Stony Brook MAT 126 Spring 2016
Lecture 01: Review of derivatives and basic antiderivatives
January 25, 2016

Start   so if you have any function and you want to take the derivative you should be able to take the derivative of any function so the basic thing is if you have some kind of polynomial remember what you do?
you.. i'll use this side you bring the power in front and you subtract 1 from the power and if you have a constant here just leave the constant for example
0:44remember that thats the easy one right?
god forbid that was on the final so you just bring the power in front multiply them together reduce the power by 1 and of course youre going to have a whole chain of these things its a polynomial so if you had
1:06something like that derivative none of my f's look the same you can guess my penmanship grades when i was young ok bring the 3 in front and you get 18x^2 bring the 2 in front you get 10x and 4 is a constant and the derivative of a constant is 0 ok?
thats the first derivative you should make sure you can do
1:39ok then everybodys favorite derivative e^x do we know what e^x is?
e^x awesome you have a constant it just stays in front okay and then you have
2:01chain rule so if you had do a little erasing if this is e^ax then youre going to multiply by the derivative ax times a ok?
so for example if you have f(x) 10e^3x
2:30f'(x) would be 10e^3x times 3 also known as 30e^3x ok?
this should be totally review for you guys nice and easy okay so thats the second one you want to make sure you know cause the antiderivatives are going to be backwards ok did you guys get to antiderivatives last semester?
you did? good so i'll do this as quickly as i can to review
3:00so you can get outside and have a snowball fight lets see..other derivatives so lets see derivative of sin... is cosine and thanks to the chain rule if x is multiplied by something multiply by a so if we had f(x)
3:31sin of 3x f'(x) cosine of 3x times 3 ok?
nice and easy?
chris i didnt see you back there weve got a whole bunch of people here that i know thought you could hide she cant just cause shes sitting all the way in the back right okay good and i see better far away than i do up close
4:03so the derivative of sin is cos the derivative of cos is -sin so for example if i had cosine of pi x by the way we put pi in problems so that more of you will get them wrong
4:30thats the main reason we use that letter okay as soon as we use pi or e or pi and e i know people are going to go down oh crap, pi alright so theres a minus sign of pi x times the derivative of pi x you get pi ok why do i have a 3 there it should be a
5:04i already posted this stuff from last semester but it should be up again i may do a new version of it, or not see how lazy i feel this week umm ok so sin and cos i may go a little fast through it but we'll have some more of these and all these use the chain rule and the chain rule is going to be important when you do the antiderivative
5:38so if f(x) is tan(x) f'(x) is sec^2x f(x) = cot(x)
6:01f'(x) is -csc^2x you guys know these i hope if f(x) is sec(x) f'(x) secxtanx if f(x) is cscx
6:32almost done f'(x) -cscxcotx okay you get all the trig ones you get e only a couple left f(x).. is lnx f'(x) is 1/x did i forget anything?
the others i think can wait
7:04you should make sure you memorize all of these product rule quotient rule dont really worry about that we're going to work a little bit with the product rule much later but i wouldnt worry about that for now the quotient rule-i mean obviously you need to retain this stuff especially if you're going to be engineers or physicist or a mathematician but other than that it doesnt really show up in this class this class is a lot of
7:30technique of integration which is what we're going to be starting on in a minute okay and then theres some word problemy kind of stuff but not too much some stuff that uses graphs but you wont have to do graphing youre supposed to know how to graph by now i think thats about it but we keep you pretty busy with techniques of integration its a lot harder than it sounds and of course here this year is very important
8:00you have to know why this stuff is true actually most of you dont care why this stuff is true alright so now the whole point of antiderivatives is working backwards alright so derivatives are easy compared to antiderivatives its very easy to smash something its very hard to unsmash something antiderivative is working backwards so when youve got the derivative youre going to go backward
8:31and find the original function which is what we call by technique of integration you can differentiate just about anything but you cannot antidifferentiate just about anything lots of functions do not have an antiderivative or if they do its extremely difficult to find so we're going to save the most difficult ones for the midterm and final and give you the easy ones in class right?
we all got this?
alright antiderivatives
9:01so now the antiderivative is working backwards now im giving you the function and i say this is somethings derivative whats the original function?
why do we want to know the original function?
any ideas?
why would you want to know the original function?
because im going to test you on it can you think of another reason?
9:33integrals are useful for all sorts of fun stuff they show up in statistics they show up in physics bio and chemistry the good news is probably not the majority of you well a few of you will get there but derivatives find tangent lines and figure out how things are changing antiderivatives you can now use to umm find totals
10:00so derivatives you can find say how fast a car is moving and antiderivatives you can find how far the car has gone what are some other types we'll get to them remember how you took the derivative of f to the x so now lets suppose i know that the derivative of a function is x^n and i want to find the antiderivative so i use capital F so im doing antiderivatives im going to write the word antiderivative
10:35look at that handwriting isnt that the handwriting of a c student i dont know why i got d's and f's i should have gotten c's anyway that says antiderivative if you cant read it
11:00what would the original function have been?
well what you do is remember you bring the power in front you reduce the power by 1 so for antiderivatives you want to go the other way youre going to want to increase the power by 1 divide by the power so if you take that in your head you say lets take the derivative of that ive got x^n+1 over n+1 were going to take the derivative bring the power in front
11:30and subtract 1 from n+1 and i have n and since this is divided by n+1 they now cancel like that and im back at x^n but as you may have learned last semester theres more than 1 function for the antiderivative so say.. f(x) is x^4 so whats the derivative of x^4?
raise the power by 1
12:03divide by the new power but theres a problem if you took the derivative of x^5/5 you get x^4/4 thats a fourth but if you take the derivative of.. x^5/5 plus 1 you also get x^4/4 derivative of 1 to 0 ok?
12:30and if you had x^5/5 plus 2 the derivative again is x^4 so every time you take the derivative of any of these functions you get x^4 so what do we do?
we just want the one antiderivative so what we do is we write plus c we say c stands for a constant we dont know what that constant is but when we take the derivative we get that x^n ok?
we just dont know what the constant is
13:01*speaking to student* so thats why we write plus c the derivative of any constant will be 0 so this is whats called an indefinite integral which we get to in a little bit couple classes but essentially if we know what our function is we know what the derivative is precisely but if we know what our function is we dont always know what the antiderivative is precisely
13:31we usually need a second piece of information so if i told you that f(x) is x^4 and i want to find the antiderivative and i told you that the original function has a value of 10 when x is 0 then you would know its x^5/5 plus 10 now if i plug in 0 this comes out 10 and if i take the derivative of this i get x^4
alright so this is the primary one you want to be good at so lets say uhh lets do an example up here i know im going kind of fast but this is supposed to be review so
14:30what would the antiderivative of that be?
so i have x^3 x^3+4x^2+10 and the antiderivative well the antiderivative of x^3 raise the power by 1 get x^4 divide by the new power
15:01you do 4x^2 raise the power by 1 get 4x^3 divide by the new power and 10 would be 10x cause whats the derivative of 10x?
its 10 right?
and then plus c ok?
soo one more
15:30just to make sure we get the idea so there i have square root of x + x^4 + x^-8 and you want to do the antiderivative so for square root of x
16:02and thats the same as x^1/2 so if i wanted to do the antiderivative of x^1/2 i take 1/2 and add 1 to it right?
so that would become x^3/2 divided by 3/2 then for x^4 thats x^5/5 then for x^-8 i add 1 to -8 and i get -7
16:32not -9 its x^-7 over -7 plus c now you can flip that fraction and make that im going to run out of blackboard here 2/3x^3/2 + x^5/5 -x^-7
17:01over 7 plus c hope you guys can see that down there that is really awful looking we'll rewrite it make sure youre very good at playing with your exponents that is 2/3x^3/2 + x^5/5 -x^-7/7 plus c okay?
17:31you all understand these?
ask questions if you get them so far this should all be review, right? yes?
for me i consider that fine ill have to check with professor T some professors are very fussy about all that simplifying stuff i dont think she is based on my conversations with her but you never know some professors say lets get it all the way cleaned up
18:05but i will certainly fight for that umm other questions?
no this will be on the midterm i can tell you that now there will certainly be at least 1 antiderivative alright what if f(x) is e^x or e^kx
18:32remember when we took the derivative of e^kx put e^kx times k so now we're going to have e^kx/k plus a constant
19:02so if i have f(x)=e^10x then the antiderivative would be e^10x over 10 so you kind of do the opposite of what you did with derivatives now that youve learned all that derivative stuff you have to unlearn it and do everything backwards so this gets very confused in your head especially with the minus signs
19:31you have to practice when you look at a function to test what that youve actually find the antiderivative take the derivative you take the derivative of this thats e^10x times 10 having the 10 in the denominator cancels that times 10 that make sense?
if youre not sure you can always take the derivative and make sure that you got the antiderivative correct what if f(x)..
20:01is sinx whats the derivative of sin cosine. so that antiderivative of sin is minus cosine we look for the plus c's on the exam by the way make sure you put that in there how about
20:31f(x)=cosx well the derivative of cosine is -sin so the antiderivative of cos is positive sin why?
because the derivative of sin is cos always remember youre going the other direction
21:02alright, im going to erase that in a minute make sure you all got it first more functions of antiderivatives ok if f(x) is 1/x then the antiderivative is lnx plus c
21:32because the derivative of natural log is 1/x not too many more f(x) is sec^2x then the antiderivative is tanx because its....thats c because the derivative of tangent is sec^2
22:02notice im not going to give you tanx because tanx is the derivative of what function?
not sec^2 soo youre going to have to learn how to find the derivative of tangent its not, its not as simple involves logarithms isnt it great, youre almost done with logarithms forever?
22:34no more unit circle almost still gotta know unit circle just a little bit longer still gotta know what sin(pi/6) is pi/6 is 30 whats sin of pi/6?
couple more
23:11okay derivative of sec is sectan so the antiderivative of sectan is secant plus c ok?
there were a couple that i skipped a little while ago when i did the derivatives
23:33f(x) is 1/square root of 1-x^2 so the original function is arc sin also known as inverse sin ok because the derivative of inverse sin is 1/square root of 1-x^2 remember this from last semester?
24:00that was one of those painful ones that probably showed up on the final inverse trig functions are very useful if you take physics or engineering how can i hand you a scalpel and say now do an inverse trig function so far ive been watching greys anatomy faithfully i have yet to see any inverse trig functions lots of funky stuff though lets see how bout..
24:37f(x) is 1/(1+x^2) tan inverse of x because the derivative of arc tan or inverse tan is 1/(1+x^2) we're almost done
25:01ok lets do a couple of more practice problems make sure we can do it
25:48ok so capital F stands for the antiderivative find the antiderivative if the derivative is x^4-4x^3+6x+2 and F(0)=4
26:02lets do the antiderivative so antiderivative of x^4 is x^5/5 4x^3 becomes 4x^4/4 ok?
and then these 4's are going to cancel think about it take the derivative of x^4, you get 4x^3
26:31ok? for 6x 6x^2/2 thats going to reduce to 3x^2 because you take the derivative of 3x^2 you get 6x and the antiderivative of 2x of 2x because the derivative of 2x is 2 so this is.. plus c right? dont forget the plus c
27:01now this helps us figure out what c is cause now we know what f(x) is in general and we know when f(x) is 0 you get 4 so you take 4 and plug it in those all come out 0
27:35those are all 0 you get 4=c therefore the function is x^5/5-x^4+3x^2+2x+4 that make sense?
howd we do on that?
nice and easy?
on the exam you can just go right to where the constant comes, sure you dont have to do the plug in if you dont want to on the other hand if you get it wrong then you dont get the partial credit but no pressure how do i make it annoying how do i make it harder log limits and logarithms
29:00there you go captial F whats the antiderivative of cos? sinx antiderivative of 3e^X is 3e^x plus a constant andi know when x is 0 i have to get 2 so 2=sin(0) plus 3e^0 plus c sin at 0 is..
29:300 good, good guess k e^0 is 1 this is 2=3+c c is -1 and that means f(x).. is sinx+3e^x-1 so far so good?
30:03ok so thats what i was supposed to cover for today for the opening class so ill see you on wednesday which will last longer