Tutor Bobble is present

Tutor Bobble is going to google Taylor Series real quick

Tutor Bobble remembers now

Let's see if you can beat my tutor

So I know that

f(x)-T(x)=0.01

I'm not sure how to get the correct answer without guessing

did you go over the Lagrange error bound?

Nope, nope

Here's the most we learned about Taylor series

Literally just the infinite equations for sine, cos, and natural logarithm

Very, very basic

so, this is the error bound

What the

Explain?

That's the first thing I thought of for Taylor series error - I think there might be another thing? This is what we used the most.

Well, can you explain this formula then?

So, n = degree of polynomial. What's the degree of this polynomial?

(the one in the problem)

5

Wait of the natural log or the series?

oops, found the more useful formula

n still means the same thing

and degree is of the T(x) given, so here it would be 5

Okay

What is x?

And also a?

well, you have to maximize all of these parts

Geez I feel like this is far too advanced for what I'm learning

so let's take them one by one

a is where the T(x) is centered, so here is 0

if the xs were all (x - 1)s, then a would be 1

Why is it zero for the T(x)?

your T(x) has an "x" term

if there was an a, it would be "(x - a)"

Ah

can you see why it's called a center?

Out of curiosity, what if it started with "1"

Like 1-x^2/2!+x^4/4! etc

a constant term wouldn't change the centering

Oh okay

you can see that it's "x^2", not "(x - 1)^2"

It would have to be (x+1)-x^2/2! etc

Ohh, I see

Hold on, let me jot this down

tell me once you're back

also Bobble Tutor will have to leave soon-ish

Okay

Continue

anyways, so you want to maximize |x - a|, where a is the a you just found, and x is any value that's in the interval given

And how do we find that?

so you pick an x value in [0, a] to maximize |x - 0|

I can pick any value?

yep, any value

Okay, for the sake of cleanliness, I'll go with 10

no, in the interval

so here you would pick a

because that would give |a|, the maximum you can get by picking from the interval

But a is zero, no?

the a in the problem is different than the variable a

yes, it's stupid they picked "a" as a given variable

So pick a number and tells me why it fits. I'm slightly confused.

if it makes you feel better, we can substitute "b"

On interval [0, b], choosing b maximizes |x - a| = |x - 0| = |x|

and I'm just going to give everything else real quick, since I have to leave like now

Ok

now you look at the next term in the series, the power-6 term

Ok

and you choose a value of x (still from [0, b]) that maximizes that term

ok..

then you plug that value in for M, and you multiply it by |b|^(n+1) - since you replaced |x - a| with |b|

Ok

and then you divide by (n + 1)!, where n = 5

Ok

and you set R_n = 0.01, since that's the error

ok

and then I think you would solve for b (which is the a from the problem, not the a of the variable you introduced)

I'm sorry I can't help more

That's fine

calcworkshop.com/sequences-series/lagrange-error-bound

see for some examples/explanation, or just google "Lagrange error bound"

I think I'm just going to go to sleep. If this kind of question comes up on my test tomorrow, it'll be multiple choice

So worst case scenario, I can just plug all the numbers in

If it's multiple choice, I guess you can get away with an even easier method

T(x) approximates f(x) up to x^5 terms, so the error between T(x) and f(x) can be approximated as the next term in the Taylor series

which is x^6 / (5×10×15×20×25)

so the bound will be close to the 6th root of 0.01×5×10×15×20×25 which is like 3.94

(I'm assuming you can use a scientific calculator)

In this case, the Lagrance bound is actually off by something like a factor of 2, and certainly isn't good enough to get the answer required to 3 decimal places. So I think they just want you to evaluate the error directly.

So for any given x you can find that just by computing x e^(0.2x) and x + x^2/5 + etc. and taking the difference.

The worst error is almost certainly going to be at the end of the range where the function is largest, increasing fastest, etc. (The worst error in a truncated Taylor series is almost always at the end of the range.)

For a multiple-choice question, you can just calculate the error at each of the x values they give and see if one of them produces an error very close to 0.01.

Oh, in this particular case you don't need to rely on any heuristic "worst error is usually at largest |x-a|" stuff, because the error is just the rest of the Taylor series for x e^(x/5), all of whose coeffs will be positive, so obviously larger x will make it larger. So the biggest error is absolutely definitely at the biggest x.

So, anyway, for a not-multiple-choice question: guess a value of x, see what the error is; if the error's too large, try a substantially smaller value of x, if it's too small, try a substantially larger value of x, repeat until you've got one x where the error is too big and one where it's too small.

Then do a binary search: pick a value somewhere near the middle of the range (literal binary search always uses exactly the middle, but you don't have to do that; pick a value with fewer digits so the calculations are easier, pick a value nearer one end if it seems like the error is nearer the target at that end than at the other end, etc.) and evaluate the error, and now pairing that with one of the two values you had before gives you a new smaller interval where the error is [... continues]

... too small at one end and too big at the other.

So e.g. maybe you start with x=1 and find the error is about 2.8x10^-6. Much too small, try a larger x.

x=2: error is about 0.00018, still too small but getting nearer the mark. Maybe don't double it this time; try x=3.

x=3: error is about 0.002. Damn, should have gone higher after all. x=4 next?

x=4: error is about 0.013. Aha, so somewhere between 3 and 4, probably quite close to 4 because 0.013 isn't much bigger than 0.01. So try somewhere in between, maybe nearer to 4. Let's try 3.8 and see what we get.

x=3.8: error is about 0.0092. Nice. So we're looking for a value between 3.8 and 4, probably nearer 3.8 because 0.0092 is nearer 0.01 than 0.013 is. Literal bisection would try 3.9 next but let's try 3.85 instead.

x=3.85: error is about 0.00993. Getting close. Now we're looking for a value between 3.85 and 4, probably much nearer 3.85. Let's try bumping it up to 3.86 and see what we get.

x=3.86: error is about 0.01009. Also close but on the other side, so now we need a value of x between 3.85 and 3.86. Let's try 3.855.

x=3.85: error is about 0.010014. Still too large; need something between 3.85 and 3.855. The error at 3.855 is quite a bit nearer the mark than the error at 3.850, so let's try something closeish to that. Maybe 3.853?

x=3.853: error is about 0.00998. So we need x between 3.853 and 3.854. Might as well try the midpoint at 3.854.

x=3.854: error is about 0.009998. Too small but very close. So for an error of 0.01 we need something between 3.854 and 3.855, probably somewhat nearer 3.854. If we want to be sure of getting the right value rounded to 3dp, we need to know which side of 3.8545 the correct value is on. I bet it's the low side, but let's check.

x=3.8545: error is about 0.010006. Too large. So the place where we cross 0.01 is between 3.8540 and 3.8545, so to 3dp it must be 3.854. Done.

That's pretty laborious when you write everything out, but you can do it in a minute or two if you aren't so verbose :-).

[You can ignore the following, but just in case you're interested:] In this case, Bubbler's cruder method gives a better estimate of the error than Lagrange's bound. That's not because Bubbler is a better mathematician than Lagrange :-), it's because of where the Lagrance bound comes from.

There is an exact formula for the error in the truncated Taylor series. It's the integral, from the "centre point" to the place where you want to know the error, of (n+1'th derivative of f) times x^n/(n+1)!. If you just pretend that that n+1'th derivative is constant and as large as it ever gets over that integral, you get an upper bound and that's the Lagrange bound.

But in practice that n+1'th derivative is very much not constant. It's much bigger at the end of the range. So the upper bound you get by holding it constant is not a good approximation to the error; it's too pessimistic.

@Bubbler Ohhh

This is so stupid

I spent so many hours just practicing

Math

And then I get a 60% on my next next because oh! I didn't have time to double-check and I forgot to subtract 180 to find an angle

What's the point of even studying? I get the same score whether I study or not

> I get the same score whether I study or not

seems to happen to me too sometimes

Happens to me ALL THE TIME

I'm already failing math as it is

And my parents have been all over me

that's never fun :(

sin(71) and sin(109) are the same value

Ugh

I forgot about the area of a sector equation

I thought that it was

r^2pitheta

It's (r^2theta)/2

Which was why I knpet getting such weird numbers

I got another question wrong because I thought it asked for even functions

I was so sure it asked for even functions

But everybody I talked to was like "it asked for odd"

I missed like 4 at most

At least 2

I need to have missed only 3 or less to maintain my 70%

What happens if you dip below %70?

That's failing the class

My parents are incredibly upset I have even a C-

oof

can I help?

I don't know

I spent so many hours studying for this unit and I get a 70%-60%

Physics is currently propaganda for data science, so I'm available to rant at if you want.

What's the point of even studying? I fail regardless

And I can try to express this to my parents and they'd be like "well you could've studied earlier"

"Studied more"

"How do you expect to even get accepted into a college with a C- in your junior year"

"You should just drop out at this point"

IT'S ONE GRADE

Yeah, it's still very possible to get accepted with one bad grade

But can I get a scholarship?

(even better if you can somehow show improvement by, for example, taking the class again. but not sure if you can do that)

Well it's two semesters

Depends what scholarships you're talking about

So if I can improve a lot for second semester

Hold on I have to take a test for APUSH

I'll be back

I came here just to tell you that if you have one bad grade, it's one grade and it's no big deal at all. I faceplanted the hell out of AP Physics and a couple other classes in high school, but I still got a really cool scholarship and did fine in college. You'll be totally fine and don't let your parents scare you!

(hey mick!)

hello :)

Did you get financial aid from the government?

Eyy 40/45

Yes, I did

Probably could've got higehr on my APUSH test if I double-checked through :/

Was super annoying but yes

I don't think I could get financial aid

why not?

Government financial aid is tough because they have like 99 rules you need to follow, so one bad grade can screw that up unfortunately.

'borg! unfortunately I have to write data science propaganda right now

@bobble ???

My dad works as a doctor. We're not ridiculously rich by any means, but he makes above what I believe the government is willing to pay for

It sucks, yeah. Middle class families do not always get aid even if they really need it

Yup. And I'd say we're upper middle class

Basically, the government views my family as "You can easily pay for your kid if you wanted to"

:|

My family only squeaked into aid for me because at the time my dad had lost his job and was between careers, so he happened to fall into the income class the year I applied

But in a normal year, we probably wouldn't have qualified.

That sucks, but I guess a silver lining

It was our silver lining for the year :p

And I get that. There are people out there who really do need financial aid

Very very bright students from poor areas wouldn't be able to make it to college without it

But it just makes my scholarship chances lower

both my parents got nice merit-based scholarships, so I'm sorta expected to get one

Financial aid is just stupid and arbitrary sometimes and unfortunately that's outside your control. Try not to let yourself dwell on it or get bummed out, just apply for as much as you can!

send out like 100 applications for as many scholarships as you can, you're bound to hit something

My parents aren't going to pay for any of my college though

They're like "none of our parents helped"

that sucks :(

My parents are first-generation immigrants though. My mom moved when she was 2 and my dad at 10

Wait hold on, can't you get scholarships for disabilities?

usually those are for specific disabilities

but a few are broader

Darn

Rarely but they do exist

and there isn't anything for "tics", everyone wants to help people with Tourette's

I would qualify for Tourette's if I had a vocal tic, since the rest of my tics are severe enough

One of the more niche scholarships I applied to was specifically for people with Asperger's/ASD who were seeking STEM degrees, didn't get it but it was worth a shot :)

I mean, I don't think my arm counts for scholarships

It might help with admission though

stick it in an essay

(Also the fact that I was adopted- great stuff to write about)

The more life experiences you have, the more potential scholarship essay topics you have :p

I was adopted at 7 1/2 years old from Korea and didn't speak any English and now look at me! 96.5 in English. My highest academic class

"What are some obstacles you've faced in your life?" Well... glad you asked

That would make an amazing essay

I just need a scholarship

Have you thought about applying specifically to second-generation/first-generation immigrant scholarships? there are lots of great ones out there!

Oh, I guess I am technically first-generation immigrant

I always forget that

You could totally qualify! :D

Don't you have to be under a certain income?

Only for some scholarships

There are lots of different types - some are merit-based, some are financial-based, and some are just generally open to everyone

Oh wow, that's a lot

Of scholarships

"Cognitive, behavioral, and emotional disabilities"

Darn

That's a pretty wide umbrella of stuff, seems weird to require that for a scholarship

what counts as an "emotional" disability?

Lots of stuff qualifies as emotional disability under ADA - bipolar disorder, Borderline Personality Disorder, etc.

it's a really wide category

Holy crap I have a 3.08 unweighted GPA

That's

Terrible

that's fine

Because I'm taking three music classes and I don't think those count toward GPA

B- average isn't too bad!

Don't mean to be racist in anyway, but I live in a pretty asian-heavy community. B- is like shooting yourself in the foot

ah, i see

That C really kills my GPA. I didn't even realize

Does your school offer any kind of retake/summer thing where you could potentially replace or lift that grade?

If I fail the class I can retake it over the summer

But failing is D, and that's just to get the credit

My teacher isn't allowing retakes at all bc of COVID

Can you retake it without failing it, or do you have to fail it?

I have to fail it

eesh, so you're kinda stuck

there's no option for summer classes or taking it at a community college somewhere?

I don't know

It's Pre-Calc

(well, Pre-calc H)

My high school allowed dual enrollment where you could take a class at my local community college for a better grade, maybe your school has a similar thing?

I don't think so

hm

there's got to be something you can do

Is this a problem mostly with tests, or with both tests and homework?

She only puts in tests

There's no grade for hw

ahh, so you didn't get a homework cushion at all? That's horrible

she's put in 20 points for participation and that's about it

No retakes

Nothing

I can't

Rephrase: are you understanding the homework?

if your grade is entirely tests, that's ... pretty crappy

The Calc classes at my school are infamous for having tests that are much, much harder than the homework, so people can feel decent on the homework and then bomb the tests

if my math classes had been entirely tests, i would absolutely have two Fs on my transcript as we speak

@bobble kids at your school on test day: [screams in pain]

the homework cushion is a necessary safety net for people who do well on homework but struggle with tests

it seems horrific not to have one

that is just mean

(meaner even than my spanish teacher's attendance system we've already covered)

Have you discussed this with anybody outside this teacher? i.e. talked to a dean/principal/whoever takes care of curriculum at your school?

I TA'ed a class that tried to give students a homework cushion - the answers were given, and we would spot-check a few problems to see if the student either got it right or corrected it. You would not believe the number of kids who didn't check their work.

key word being "tried"

the class I TA'ed did a similar thing, we just spot checked some problems and then gave credit for doing the homework. It was funny how many students counted on me getting lazy later on and tried to slip stuff by me

also, <lunch>

g'bye

they would just write some nonsense equations and then write the answer and circle it and hope i wouldn't notice

@bobble Yeah, I'm fine on the HW

are the test problems harder, or is it just the time pressure & environment?

Both, I think

Time is the main problem

Is it that you get stuck and don't know how to approach the problem, or that the calculations take a while to do/write out?

I don't know! Half of my errors are because I plugged a number wrong or something

It's multiple choice, so no points for doing to steps correctly

So you know how to do it, and it's just getting dinged for silly mistakes, basically?

We're just trying to figure out what part you struggled on

these are the standard questions I ask people I tutor who are struggling on tests

It could also be that you feel really prepared going into the test, and then you realize the test isn't what you expected and as a result you struggle?

@Sciborg Mostly, yes

Do you feel like you make enough silly mistakes on enough problems to warrant getting the poor grades, or are there other factors that lead to the poor grades?

I don't know

Do you make silly mistakes on the homework?

Sometimes

how are you solving these problems on the tests? if you write out all the steps, you can visually check for errors

also, the way I mostly avoid calculator errors: plug in the same equation 3 times.

if it comes out the same way at least twice, that's probably your answer

I don't know

Sigh

is this helping, or do you just want some virtual hugs? or both?

Just some hugs, please

virtual hugs

okay, hugs and we will stop. we love you and you'll be fine <3

I'm just not good at anything enough to apply for any real scholarship

you are good. you are valuable.

you are the bestest tree and it's okay to struggle sometimes

"Who gives a shit about that special snowflake in the middle of a storm?"

Is how I perpetually feel

the snowflakes that happened to end up near it

You're in a feelings stage right now and it's important to work through the feelings, so give it time and let yourself process. We can talk about solutions/ideas again when you are in a solutions stage :)

I guess so. I'm sorry. I don't mean to be all doom and gloomy

speaking of doom and gloom: 'borg, I have a very long rant about my tics written out that I was planning to email you. Would that be okay?

No worries, we are happy to listen

and sure, you can email me anything :)

The most important thing I learned about processing feelings is that you go through a feelings stage and then a solutions stage, and it's important to focus on what stage you're in and not try to think about solutions in a feelings stage.

I don't know. It's just. I look ahead, and all these special applications for these amazing scholarships (I'm looking at two in specific: one for philosophy and another for composition) require so many work to have already been done

@PrinceNorthLæraðr trees are much bigger than snowflakes. you are smart.

@matt "If a tree falls down in the middle of a forest...."

... we'll help you back up

the bobblies will make a giant tree-sized pulley and sling and lift you up

<3

(@Sciborg, Doodle Request: draw that)

I don't have any paper with me but I can do an MS Paint doodle!

sure!

hmm, drawing pulleys is hard

something like

o`
 |
 `o

i guess?

wonderful!

Bobblie Inc. Tree-Lifting Services

Awww

@bobble Let's solve the amount of work required to lift the tree!

Hehe

shouldn't that bobblie in the tree get down so the lifting is easier?

she's trying her best

I thought that was my crown

your crown is a bobblie

it's a parasitic mutualistic relationship

Is that just a random tree that fell down?

i can draw eyes on you, hang on

Hehe

What are the ground-bobblies doing?

Stabilizing

is the bobblie on the platform doing anything but yelling?

Isn't that you

hang on, i'll make them be operating pulley controls

Amazing

They're very good at tree-lifting

They are professional Tree-Lifting Bobblies

This is a good use of your time

good, productive use

highly productive

indeed

also dang nice ms paint skills

idk

That makes no sense

as do most things i say, tbh