Stony Brook MAT 126 Spring 2016
Lecture 08: Midterm 1 Review
February 22, 2016

Start   are we recording this now?
theres no seats, wait there some seats in the middle.
give you a couple minutes see if you can do that.
this one.
we wouldnt give you numbers this hard on the test..thats not me.
1:37just in case you should know how to set it up there you go. alright.
there were 2 of these.
and they were both-so now we have 1.
it works. alright.
for now.
we wouldnt actually give you numbers this difficult to work out but the numbers we give you were going to expect you to be able to do.
2:04so first you want to approximate the area from 2 to 10.
so 5x^3+1 is just one of these sort of upward tilting, polynomial thingys.
and you go from 2 to 10.
i recommend you draw a picture because you might get a point for that i dont know.
ill put it this way if you draw a picture and you can see where you went wrong its better than if you sort of just write stuff down.
2:33okay the idea of partial credit is that we can see where you made your mistake if we think your mistakes not a big one you get most of the credit.
if we cant see what you did if you just scribble stuff you dont get partial credit for scribble.
so the right hand rectangles would look like this.
theyre going to overestimate.
and lets find the area of each of those, well each of them is 2y.
3:00and then you would do f(4) + f(6) + f(8) + f(n).
what is f(4)? its 5(4)^3 +1.
then 5(6)^3 +1.
5(8)^3 +1.
3:31and 5(10)^3 +1.
that is 2 times 321 plus uhh..1,081 plus uhh 2,561.
plus 5,0001.
is that correct?
alright now i have to add them up.
uhh lets see 1,402...3,963...8,964 times 2
4:07alright which is 17,890 nope nope 17,928.
that right?
good huh?
just to show i can actually do it.
do you want to get that again? ready?
4:37now how would we write the actual riemann sum?
he didnt get a chance to do that a lot in the superbowl did he?
alright im going to erase this.
5:20so when you do the riemann sum you need 2 things you need the width of each of these right?
and then youre going to need the f of them.
so the width while its going from 2 to 10
5:32so 10-2 is 8.
this is 8/n.
and then we have to do f.
so you take the starting point which is 2.
and then you add 8/n(i).
okay thats reimann sum thats all thats involved and then youre going to take that and plug it into the equation.
6:01its going to be 8/n sigma i=1 to n of [2+(8i/n)]^3 +1.
and 5.
thats the reimann sum, yes?
you always start at 1.
i always goes from 1 to n.
this is always the left end point
6:31this is width times i and so this is the same as this.
will i get points off if you take 8 out?
if i leave the 8/n inside the sigma thats fine.
you could put it either way.
okay because since theres no i in the term you can take it out of the sigma or put it in the sigma.
7:04umm now if you wanted to compute the riemann sum you could either do it the hard way or the easy way i recommend the easy way.
okay so you would find the integral from 2 to 10 5x^3+1
so what is the integral of 5x^3+1 dx?
well lets see..integrate you get 5x^4/4 +x.
from 2 to 10.
plus in 10, plug in 2 and subtract.
so lets see thats uhh 12,510
8:00minus.. 22.
okay which is 12,488.
so far so good?
its not perfect, sorry i'll choose a better one.
we like that?
of course as i said you can work it out long hand but i cant imagine any of you actually want to do that to yourselves.
alright we all understand riemann sum stuff?
hello yes?
2 is the left end point youre starting from x=2.
youre taking 2 and first youre moving 8/n to the right of it.
okay so thats why you start at 2+(8/n)
9:00because i=1.
if i was 0 youd be doing the left end points.
you do not have to if you dont want to.
why is 8/n?
well how far is it for 2 10s 8 meaning you cut it into n rectangles so the width of each rectangle is 8/n.
9:30now why is the 8/n in the outside than in the inside?
it doesnt matter it could be either place.
you can move it out of the sigma and put it in the sum.
do you need to write the limit? i dont think you need to.
but if you want to, just in case, write.. the limit n goes to infinity.
the important thing is the sigma not the limit.
10:16other questions?
i could do another one of these if you want but theyre time consuming.
lets do some other stuff we can always come back.
alright we can do some stuff on the review sheet or i can make up stuff similar to the review sheet?
10:32new stuff? okay.
sorry about the minus signs, you know thats why i dont punish you guys if you occasionally miss a minus sign.
i was doing this late at night.
you know..sitting in the arm chair.
kind of mellow.
and i said maybe i should work out some of these.
and instead of falling asleep i decided to do them.
this is the result.
let see.
11:07b says f(x)= e^x+ (4/x) and x at 1 =0 and i should say f(1)=0 sorry about that.
that says e^x.
11:30f(1)=0 repeating 1 more.
12:00alright next one now.
f'(x) is x^2- radical x+4 if you want to find f(x) we integrate that.
the integral of x^2-radical x +4 dx is x^3/3 what is square root of x? well thats x^1/2
12:31so this becomes x^3/2 over 3/2 +4x plus a constant.
in dividing by 3/2 its like multiplying by 2/3.
so its x^3/3 minus (2x^3/2)/3 +4x +c.
okay thats f(x).
okay? again, were just integrating.
and then
13:00we know that f(1)=8.
so who knows how that comes out, 8 is 1/3 minus 2/3 +4 +c.
so lets see that is uh, 11/3...8..take it away, 13/3 equals c.
soo f(x) is (x^3)/3-[(2x^3/2/)3]+4x+13/3.
13:35so far so good?
you like that one?
how about the second one. so were going to integrate e^x.. +4/x dx the integral of that is e^x
14:03+4... ln of the absolute value of x+c and we know that f(1) is 0.
thats f(x) right?
so 0 is e^1 + 4ln(1), ln(1) is 0.
c is negative e.
14:30so this would be e^x +4lnx-e.
did i do that wrong?
e^1 is e, right?
e^1 is e because anything to the 1 is itself right?
pi^1 is pi..good.
We like to put that stuff in.
You put in pi, immediately 20% won't get the question.
now ive got sinx +2cosx.
15:02sinx +2cosx dx so f(x) would be.. the integral of sin is -cos the integral of cosine is sine +c.
and f(0) is 1.
so 1 =....the cos of 0 is 1.
so this is -1
15:31+0+c so c is 2.
on the exam dont just write +c rewrite the answer so here you would have f(x) is -cosx +2sinx +2.
dont forget that last step okay?
here you have to write out e^x +4lnx-e.
and so on, ok?
so far so good, alright.
lets give you some more stuff.
let me know when i can erase.
its good?
how about..
17:04and lets make up one more.. there ya go, theres 3.
lets do the first one.
the integral of x^3 square root of 5+x^4 dx
17:37you look for function and its derivative.
well the derivative of x^4 is 4x^3 you have an x^3 thats pretty close.
so lets let u equal 5+x^4.
then du is 4x^3.
18:04so i look at the integrand i dont see 4x^3 i see.. x^3.
so i divide both sides by 4.
and you get 1/4du is x^3dx.
so now i can rewrite that integral 1/4 the integral of square root of u du.
18:33now square root is to the 1/2 power.
so the integral of that is (1/4u^3/2)/(3/2) plus a constant.
flip the fraction and you get 1/6... u^3/2 is 1/6 of... 5+x^4 3/2
howd we do on that one?
that one is okay?
not too bad?
lets do another one.
the integral from 0 to pi/2 xcosx^2 of dx.
two ways you can do when you have limits of integration.
one is after you do substitution you can change the limits
19:32as well then you have to substitute back.
or you can sort of ignore the limits.
when youre done with the integration you substitute back and then you put the limits back.
so for now thats the only way ive taught it so well do that way for the moment.
i look and i see x^2 and an x so im going to let u= x^2 the du would be 2x dx.
20:01so 1/2du is xdx.
so ignoring the limits i would just have 1/2 cosu du which is 1/2 sinu which is 1/2 sin(x^2)
20:30now we go from 0 to pi/2.
that equals 1/2 sin of.. pi^2/4 which i have no idea what that is minus 1/2sin0 which is 0.
so far so good?
how did we do on the second one?
21:03eh? good?
alright my goal is to send some of you to med school.
no its not the sin thats squared its the x thats squared.
the pi/2 is squared.
21:31its not the sin thats squared its the pi/2 thats squared.
you got that?
youre supposed to know the unit circle.
you got a little more than 24 hours to rememorize it.
its up on the insta.
you can find it i put it up there about a week ago.
got a lot of likes for that.
i mean who puts a unit circle on their instagram and gets likes?
22:06alright lets do this one.
i could let u equal e^x+5 why would i let u equal the denominator?
well if i let u= the numerator du is going to end up down here and have like 5 plus du's.
have no idea how to do it.
but if i let u=e^x+5 then du is e^xdx
22:31and this just becomes du/u.
the integral of du/u is just the natural log the absolute value of u plus a constant.
so this would be the ln of.. e^x+5 +c.
you dont actually need the absolute value bars because e^x+5 is always positive.
23:03okay e^x is always positive.
never 0, never negative.
alright i had a request for a specific problem.
this one i was kind of proud of its slightly nasty but thats okay.
23:45i have f(x) is... this is number 2c on the review sheet.
cosx sinx square root of 1-t^2
24:01dt we have to find f'(3) and f'(5pi/4).
but first lets just find f'(x).
this again, this is on the review sheet number 2c.
okay so the derivative of this youre going to get the square root of 1-sin^2x.
24:30you plug in the top function.
times the derivative of sinx which is cosx.
minus.. the bottom which is 1-cos^2x times the derivative of cos which is -sinx.
so then you look at this and you panic because you say what am i going to do i dont know how to find sin of 3.
but remember in integral trigonometry you learned all those trig identities for a reason.
trig identities make your life much much happier.
25:02so 1-sin^2 is cos^2.
and 1-cos^2 is sin^2.
and whats the square root of cos^2?
cos. and cos times cos is?
square root of sin square root of sin?
25:30sin times sin is sin^2.
this becomes plus sin^2 equals 1.
it doesnt matter what you plug in youre going to get 1 no matter what you do.
isnt that annoying?
i mean you hate problems like that. remember when i told you when i doubt put 0?
you could also put 1.
just look at this and write..its obvious..its 1.
lets move on.
make a little contempt face on the test.
how dare you insult me with such a simple problem?
26:05even a fool could see that this is 1.
alright lets try something else.
you guys were able to do everything on the review sheet with my explanations and all that?
thats pretty good. lets try something then.
how about that integral?
this chalk is sort of skipping on the board.
sin of the cube root of x
26:32over the cube root of x^2.
dont be intimidated its not that hard.
you can do it.
howd we do on this one?
we ace this one?
its not hard come on folks.
actually it is hard.
take it back.
thats alright we can do it.
the cube root of x^2 come on thats gotta be somehow related to the cube root of x.
27:01so if u is the cube root of x also known as x^1/3 then du would be 1/3x^-2/3.
which is the same thing as 1/3 times the cube root of x^2.
gotta have a little faith.
gotta figure this.. how do i integrate this?
27:30so 3du oh this is dx.
would equal dx over the cube root of x^2.
so this would be 3sinu du.
and the integral of sinu is -cosu
28:01this is -3cosu + a constant.
=-3cos(x^3) +c.
so far so good?
why is the dx the numerator?
put the dx in the 1 okay?
dont be afraid, ask questions.
28:38oh yeah im sorry you know that.
i need you guys to proofread my thoughts.
be very helpful.
lets give you another question.
29:42okay so g(x)
heres 3 fun questions.
30:33thats a minus.. minus 4.
i dont know if any of you are watching the videos.
are you watching these at home?
31:24its hard to make this stuff up in class-its gotta be 2 right?
otherwise its not a semicircle.
31:30did that change your answers?
want to make it 4?
it screws up the other part..alright.
so g(0) well the integral from 0 to 0 is just 0 right?
cause youre not going anywhere.
any time these 2 numbers are the same you get 0.
cause you havent integrated anything.
you start and finish at the same spot.
when you get the top and when you get the bottom you subtract and get the same answer.
alright we're going from 0
32:01to 8.
thats what g(8) is.
from 0 to 8 of f(t) dt t is a dummy variable doesnt really matter what it is.
well that would be the area of the semi circle.
which would be pi(2^2)/2 feel free to plug in 4.
minus.. the area of this triangle.
well that has a base of 4 and a height of 4.
32:32so thats got an area of 8.
so that would be the answer so it would be 2pi-8.
but for those who used 4 it would be 8pi-8.
now where is g(x) a maximum?
well you could do it 2 ways 1 is you could do it intuitively.
by just looking at the areas.
the other thing is say well look i get some area here right?
then i subtract a bunch of stuff and then i add some stuff back
33:00so the question is have i added back more here than i subtracted here?
no so it has to be bigger here.
that make sense?
i had to get my maximum area here because then i subtracted 8 and i only added back uhh 2.
the other is you take the derivative.
because g is a maximum well were going to want to find where is g prime equal to 0?
well what is g prime?
33:30g'(x) is f(x).
if g(x) is the integral from 0 to x of f(t) then g'(x) is just f(x).
so f(x) is 0 in 2 here. and here it goes from + to - its going from up to down so thats going to be a maximum.
here it goes from minus to plus so it has a maximum at x=4
34:00and itd be a minimum if we cared at x=8.
if i ask about ???? you take the derivative again.
did i skip b?
oh i forgot g'(4).
well sure, whats g'(x)?
g'(x) is f(x).
so g'(4) is 0.
remember what i said when in doubt what do you guess?
there ya go.
34:33not really a great strategy.
right its like baseball trivia when in doubt guess satchel page how did i get g(8)?
its the integral from 0 to 8 so its going to be the area of the semicircle which is 1/2 of (pi)r^2 so itd be 2pi.
minus the area of this triangle.
35:00the area of the triangle is 1/2 base times height is 8.
so itd be this minus 8.
thats good?
alright everybody study hard.
see you tomorrow night in javits at 8:45.