here's what happens when you look at a screen with pixels if you look very closely and computer software screens of course is very hard to see this but I'm graphing calculator see screens you can see that pixels are these little dots that make up the screen and that's resolution that you can get so what kind of problems happen there well this is a fairly crude resolution one problem can happen in an example like this we have our pixels again here and we're trying to graph a function that has a very very steep slant then here's what will happen you might get a pixel here one here and one here now there are gaps there there are no gaps in the actual curve you're graphing but there are gaps in this picture why because this is also determined by the definition of a function by the vertical line test a function which is what the calculator is trying to graph a function only allows at most one pixel which represents one point in the image one pixel in each vertical column because the vertical line test has to be satisfied for a function so if you're being slanted like this and your pixels are these fixed sizes what happens is in this column there's only one point in this column there's only one point represented by pixel and in this column there's only one point but visually it looks like there are gaps so you should be aware of that when you're using a graph er here is another thing to be aware of that often happens sometimes you have line segments appearing that you didn't intend false line segments now there are ways to get around this but you should know mathematically that these are not correct and be suspicious of them when you see them so here's an example and I'll draw both the way it should look and the way it might look if you're not careful suppose you have a curve that has an asymptote here and I'll draw this it comes out of the space so you see that that's not part of the graph this is something I added in order to see the asymptotes what will often happen is this the graph er will not draw this correctly like this but will add a line in here because it's trying to connect the left and the right parts which it shouldn't do so this is a false line a line that should not be there but which the graph R will put in so you have to know mathematically that this makes no sense it is a nice way of representing an asymptote but it is not part of this curve and should not be there so you have to remember that graph errs suggest correct graphs but will often be an error if you look too closely if you look at the details so use graphs wisely understand the

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Length: 46min 50sec (2810 seconds)

Published: Tue Jun 16 2009

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[u/cjmancil88]

Here is video 1 from him, and every video after goes in order and video 1 starts as a short review of precalc. I really hope this helps someone as much as it has helped me :) P.S. sorry about my writing skills, it's the thought that counts, right?