- Silvana Mirkning: So far, so good. But why did Tibor call you a “genius”?
- Tibor: Just ask her to calculate a derivative and you’ll see.
- Silvana Mirkning: You’re joking, right? <tube> What’s the derivative of… hmm…… 7 to the power of 2 times x? I know this is--
- Scarlet: Easy! The original term multiplied by the natural logarithm of 49!
- Julius: What’s a derivative, mom?
- Silvana Mirkning: Nothing for small children.
- Scarlet: Come on, another one!
- Silvana Mirkning: Let me think for a moment…
- Scarlet: Just take care that the function is differentiable.
- Silvana Mirkning: Can’t you calculate the derivative of any function?
- Scarlet: No, that’s only true for continuous functions, as Zolt Cator stated.
- Scarlet: However, I don’t remember reading a proof for it. Hmm…
A Sky Full of Stars 030
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27 thoughts on “A Sky Full of Stars 030”
Math guy
Is any continuous function differentiable? Not in the classical sense, e. g. |x| does not have a derivative at zero. But maybe a. e.? Not sure with very irregular functions though… need to check this…
Wait! That’s a web comic! Why does it make me think about university math!?!
Nerd snipe successful.
+1 for Oliver! 😊
No one of conseqence
You can construct a continuous function that is nowhere differentiable. One simple example is the Weierstrass function – there is a Wikipedia article on it.
Sekhmet
Obviously, Scarlet is a genius, but she does not know (yet) about continuous functions that are nowhere differentiable. 🙂
She has the right to make mistakes! She’s only 9 years old!
thysi
Actually, she didn’t make a mistake. She is correct: discontinuous functions are not differentiable.
Intrebute
I don’t know why I can’t reply to Thysi directly, so hopefully they see this reply instead.
Scarlet specifically said that “you can find the derivative of any function” is true in the case of continuous functions.
“Can’t you calculate the derivative of any function?”
“That’s only true for continuous functions.”
She’s saying that in the continuous functions, every one of them is differentiable. People have already mentioned the Weierstrass function, which is continuous but not differentiable. So yes, she did make a mistake. Can we really blame her though?
Math guy
Should have know that:
https://en.wikipedia.org/wiki/Weierstrass_function
There are continuous functions that are nowhere differentiable!
Hegel-Marx
Meh, I hate math freaks! 🙄
Except Scarclet. It took me ten minutes to calculate it. However, it’s been half a century since I studied higher mathematics. 😫
Crystalgate
This kind of problem you usually don’t calculate, you learn the rule and then you solve it as quickly as Scarlett did. There is no way she calculated it that fast. Noted of course that you calculate the problem when learning the rule, but once you know the rule, you just use it rather than doing the calculation you did every time. Of course, that means Scarlett probably did calculate it at some point, but I do not think she did the calculation this time.
Hegel-Marx
Bleh… As I said at the first place: I hate math freaks! 😏
MacareuxMoine
I don’t remember the rule for derivation of exponential functions (for me higher mathematics was 20 years ago), but from Scarlet’s reply I deduct it’s probably:
dx( a^(bx) ) = ln(ab) a^(bx)
MacareuxMoine
Dang, somewhere between deducting it and writing it down I got it mixed up.
Correct rule is: ( a^(bx) )’ = ln( a^b ) a^(bx)
I also did the calculations to prove it now (with the help of my high-school analysis book – still took me three attempts until I got it right and realized which step went wrong on my first two attempts 😉 )
BlackDragonSlayer
Good to know my college education places me at the same level of knowledge as a particularly-determined child 😛
Hegel-Marx
Bleh… As I said at the first place: I hate math freaks! 😋
Hegel-Marx
WRONG PLACE FOR A PEPLY, SORY!
DavidK
The “Hmm…” is Scarlett realizing the book is wrong and there are continuous functions not differentiable everywhere. Or maybe it’s Scarlett constructing her own “Weierstrass function.”
Crystalgate
Unless Scarlet is supposed to be a super genius instead of a genius, I’d say she’s not.
Kazuma Taichi
I know the brainrot is setting in when out of nowhere a thought of “dang, Silvana is really cute” pops in just from her raising an eyebrow
Vicious Sand
Great, she’s a Mentat. Maybe Tibor can rent her to a tax office or something.
No one of conseqence
It is by will alone I set my mind in motion…
Vicious Sand
It is by will alone I throw the beets in motion. It is by the bandits in the hills that beets acquire speed, skirts acquire stains, stains cause me a scolding. It is by will alone I throw the beets in motion.
BadRoad
Lots of calculus problems are easy if you know the shortcut. There’s just a lot of different shortcuts, and frequently the problem you want to solve doesn’t obviously fit into one.
Of course, she’s about half the age of most people I’d expect to know this one…
LurkerAbove
Silvana must be a math teacher or scientist.
I cannot think of anybody else comiing up with such questions, on the spot.
Clevershitter
I totally love the idea of a fictional world where the average person has college level math literacy.
Somehow I love it even more than a “regular” fictional world with wizardry.
Graybeard
I’ve long thought that basic (not advanced!) calculus should be taught a lot earlier in the math curriculum than it is. Basic calculus is intuitive and we do it all the time. Catching a thrown ball? You’re solving a calculus problem without thinking about it. Gauging how early to apply the brakes so that you stop your car at the stop sign? Same. Judging how much wrapping paper it will take to wrap a present? Same. And so on. Remove the fear factor that goes with the name, and most anyone can do it and does.
This said, Scarlet obviously isn’t “most anyone.”
JoB
Not addressing the question of whether math is properly placed in nowadays’ curriculae, but I’d like to point out that just because we solve problems that *can* be described as mathematical problems doesn’t mean that we *do* solve them in a mathematical way. To the contrary, your motoric (both self and car) examples we’re much better at by leaving them to unconscious mechanisms, rather than having our math-capable consciousness figure them out. To take that to an extreme, I once heard a lecture from mathematicians who tried to find optimal algorithms for aerial dogfights; they found that to be an intractable problem mathematically (too many bifurcation points), yet birds, human pilots, and (as far as they’ve gotten yet) AI can master that task rather well.
Graybeard
The idea isn’t to turn catching a ball into math, but rather, to start the process of teaching. As I said, remove the fear factor first.
Crystalgate
Tibor somehow knew Silvana understood derivatives. He was also not wrong about Scarlet getting smarter than him and his co-workers in a few years, she is now three years later.
Characters
Ariana Carolus Athamas Baako Bolin Clara Graybold Cyril Darius Leyh Elwin Flammt Evelina Ferrus Gale Nenawa Gavin Carolus Iona Jehona Shardyk Julius Mirkning Kala Silverspring Kenan Orege King Protos Lauro Levon Goldwood Leyton Wayt Ludis Graybold Luitpold Marik Marton Princess Elodie Raymond Reto Mirkning Reva Rodar Rosetta Greenleaf Scarlet Carolus Severin Silvana Mirkning Tibor Frey Vidal Viola Vreni Xaver Yerim Zach Ilkin