u/Chance_Rhubarb_46

I am failing to connect the relationship with derivatives and the definition of limits.

When we have a limit, its defined as a constant that a function converges to.

However,

lim x->a f(x) = L

does not guarantee

f(a) = L

For example,

lim x-> 1 f(x) = (x - 1)(x + 1) /(x - 1) = 2

However, it is undefined for x = 1. That is,

1. Limits are used to show what the functions converges to.
2. The function never reaches the limit.

For a derivative given the definition,

lim h -> 0 (f(x + h) - f(x))/ h

derivatives are based from limits and therefore should follow the same restrictions imposed on limits. From this definition, the derivative therefore originates from congruence behaviour. It is not the slope of one fininte interval, but the unique value toward which all sufficiently small nearby interval slopes converge under infinite refinement as h approaches 0.

So being picky here, the derivative at point f'(2) = ... should not be that value, but rather a derivative at a point is the exact limiting value of nearby secant slopes as the interval width approaches 0. It is not guaranteed to be that value from the previous example above.

So I am having difficulty grasping the connection being derivatives being used given that they're based off of limits. If we take any basic example f'(2) = ..., this based off of a limit. It shows what it converges to, it is not guaranteed to have a specific value because it is not a function at a point, it is a limit.

Does anyone understand what I am trying to ask here? I have asked AI several times but I have given up with their responses. My confusion is that derivatives are based off of limits, but limits are defined to never truly reach a value, it shows where it converges to. Converges to is not identical as that value but we use derivatives in calculations as its a slope at that exact point whereas its a slope from the given point as a limit from h-> 0 from the formula given above. So shouldn't that mean that we cannot use derivatives directly in a calculation, especially in real world applications as they're based from limits?

I am still trying to practice employing units into questions in order to validate my answer. I attempted to employ my techniques in a question I found but it failed to translate correctly. I was hoping someone could advise me on how to properly create functions and units to validate the answers dimensionality.

The question was "You're in a stadium of unknown size. Every second the amount of water in the stadium doubles. It starts with a single drop. The stadium is full after 60 seconds. When is the stadium a quarter full?"

I understand the question and frequently given answers. However I am trying to practice function creation and unit placements. So I did the following,

V0 = 1 m^3 * 2^0
Vf = 1 m^3 * 2^60
Vf/4 = (1 m^3 * 2^60)/2^2 = 2^58

Therefore, 1*m^3 * 2^t = 2^58m^3, it follows t=58.

My confusion comes from the phrasing, in particular "seconds" where no seconds unit is present. When first trying to construct this equation I thought,

V m^3 = 1 * m^3 * 2^t

As conceptually I thought,

1. The volume had to be in m^3, thus the result is in m^3
2. The single drop, which is the base volume has to be in m^3 as it scales up
3. The time, which is in the time passes has to be in seconds.

However, given m^3 *s != m^3, this fails to work.

Can anyone help me understand when do I employ units correctly as I try to use them extensively now to check for dimensionality and answer validation and in some cases use them solely to solve questions provided the information.

This question seems to not use the "seconds" units here, despite being phrased as such. In physics, such as acceleration m/s^2, we still multiply by s to maniuplate units when counting forward in time, why not here?

I have been studying a lot about the pendulum serve, but can someone help validate for me the exact contact point? For example this video discusses how you flick your wrist in an upward motion upon contact.

However, I am starting to think if I should be grabbing the ball at the very bottom or it would be easier to grab it on an angle where you contact around the centre, as to grab it easier and thus have more control when flicking it upwards with your wrist.

I will try to illustrate with images. So my first thought if mimicing Ma Long, where he does an upward motion to a downward motion with his arm. I managed to capture a frame below. Despite being backspin (With side), it doesn't make contact at the very bottom. Can that only mean that contact is ideally shown on the left, where you make the contact here and this perform a wrist rotation upwards to add additional backspin.

https://preview.redd.it/1p7qtgnu99zg1.png?width=2235&format=png&auto=webp&s=011762db097688d287048b6ec59aeaf5a84dce30

OR

https://preview.redd.it/a920a8snb9zg1.png?width=1016&format=png&auto=webp&s=131a2ddd8b2074778bebac4fa1a147c346091fe8

Are you meant to make contact around this point and then rotate your wrist upwards? I really cannot seem to find a consistent answer, never mentioned in the video above.