Stony Brook MAT 125 Spring 2015
Lecture 06: The function f'(x); what does f' say about f?
February 18, 2015

Start   okay things you should know for the exam you should know when the exam is its next thursday at 8:45 pm i dont have a room yet but once we know well put it up on blackboard it may be waiting in my mailbox ill make the announcement and professor sutherland will make the announcement you should make sure you know where you are going try to not wait to the last minute to find out where the room is this would just cause you unnecessary stress as a policy for we and most courses
0:32we do not give make ups you should not miss the exam if you have a legitimate excuse thats fine, we had some of you get in a car accident last semester and had a note from the tow truyck service were gonna ask you for a legitimate excuse you cant just say yes im getting my period its not an excuse for missing the midterm we cant give makeups because well end up giving hundreds of them youre suppose to be here and when the course started i said this weeks ago
1:04that we dont give makeups and this is when the midterms are and yu should know this sorry its a hard rule but its how it goes and really try not to miss the final what will be on the exam? limits continuity
1:30the definition of the derivative and if you have a functio if you have a graph of a function what does the graph of the derivative look like if you have a graph of the derivative of the function what is the function of it look like\ so the webassign problems should give you a reasonable approximation of what the exams look like other than thsat i have not seen the exam other than whats general going to be on the exam the simple questions and once i do im still not gonna tell you what the questions are cause that would be wrong
2:03alright so lets learn some new stuff but first lets practice one again so lets practice finding the derivative of something
2:44find the derivative of f of x is x cubed plus x at x equals 2 no you can not use the short cuts if you already took calculus in high school you have to do it the hard way i am sorry all that means is youll know that youve got the answer right
3:04alright i think thats long enough, you guys can follow along we want to find the derivative at x=2 say we have to find the limit as h goes o 0 of f of x plus h minus f of x all over h so the annoying part
3:31you have to figure out what x cubed is so in another words this becomes the limit as h goes to 0 of x plus h cubed plus x plus h minus x cubed plus x all over h okay as i pointed out on monday
4:01the typical way to mess up on the exam is to forget to plug the x plus h here remember this is x cubed plus x everywhere you have an x you plug in an x plus h you plug it in here you plug it in there right h plus h cubed and x plus h dont just put in the first thing that has an x everything okay now you have to expand that out so lets write these down and memorize them
4:32if you multiply out x plus h squared you would get andyou get x squared plusb 2xh plus h squared and if you did x+h cubed you get x cubed plus 3x squared h plus 3xh squared plus h squared, so your terms are 1 3,3,1
5:01and you go from x cubed to x 1 x to nothing and then you go from nothing to x cubed plus h squred plus h squared plus h cubed, you see the patter the x's go down and the h's go up and your co efficients go 1,3,3,1 i recommend memorizing it cause you should figure out where it comes fom x plus h tomes x plus h times h plus h but you dont have to do that over and over again you should memorize that, if you know the binomial expansion or pascal triangle which is easier
5:32for most of you i recommend this so that means we can now multiply that out x plus h cubed is x cubed plus 3x squared plus h plus 3xh squared plus h cubed you get that from taking x plus h
6:01multiply by x plus h multiply by x plus h plus x plus h now you got the minus term here so you have minus x cubed minus x the whole thing over h another pointer when you have a polynomial like x cubed plus x and you do the definition of the dirivative so you have thee terms on the right they will all cancel with the term on the left
6:30if they dont you made a mistake you made an algebraic mistake usually you mess up a minus sin so when you have these terms over here its a polynomial all of them will cancel with something over here if that doesnt happen you made an error knowing you made and finding the error is not the same thing but at leats you know you made one alright cancel the x cube cancel the x and now we have
7:01the limit as h goes to 0 of 3x squared h plus 3xh squared plus h cubed plus h all over h the problem is when you plug in h equals 0 everything on top has an h so its 0 everything on the bottom has an h so you get 0/0 but wait everything on top contains an h
7:31so we can factor an h out of it and we get 3x squared plus 3xh plus h squared plus 1 over h and then this is times h so i guess ill put it over there so you take an h out of every term and then you can cancel
8:00why can you cancel because h isnt exactly 0 its very close so if you had the same thing on the top and the bottom you can cancel these terms now you have the limit h goes to 0 3x squared plus 3x h plus h squared plus 1 okay now you can take the limit and let h equal 0 this will go away and this will go away so they both equal 0
8:32and youre left with 3x squared plus 1 alright i didnt say to do it in x i said do it in 5 you couldve, im sorry 2, you couldve put the 2 in the beginning or you could put the 2 in the end so that becomes a 2 add x equals 2 you get 13 so thats the answer
9:00all that work just to get 13 and if you just write 13 youre not gonna get any credit if you just write 13, no you do the short cuts youre gonna get nothing on the other hand if you get lost somewhere in the middle youll get a fair amount of credit if you write nonsense you wont get credit you get credit for working at it and doing the best you can
9:31but im not in charge so at some point will find out how the partial credit works ah good question, do you have to write the limit at each step im gonna say you dont have to but i would say you should cause its a good habbit but if you forget the limit somewhere here and you keep going thats okay with me ill double check that with professor sutherlan
10:09yes you couldve done the 2 at the beginning and then you would have 2 plus h cubed 2 plus h the advantage of doing it this way we now say find it in a different number the advantage of doing it the other way is its generally easier to do it algebraically if you only have to find it once but this is the general derivtive 3x squared plus 1
10:30and i did it that way for a reason can you start with h on the bottom, yes you dont have to put x+h -x but you should write this to show us you know what the derivative is so you get partial credit demostrate you know what you are doing
11:00write this step so we can see that youre going in the right direction even if you mess up you get something now lets play with this a bit im gonna leave part of this up here erase the rest
11:38its alright you can catch this from the youtube channel
12:07never mind im gonna erase the whole thing now lets think about what the derivative tells us so lets go back and think of a simple expression
12:39so we have the function f of x equals x squared this is x this is f of x im just gonna relocate this a little bit thats just your basic parabula so lets think about the graph for a minute
13:03okay what does it mean what is the derivative telling us, the derivative is telling us the slope of the curve and the slope is constantly changing remember a couple of things about slopes wheres the place to putit a line with tat slope has a positive slope right thats positive
13:30thats a negative slope thats a zero slope so if we think of what the tangent line would be doing the tangent lines are gonna sort of have negative slopes then get flatter and flatter till you get to right there, the slope is 0 so if we were to graph the derivative
14:00lets think about whats happening, well right here at the minimum the slope is 0 the slope would be 0 as we approach that 0 we start with a big number very negative a big negative number and then it gets closer and closer to 0 so the slopes will be doing something like that, the slopes actually suppose to be striaght and from here forward the slopes
14:31get steeper thats a terrible picture thats a line so notice if we have the equation we can graph we can graph the derivative thats not so hard right, its easy for me thats why im standing here with chalk lets do another one
15:09say thats a graph of f and x lets think about what f prime would look like cause we can ask you to do something like so you can expect to see something like this on the exam the first thing you do when you see this type of graph you say where
15:34is the slope 0 the slope is 0 at the top and bottom cause thats where the tangent line will be horizontal now horizontal tangent line is the slope of and that line is flat how do we know its flat well you imagine youre on a rollercoaster, right here you stop and you turn around and come back down your tangent lines are getting closer and closer to flat and right there it stops and come down and right there it stops again
16:02if you had a zero i dont know there and a zero there so far so good?
k loving this good alright now those slopes they start the pointing upwards so its positive very steep and approaching 0 a positive number approaching 0 in between here and here
16:30they become negative then stop about there and become to negative and then turn around and head back to 0 again something like that and then the slopes are positive again and they get steeper thats what your derivative would look like people must have questions cant be this easy
17:00question well review that people didnt hear that so thats okay this is a positive slope, notice ow its pointing upwards and its positive and its getting smaller and smaller and closer to 0 because your line first is tilted a lot and then its tilted until its almost 0 so positive slope means that youre just above the x axis
17:30cause this is the graph of the derivative so by positive we mean by the x axis so dont confuse the original graph with the derivative graph the original graph you look at the slopes the derivative graph you just look at the positive and negatives here we have positive slopes that get closer and closer to 0 so here we have positive numbers that get closer and closer to 0 and here we get negative slopes and then turn around, there still negative but they go back to 0
18:03they start at 0 they step a bit and then go back to the 0's and start at 0 and you get sort of negative stay negtaive and get back to 0 now we have positive slopes agin okay so we have positive numbers again a lot of you have i hate this face
18:37thats a good question okay with the original graph we have a derivative graph the derivative graph is always one power less yes almost always yes for polynomials in other words if you see 2 maxima minimum here then youll see 1 on a derivative graph so if you have 10 on this one youll have 9 on this one
19:00and youll learn when you do derivatives why thats true okay lets do another one
19:31okay say thats our original graph thats gonna be our derivative graph so first thing you always do is try to figure out where the derivative is 0 thats the first helpful thing to do so the derivative is going to be 0 here here and here, thats where we have maximums and minimums the tangent line will be horizontal, you have a 0 here
20:01here and here so thats what there there and there doesnt really matter its just a sletch and the derivative is sort of 0 there cause we have a horizontal asyptote its fkattening out so its kind of 0 so somewhee out here its gonna be 0 but we have to think about that so leave that alone for a little bit now think about the direction that this is going
20:34the slope is positive and then it gets to 0 so the graph is increasing and it gets to 0 so if you have a positive slope like that, rthen over here were gonna have positive numbers and getting to 0
21:03alright now we have a 0 slope and we get negative slopes until we get back to 0 oh well the question is how much does it go down
21:30its very hard to know because its negative and its suppose to come down but then it has to turn around and come back to 0 so it doesnt really matter so you can go down deeper its not important the idea is just a sketch kind of see a pattern now were going back up again then when we get to 0 so when we have positive numbers and we turn around and go back to 0 now we have negative numbers again
22:00but then they sort of flatten out so first they start off steep and then turns around come back close to 0 to something like that notice a few negative slopes though were negative numbers, were heading downwards and we never quite get to 0 cause its an asymptote we never quite get to 0 here
22:33lets give you guys one to work one then were gonna go backwards
23:08there you go thats a graph alright lets do this one so this is kind of a fun graph so ignore the graph for a minute and look where the max and the mins are theres a max and a min here
23:32minimum means the tangent line is 0 so were gonna have a 0 there and were gonna have one here so there so lets just work between them for a moment so we have positive slopes and then back to 0 so the middle part is gonna do something like that so far so good now whats going on here
24:00well notice the slopes starts at 0ish and then get negative and come back to 0 and its an asymptote something like that and its negative so its got to be below the x axis alright what about here well again the slopes are native and then 0ish so something like that howd we do?
24:35so far so good lets do another one of these then were gonna do it in the other directions
25:17think about that one for a couple of minutes sure lets review this one more time
25:31it starts at aproximately 0 its a horizontal asymptote so 0ish and gets negative until we get to here hwere its 0 so this is 0ish so we have negative numbers we come back to 0 now we have 0 slope positive numbers and back to 0 so 0 positive numbers than back to 0 and here at 0 negative slopes and then back to 0 0 negative slopes back to 0
26:02now for a fun one alright thats long enough so couple things to remember if the function has an asymptote the derivative does too because the function is not defined at whatever that number is so the derivative cant be defined there first you can do is take those 2 asymptotes and you just transfer them to your derivative graph
26:32not defined ther cant be defined here the other way around is not necessary true the derivative can be undefined but the function can exist but if the fundtion is undefined the derivative does not exist remember the stuff contenuet if a function if a function has a derivative it has to be continuous you cant take te derivative if its not continuous however if its continuos it does not have to be differential
27:04now other way around the same thing if the function exist the derivative could not exist but if the function does not exist neither does the derivative okay i know i confused all of you with that dont worry about it youll figure it out so the asymptotes here and the asymptotes here second you look for maximum and minimum there are none there arent any 0s here oh well now we go to step 3
27:34and then we get very steep so were gonna have a horizontal asymptote be very steep these are positive numbers and those are positive numbers its looks the same, it isnt actually the same but it looks the same its actually a little flatter and a little steeper but no one cares we wouldnt take off for that and what about here in the middle well the slope is positive
28:01flattens asmuch as its going to flatten there and then get steep again so its a positive very big number looks something like that how do we know] ots a very big number its pointing straight up it hits bottom somewhere around here so whatever that slope is say that slope is 1 then this will be 1
28:30that slope is 2 then the bottom here is 2 and then it goes back up again now here the slopes are positive and they approach 0 so your positive numbers and oyu get to 0 so the graphs are similiar but not the same that was kind of nasty that felt better though i enjoyed that
29:01now lets try the other direction alright lets think for one more minute just think of a curve and this is the maximum right here
29:35whats going on notice the derivative is positive and it gets closer and closer to 0 then 0 and its negative on the other side so think about that its positive 0 then is negative when you see that on a derivative graph that tells you you have a maximum so if you have a derivative graph theres a place where its crossing the axis
30:01when its crossing from positive to e=negative like that thats the derivative graph and the must be a maximum at that point because notice theres a positive here 0 then its negative so if you start with this graph you know that you have a maximum there okay a maximum right here again because youre positive before it youre negative after it and 0 at that spot
30:36what about if we had a minimum well minimum lets see what happens so the graph is going down we know its negative
31:010 then its positive so here the graph would do something like that notice at 0 we have a 0 here to the left of the graph we are negative
31:30the right of the graph were positive this is the original graph this is the derivative graph you can find maximum and minimum from the derivative to go backwards cause you just look at the derivative graph and see where it goes backwards its negative on the left and positive on the right that must be a minimum spot
32:01if its positive on the left and negative on the right it must be a maximum that makes sense so lets write that down so if you go from positive to 0 to negative thats a maximum
32:33if you go negative to 0 to positive thats a minimum those came out really well usually they are just squiells
33:16alright so heres the fun question if youre wondering by the way this is chapther 2.8
33:31what does f prime say about f doesnt say much of anything according to the book
34:06so lets say that this is the graph of f prime what does the graph of f look like so now were going the other direction so if we know thats what the derivative looks like what does the original graph look like
34:32well do this one as a team some fun is going on at -5 or i wouldnt have labeled it theres a 0 ther e notice the left of the graph is negative and the right the graph is positive so the graph must be going down on this side 0 is going up so thats means we have a minimum so on this graph minimum there doesnt really matter where you draw it okay you can draw it down here doesnt amtter
35:05and at this spot at 3 whats going on well we have a 0 so slope is 0 slope is positive before hand slope is negative afterwards so we look over there that means we have a maximum there the graph doesnt hve to look like that it could look like all sorts of things but has to have a minimum there has to have a maximum there
35:34what about whats going on at negative 1 well negative 1 is sort of where it changes the derivative of the graph goes from increasing to decreasing thats gonna be minus one all we need to know
36:02the maximum means a place where the curve makes a mountain the minimum is where the curve makes a valley these are what are called local maximums and local minimums this isnt actually the smallest value on the graph cause this keeps going down and this isnt the bigges value of your graph cause this is going higher but in the region its a minimum and maximum so lets do another one of these
37:01so if thats the graph of the derivative what is the original function look like when you watch the video theres these nice long pauses im sort of off here doing something npw we have the derivative now we have to go backwards remmeber when we found them going in the other direction
37:30first we looked at the max and min and put them on the derivative but now you go so now you go ehere the 0 are the derivativ and youre gonna put maximum and minimums so the derivative is negative here, then its 0 0 and then positive, negative means the graph is going down then at 0 turning up somewhere around there it has a minimum that looks like a good place to put a minimum
38:02okay now we come over here and its positive and its 0 then its negative so it must be a maximum there that looks like a good spot okay because its positive its 0 then its negative, again its negative 0 positive then it must be minimum somewhere around there
38:32and then it only has positive slope so it just keeps going up from there by the way its a polynomial thats a 3rd power and thats a 4th power thats a 4th power question is the graph always drawn abbove the x axis nope thats a perfectly fine
39:01equal point equal valu it can touche, you can do all types of fun stuff no okay so this is just a sketch
39:31you can make nice big max and min you can make sallow max and min you can be above the x axis you can be below the x axis, were just giving us a sketch okay its not important we would have to give you another piece of information so suppose we told you that f of 0 was 10 well they can have an idea where to put this graph becasue it would have to cross the x axis at 10 if in told you that 0 was at -5
40:00you would have to cross down here at negative 5 so if we dont give you any information you just sort of draw a picture so far so good so lets do one other thing cause i want to make sure youre ready for this for homework/webassign that way once i covered this then we can do lots of review next week okay so you got the idea to graph it well review this some more on momday
40:34right you pick your own min amd max its just a sketch okay this w could slide anywhere you want and you get the same derivative graph, the derivative graph will move too but we didnt give you any information so its very hard to do youre just sort of getting the feel its called qualitative just getting the sketch
41:45suppose we asked you for something like this so remember we told you the derivative is the slope of the tangent line suppose i asked you for the actual equation of the tangent line how would you do it
42:01should we do it together why not so look what class is this this is calculus so what are we gonna ask you to do, your gonna have to find the derivative its calculus what else would we ask you to do its not asking you for reaction rate, thats chemisrty by the way reaction rates are derivatives but thats a whole other thing caause there changes and yea
42:30so lets find the derivative of that at x equals 1 what are we going to be looking for the equation of a line, what is this sorry thats f of x is 2x squared plus 5 kind of looks like that, doesnt really matter we want to find the equation of the tangent line there at x equals 1 so far so good? we want to find the equation of the line
43:01now what is the equation of a line y=mx+b but theres also y minus y1 m times x-x1 okay thats the equation of a line much better way to learn it we know x equals 1 when x equals 1 we could already found out what y is we know that f of 1, you plug in 1
43:30you get negative 2+5 is 3 so this is the point 1,3 so this means the equation of our line would be y-3 is m times x-1 so now we just have to find m and thats the calculus part, yay so how do we find m we find the derivative at x equals 1 so were gonna do the limit
44:03h goes to 0 of f of x plus h minus f of x all over h so were gonna do it at x equals 1 so were really gonna find the limit h goes to 0 f of 1 plus h minus f of 1 all over h so far so good
44:32now lets do a little algebra so what os f of 1 plus h well it is negative 2 times 1 plus h squared plus 5 minus f of 1 we found f of 1 its three all over h
45:03that equals the limit h goes to -0 is negative 2 times 1 is 2h squared plus 5 minus 3 all over h
45:34that becomes the limit as h goes to 0 minus 2 minus 4h minus 2h squared plus 5 minus 3 over h not so bad because 5 minus 3 minus 2 0 is now the limit
46:01h goes to 0 minus 4 h minus 2h squared all over h okay its just the algebra thing just like before factor out an h you get an h times negative 4 times 2h over h cancel and the h goes to 0
46:32you get negative 4 so that means the equation of your tangent lin is y minus 3 equals negative 4 times x minus 1 you do not have to rearrange the line into some other form you can leave it in that form, that is the equation of a line oh webassign might make you rearrange it but that i cant answer
47:06what happens when you do the limit as h goes to 0 what happens to h it becomes 0 okay y equals m x plus b formula is not like the better form of a line thatn some other form however webassign you may have to put it in a particular form, that i cant control okay but we dont care i cannot hear you
47:31on the exam this will be the full credit answer were looking for can you find it the x value can you find the y value, can you find the slope in order to find the slope you have to do the definition of the derivative you cant just say oh i took calc in high school so i know the answer is -4 you wont get points for that lets do one more of these to make sure we understand it
48:03im gonna erase in a second so i hope you guys all wrote this down take a picture it last longer
48:51find the equation of the tangent line to f of x is x squared minus x plus 1 at x=2
49:11alright im getting ready to do this so remember youre gonna need to find the equation of the line y minus y 1 equals slope times x minus x1 we know x1 is 2
49:30and we can find y by plugging in 2 f of 2 4 minus 2 is 3 you got y minus 3 m times x minus 2 half way home now we just need to find the slopes so got to find the derivative lim as h goes to 0 of f of 2 plus h
50:02minus f of 2 over h so far so good plug in 2 plus h you get the limit h goes to 0 2 plus h squared minus 2 plus h watch your minus signs plus 1
50:30minus f of 2 which is 3 all over h lets do a little multiplying out limit as h goes to 0 of 4 plus 4 h plus h squared minus 2 minus h plus 1 minus 3 all over h, make sure you distribute the mius sign
51:11limit as h goes to 0 so notice 4 minus 2 is 2 minus 3 is 3 thats all 0 im sorry minus h all over h
51:31now you can pull an h out you get 4 plus h minus 1 over h then cancel, these all sort of look the same dont they thats the idea now you plug in h equals 0 and you get 3
52:01kay so the slope is the so the equation is 3 y minus 3 equals 3 times x minus 3 or if youre doing webassign y equals 3x-3 why would i subtract what?
you can subtract h from the 4h it doesnt make a difference
52:32alright ill see everybody sunday night or monday