|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
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
|1:06||something like that
none of my f's look the same
you can guess my penmanship grades
when i was young
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
thats the first derivative you should make sure you can do
everybodys favorite derivative
do we know what e^x is?
e^x awesome you have a constant it just stays in front okay and then you have
so if you had
do a little erasing
if this is e^ax
then youre going to multiply by the derivative ax
so for example if you have f(x) 10e^3x
times 3 also known as
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
|3:00||so 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)|
|3:31||sin of 3x
cosine of 3x
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
|4:03||so 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|
|4:30||thats 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|
|5:04||i 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|
|5:38||so if f(x) is tan(x) f'(x) is sec^2x f(x) = cot(x)|
|6:01||f'(x) is -csc^2x you guys know these i hope if f(x) is sec(x) f'(x) secxtanx if f(x) is cscx|
okay you get all the trig ones
you get e
only a couple left
did i forget anything?
the others i think can wait
|7:04||you 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|
|7:30||technique 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|
|8:00||you 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|
|8:31||and 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
we're going to save the most difficult ones for the midterm and final
and give you the easy ones in class
we all got this?
|9:01||so 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?
why would you want to know the original function?
because im going to test you on it can you think of another reason?
|9:33||integrals 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|
|10:00||so 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|
|10:35||look 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|
|11:00||what 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
|11:30||and 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
f(x) is x^4
so whats the derivative of x^4?
raise the power by 1
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
you also get x^4/4
derivative of 1 to 0
|12:30||and if you had
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|
|13:31||we 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
|14:30||what 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
|15:01||you do 4x^2
raise the power by 1 get 4x^3
divide by the new power
would be 10x cause whats the derivative of 10x?
its 10 right?
and then plus c ok?
soo one more
|15:30||just 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|
|16:02||and 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
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
|16:32||not -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|
|17:01||over 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
plus c okay?
|17:31||you 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
|18:05||but i will certainly fight for that
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
|18:32||remember 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|
|19:02||so 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|
|19:31||you 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)..
|20:01||is 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|
well the derivative of cosine is -sin
so the antiderivative of cos
is positive sin
because the derivative of sin is cos always remember youre going the other direction
|21:02||alright, 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|
|21:32||because 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|
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?
|22:34||no more unit circle
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?
|23:11||okay derivative of sec
so the antiderivative of sectan
there were a couple that i skipped a little while ago when i did the derivatives
|23:33||f(x) is 1/square root of 1-x^2
so the original function
is arc sin
also known as inverse sin
because the derivative of inverse sin
is 1/square root of 1-x^2
remember this from last semester?
|24:00||that 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..|
|24:37||f(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|
|25:01||ok lets do a couple of more practice problems make sure we can do it|
|25:48||ok so capital F stands for the antiderivative find the antiderivative if the derivative is x^4-4x^3+6x+2 and F(0)=4|
|26:02||lets do the antiderivative
antiderivative of x^4 is x^5/5
4x^3 becomes 4x^4/4
and then these 4's are going to cancel think about it take the derivative of x^4, you get 4x^3
|26:31||ok? 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 2...is 2x because the derivative of 2x is 2 so this is.. plus c right? dont forget the plus c|
|27:01||now 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|
|27:35||those are all 0
you get 4=c
the function is
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
|29:00||there 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..|
|29:30||0 good, good guess
e^0 is 1
this is 2=3+c
c is -1
and that means f(x)..
so far so good?
|30:03||ok so thats what i was supposed to cover for today for the opening class so ill see you on wednesday which will last longer|