The paradox of the derivative | Chapter 2, Essence of calculus

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Recommends Komninos · 2020-04 · Mathematics Stack Exchange · on “Derivatives car velocity paradox”

This is my first post on the Maths section of StackExchange. I am a high school student from Greece studying for IAL Math and got a bit confused on derivatives while watching this [this video] video; specifically on minute 14:29. The author gives an example of a car traveling on the curve $x^3$ where $x \in [0, 3]$ . He then asks if the car is moving or not at $x = 0$ Let's use $x^2$ instead of $x^3$ here for simplicity. Through the use of the derivative of $x^2$ , $\frac{dy}{dx} = 2x$ we see that at $x=0$ the velocity of the car is $\frac{dy}{dx}=2(0)=0$ which suggests that the car is not moving. I can see, however, the point the author is making, arguing that the car is actually moving, since t

Recommends awkward · 2020-04 · Mathematics Stack Exchange · on “Derivatives car velocity paradox”

You might be interested in this youtube video by 3blue1brown, in which he discusses this apparent paradox: [this video]

Recommends Nitin · 2021-07 · Mathematics Stack Exchange · on “Need help resolving the paradox of limits in proof of differentiability implies continuity.”

I have been following Professor David Jerison’s 18.01, Single Variable Calculus on MIT OCW. In Lecture 2 ( [LINK] ), he outlines the proof for “differentiability implies continuity”. Proof: [LINK] Now, he mentions that we are multiplying and dividing by $x-x_{0}$ which should be $0/0$ and is not a logically permitted operation. Then he goes on to mention that it is permitted in this case as technically the limit is not $0$ but a very small number. But in the next step we substitute $lim_{x->x_{0}} x-x_{0} = 0$ . My question is: Doesn’t this mean that $x=x_{0}$ ? The way this entire paradox seems to have been resolved relies on some sort of arbitrary loophole that was constructed when we ran

Recommends soupless · 2022-01 · Mathematics Stack Exchange · on “What does the derivative really mean?”

You can try to watch this video [this video] from YouTube as it discusses your point where the derivative is an instantaneous rate of change.

Cites Rajdeep Sindhu · 2020-08 · Mathematics Stack Exchange · on “Does 3Blue1Brown's series on Calculus : Essence of Calculus approach it via limits or infinitesimals (or both)?”

I was introduced to Calculus by the online series on it by Grant Sanderson (3Blue1Brown's owner) called Essence of Calculus . In his videos, he treats $dx$ as $\Delta x$ that approaches $0$ and $dy$ as the corresponding change in $y$ ( i.e. $\Delta y$ ). He especially mentions in one of his videos [this video] that he doesn't like to treat $dx$ and $dy$ as infinitely small quantities but rather finite quantities that approach $0$ , which is similar to the idea behind limits rather than infinitesimals. In that very video, he defines $\dfrac{df}{dx}$ as what the slope of the line joining $(x,f(x))$ and $(x+\Delta x, f(x+\Delta x))$ approaches as $\Delta x \rightarrow 0$ , which is another way of sa

Cites Agustin Saenz Anile · 2022-04 · Mathematics Stack Exchange · on “When it's better choose an specific Dx than calculate the Derivative?”

The problem is that to calculate velocity some amount of time must pass. Because of the definition of velocity, instantaneous velocity is an oximoron. It feels weird, isnt it? In this sense, the derivative isnt exactly velocity but a constant aproximation, then maybe i want a Dx that aproximates some velocity to tell me another thing about the movement. This is some cool video that inspired this question: [this video]

Cites CalculusisCONFUSING · 2023-02 · Mathematics Stack Exchange · on “Is 3Blue1Brown not completely true about “instantaneous rate of change” being nonsense?”

In 3B1B's video [this video] on the paradoxes of derivatives, he mentions that the definition “instantaneous rate of change” is nonsense. Why is it no good, isn't the value that we are approaching as dx approaches 0 not actually the rate of change of a line tangent at a point? Computationally, we are just not measuring the rate of change at an instant is all, doesn’t mean our approximation isn’t the rate of change in an instant. Instead we are measuring constant approximation of rate of change or the rate of change that is being approached as the time becomes smaller and smaller and approaches zero but the value that we approach is rate of change at an instant (so I thought).

Cites Princess Mia · 2023-03 · Mathematics Stack Exchange · on “if the derivative is just an approximation, how can it be so accurate in analysis?”

In ThreeBlueOneBrown's video on the paradox of the derivative, it is stated that the derivative is "the best approximation for the rate of a change around a point" ( [this video] , around 9:44). Given this, it confuses me as to how the derivative can be so accurate in our analysis of our functions if it is a mere approximation; for example, in determining whether a curve is concave up or concave down, the derivative is always accurate as to this matter. I am wondering if there are any instances of the derivative telling us non accurate information as to a function due to it being an approximation and not something exact, and also why it possesses such accuracy even while it is an approximation?

Cites wasu · 2024-04 · Mathematics Stack Exchange · on “Can someone clarify what d(t) means?”

In the paradox of a derivative [this video] , by 3b1b, he says at 1:49, "it's pretty common to name a function like this s(t). I would use $d$ for distance but that guy already has a full time job in calculus". What does he mean by this? I know d $x$ means with repsect to $x$ but that's different from $d(x)$ . I tried looking for answers by searching up "d(x) math" but I only got results about d $x$ . I think he meant d $x$ but I just wanted to clairfy to make sure I'm not missing anything.