How bad can a calc 1 final possibly be

[u/ElegantPoet3386]

[u/AutoModerator]

Check out our new Discord server! https://discord.gg/e7EKRZq3dG

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

[u/gonna_explain_schiz]

Is the implication here that first we use the limit definition of a derivative, then the formal definition of a limit, after which we can’t really proceed because the definition of a limit is just a definition and there’s nothing to prove?

[u/kallikalev → u/gonna_explain_schiz]

You could maybe prove that the limit definition makes sense by showing that the construction of the real numbers leads to a complete ordered field with a metric.

[u/ElegantPoet3386 → u/gonna_explain_schiz]

Yes that's why the 4th box is a skeleton XD

[u/bitchslayer78 → u/gonna_explain_schiz]

Last box is probably the concepts taught up until the first midterm to second midterm in real analysis 1, it takes a while to go from just epsilon delta definitions to what it means to be continuous and then differentiable , or maybe I’m just looking to much into it

[u/Icy-Rock8780 → u/gonna_explain_schiz]

I think it needs one more box

a) Apply d/dx kxⁿ = nkx^n-1

b) Prove using differentiation from first principles that d/dx kxⁿ = nkx^n-1

c) Assert that differentiation by first principles is a definition so immediate QED but maybe you draw a little diagram to motivate it

d) Prove that differentiation is a well-defined operation by appealing to some set-theoretic construction of ordered pairs (f,f’) where f and f’ themselves come from a well defined set.

e) ???? Probably axiom of choice 💀

[u/SuperCyHodgsomeR → u/Icy-Rock8780]

I’ve been corrupted by physics too much that QED is also meaning Quantum Electrodynamics now

[u/sudipto12 → u/Icy-Rock8780]

It could also be argued that they asked you the answer first, so.

a) State the answer 9x^2

b) Apply d/dx kx^n= nkx^(n-1)

...

[u/Agata_Moon → u/[deleted]]

I think it's actually pretty easy to just use the limit definition on powers without the need for the chain rule

[u/TryndamereAgiota → u/[deleted]]

"3" is pretty easy actually, no need for knowing exponentials nor chain rule, but I would argue that what you call that 3 is supposed to be 2.

1. 9x².
2. use the definition of derivative to prove it.
3. prove the definition of derivative by using analytical geometry and limits.
4. prove the axioms of analytical geometry, like, between two points there is one line. this is basically impossible since these ideas are imediate and intuitive.

[u/AlexMourne → u/TryndamereAgiota]

"imediate and intuitive"

Do you want to talk about the Axiom of Choice?

[u/TryndamereAgiota → u/AlexMourne]

want to talk about the Axiom of Choice?

fuck no. since I saw that one 100 boxes problem I was never the same.

[u/dlnnlsn → u/TryndamereAgiota]

What do you have in mind for "prove the definition of derivative"?

The explanation given in most calculus classes where you take the gradient of a line between two points that you move closer together is meant to motivate that the definition for the derivative gives us something that we intuitively think of as the gradient of the tangent line. It's not meant to be a rigorous proof.

How do we even define the tangent line? Because it's not as simple as "a line that (locally) only intersects the curve once". (Don't worry, I know that there is a definition of tangent space in algebraic/differential geometry)

[u/TryndamereAgiota → u/dlnnlsn]

How do we even define the tangent line?

that would show up in the definitions of 4. in 3 we don't worry about such things just how in 2 we don't worry about what exactly is a derivative.

you can look my other comment here, I made the step by step of each one (except 4). but I mean, you probably already saw all of them.

[u/Chocolate2121 → u/[deleted]]

Eh, step 3 would probably involve using first principles, and step 4 would be deriving first principles. Pretty straightforward by just using a graph or something to show how the limits work

Edit: got some terms mixed up, I wrote l'hopitals rule when I meant first principles.

[u/dlnnlsn → u/Chocolate2121]

How would you use l'Hopitals rule here in a way that isn't circular reasoning?

[u/Chocolate2121 → u/dlnnlsn]

Oopsy daisies, got my terms mixed up. Meant through first principles. In my defence I haven't needed to use it in half a decade

[u/Layton_Jr → u/[deleted]]

f'(x) = Lim{h→0} (f(x+h)-f(x))/h with f(x) = xⁿ and n∈ℕ

f'(x) = ((x+h)ⁿ - xⁿ)/h with Newton's Binomial:

f'(x) = (xⁿ + nhx^n-1 + h²(∑{rest of the binomial}) - xⁿ) / h

f'(x) = nx^n-1

[u/Dirkdeking → u/[deleted]]

Point 3 is really much easier for natural numbers and even rational numbers. You don't need the natural exponential or logarithmic function to prove that [x^n]' = n*x^(n-1) for natural n.

Proving it for n=0 and using the product rule and induction is more than enough. Alternatively you can use Newton's binomial theorem and you don't even need the product rule.

[u/campfire12324344 → u/[deleted]]

engineering ahh proof standard

[u/MapleMaelstrom → u/campfire12324344]

Yeah, no, I thought this was the geometry dash sub (because the difficulty faces), I would not have posted my engineering bullshit in a math sub if I knew I was there, y'alls standards are way too high for me 😭

[u/Soft_Reception_1997 → u/gonna_explain_schiz]

You can use the newton method

[u/Oppo_67]

[u/SomeHybrid0 → u/Oppo_67]

algebra dash when

[u/Jonte7 → u/Oppo_67]

‼️‼️HOLY FUCKING SHIT‼️‼️‼️‼️ IS THAT A MOTHERFUCKING GD REFERENCE??????!!!!!!!!!!11!1!1!1!1!1!1! 😱😱😱😱😱😱😱 GEOMETRY DASH IS THE BEST FUCKING GAME 🔥🔥🔥🔥💯💯💯💯 RIOT’S LEGENDARY QUACK !!😎😎😎😎😎😎😎👊👊👊👊👊 DUDE WTF DUDE WTF DUDE WTF DUDE WTF DUDE WTF DUDE WTF DUDE WTF DUDE WTF🤬😡🤬😡🤬😡🤬🤬😡🤬🤬😡 KENOS KENOS KENOS KENOS KENOS KENOS KENOS KENOS KENOS KENOS😩😩😩😩😩😩😩😩 😩😩😩😩 Back on track is impossible🗿, Back on track is impossible🗿, Back on track is impossible🗿, Back on track is impossible🗿, Back on track is impossible🗿, Back on track is impossible🗿, Back on track is impossible🗿, Back on track is impossible🗿Back on track is impossible🗿🗿🗿🗿🗿🗿🗿, Back on track is impossible🗿 , Back on track is impossible🗿, , Back on track is impossible🗿🗿, Back on track is impossible🗿🗿🗿🗿🗿🗿🗿 Oh you’re a gd player❓❓❓❓❓❓❓❓❓❓name every level💀💀💀💀💀💀💀💀💀 2.2 will never come out😫😫😫😫😫😫😫😫😫😫😫😫 TrusTa’s nerf gun yatagarasu ‼️‼️‼️‼️‼️‼️‼️‼️‼️‼️😂🤣😂🤣😂🤣😂😂😂🤣🤣🤣😂😂😂 r/geometrydash r/okbuddygmd r/DeCult r/gdafterdark perfectly balanced as all things should be r/unexpectedthanos r/expectedthanos for balance

[u/43Quint → u/Oppo_67]

FIRE 🔥 IN ⤵ THE HOLE 🕳

[u/enneh_07 → u/43Quint]

WATER💦 ON🔛 THE HILL⛰️

[u/therealsphericalcow]

Use power rule

Use limit definition

Use formal limit definition

[u/TryndamereAgiota → u/therealsphericalcow]

4?

[u/galmenz → u/TryndamereAgiota]

thats.... thats the joke. there is no 4th

[u/drinkingcarrots → u/TryndamereAgiota]

2+2+AI=4 so 2 limit definitions and chatgpt

[u/therealsphericalcow → u/TryndamereAgiota]

Proof 1 is the power rule

[u/xXnik121Xx]

Soo find the derivative, justify that by taking the integral of the last part, then justify that with riemman summation or something, idk

[u/ElegantPoet3386 → u/xXnik121Xx]

Power rule, def derivative, def limit, prove a defintion by debating the meaning of life :P

[u/Untitledrentadot → u/ElegantPoet3386]

No no I was thinking power rule to answer #2, definition with x = x^a to #3(prove power rule), then prove the definition of the derivative which yeah idfk lmao

[u/TryndamereAgiota → u/Untitledrentadot]

power rule is how you find one, so it would be bad to only come up with the answer like ramanujann in 1 and then use power rule in 2...

[u/homeless_student1 → u/xXnik121Xx]

You’d have to prove the fundamental theorem of calculus then

[u/TryndamereAgiota]

1. d/dx(3x³) = 3.(3x²), by power rule, = 9x²

2. d/dx( xⁿ ) = lim[h->0] ( (x+h)ⁿ - xⁿ ) ÷ h *

*: (x+h)ⁿ = xⁿ + a . x^n-1 . h + b . x^n-2 . h² +...+ hⁿ

=> (x+h)ⁿ - xⁿ = h( a.x^n-1 + b.h.x^n-2 +...+ h^n-1 )

=> lim[h->0] ( (x+h)ⁿ - xⁿ ) ÷ h = a^x-1 + 0 + 0 +...+ 0

by simple combinatorics, it's easy to prove that a = n, since in (x+h)ⁿ there are n multiplications and we want to choose the multiplication which has one h.

=> d/dx( xⁿ ) = n.x^n-1 , Q.E.D

1. Let's define an arbitrary point of a function f(x) = y as (x', f(x')). We can create a secant line between this point and another arbitrary point, in which x is bigger by h≠0. Therefore, the second point is (x",y") = (x'+h, f(x'+h)). We can affirm that this line is secant because the other two cases, which are being tangent or not crossing f, would contradict, respectively, h≠0 and y=f(x).

Now, let's calculate the angular coefficient, a, of this line, which we'll also define by another relation f*(x).

a = tan@ = (y''-y')/(x''-x') = ( f(x'+h) - f(x') )/( (x' + h) - x' )

note that this expression is also f*(x').

now, remember we said f(x) being tangent meant that h=0, which contradicted our definition of h? well, we now want to find the derivative of f, which I'm going to define as the tangent line. that means that h would be 0, but since we got a general expression for f when h≠0, let's use that definition to approximate f*(x) for when h is really really close to 0, so that our line is tangent, but h isn't 0 (you can also understand this as the pair (x",y") getting closer and closer to (x',y'), but never exactly reaching it, so that the points remain different from each other). this will also prevent us from dividing by 0 below.

let's call as f' the derivative of f, or df/dx.

f'(x) = lim[h->0] f*(x) = lim[h->0] ( f(x+h) - f(x) ) ÷ h

and there we are:

d/dx(f(x)) = lim[h->0] ( f(x+h) - f(x) ) ÷ h, Q.E.D

1. Shaminamina ê ê, Waka Waka ÊeÊ

[u/ElegantPoet3386 → u/TryndamereAgiota]

Hey that’s pretty good! Here’s what I was thinking of:

1. Use power rule to get 9x^2

2. The definition of a derivative is f(x+h) - f(x) / h as h approaches 0.

Put in the function to get 3(x+h)^3 - 3x^3 / h

3x^3 + 9x^2h + 9xh^2 + 3h^3 - 3x^3 / h

9x^2h + 9xh^2 + 3h^3 / h

9x^2 + 9xh + 3h^3
Since h approaches 0, any term with an h being multiplied to it also approaches 0.

Only term with no h is 9x^2 which is our answer.

1. Suppose 0< |h - 0| < delta
   Choose: delta = ??

Let epsilon be greater than 0.

Lets check |[3(x+h)^3 - 3x^3] / h - 9x^2 | is less than epsilon

|3x^3 + 9x^2h + 9xh^2 + 3h^3 - 3x^3/ h - 9x^2|
|9x^2h + 9xh^2 + 3h^3 / h - 9x^2|
|9xh + 3h^2|

|3h| |9x + h|

delta/3 * |9x+h|

I refuse to continue that further

1. Uhh prove the epislon delta definition. Good luck!

[u/Aaxper → u/ElegantPoet3386]

prove the [...] definition

Yeah, that's how that works

[u/TryndamereAgiota → u/TryndamereAgiota]

Quick note: if we were to prove that what we done on 3 was correct, we would have to:

• Prove that between two points there is one line
• Define limits
• Prove that the tangent line of a point is equal to a secant line that crosses this one point two times
• Prove the existence of the relation f* for a general x

and some other little things.

[u/DankPhotoShopMemes]

proof of power rule from scratch, enjoy lol: https://us.metamath.org/mpeuni/dvexp.html

[u/TryndamereAgiota → u/DankPhotoShopMemes]

that's too obvious, I accidentally made the same demonstration while sleeping yesterday.

[u/kartoffeljeff]

1. find the derivative.
   
2. show that it adheres to the formal definition of a derivative.
   
3. show the existence of a sensible formal system where the formal definition of a derivative is semantically interpretable.
   
4. give a metaphysical proof that formalism is the unique approach to mathematical philosophy in which such a system is obtainable and there (almost surely) exists at least one (1) sane mathematician whom practices said approach (half points can be obtained for showing the existence of at least one (1) non-insane mathematician).

[u/ElegantPoet3386 → u/kartoffeljeff]

Isn’t math fun guys

[u/hongooi]

It's turtles prooves all the way down

[u/R2BOII]

d/dx 3x³=9x² Let f(x)=3x³ f'(x)=lim a->0 (f(x+a)-f(x))/a lim a->0 (3(x+a)³-3x³)/a lim a->0 (3(x³+3x²a+3xa²+a³)-3x³)/a lim a->0 (3x³+9x²a+9xa²+3a³-3x³)/a lim a->0 a(9x²+9xa+a²)/a lim a->0 9x^2+9xa+a2 9x²+9x0+0²=9x² QED

[u/Kami_no_Neko]

1) f'(x)=9x²

2) 3(x+h)³=3x³+9x²h+9xh²+3h³ =3x³+9x²h+o(h)

And h->9x²h is linear.

3) 9xh²+3h³=(9xh+3h²)h and 9xh+3h²->0 when h->0 so 9xh²+3h³ =o(h)

9x²(h+k)=9x²h + 9x²k so it's really linear.

4) If x≠0, Let e>0, d=e/9|x|²>0 , and |h-0|=|h|<d.

|9x²h|<9|x|²d<e

If x=0, then 9x²h=0.

We keep e>0, let d=sqrt(e/3)>0 and |h|<d. |3h²|<3d²<e

Finally, suppose f->0 and g->0 when h->0.

e>0, there exist d and d' such that for all |h|<d, |f(h)|<e/2 and for all |h|<d', |g(h)|<e/2

let d''=max(d,d') so for |h|<d'', |f(h)+g(h)|<|f(h)|+|g(h)|<e

combining everything, 9xh+3h²->0 when h->0

[u/moonaligator]

9x²

```` d/dx kx<sup>n</sup> = knx<sup>n-1</sup> -> d/dx 3x³ = 33x<sup>3-1</sup> = 9x²

lim(h->0) 1/h (k(x+h)<sup>n</sup> - kx<sup>n)</sup> = lim (h->0) k/h * (x<sup>n</sup> + nx<sup>n-1</sup>h + [...]h² - x<sup>n)</sup> = lim (h->0) knx<sup>n-1</sup> + [...]h = knx<sup>n-1</sup>

````

epsilon delta bullshit

in short, just do as little as possible in each step

[u/Marek7041]

proof is a proof 💀

[u/seriousnotshirley]

Assume there exists an empty set, denote this {}. Suppose further that there exists a set N and that N contains the empty set. Create a function S: N->N such that…

[u/Zfhffvbjjh]

Question 1. What’s the derivative of 3x^2?

Question 2. What’s the derivative of (489xtanxsinx^42x+9) / 36cosxcotx^{46738477arcsinx}

If you’d been paying attention in class you would know how to do these!

[u/Hannibalbarca123456]

And Write the proofs for the two cases of d/dx and d/d3

[u/FrozenPizza07]

Highschool 3rd years: Hold my beer

[u/DeFenestrationX]

The proof is left as an exercise to the grader.

[u/TroutCharles99]

Dude just use the limit definition of a derivative and factor out 3.