Really understanding how to apply calculus.

• IronPlate
In summary: You'll need to use the chain rule to find \frac{d^{2}u}{dt^{2}} in terms of \frac{du}{dt}.In summary, the conversation discusses the application of knowledge on solving differential equations and taking derivatives. The topic of Newton's Second Law is brought up, with equations for rate of change in velocity and force being mentioned. The conversation also touches on the assumption of constant mass and the added complexities when mass is not constant. The notation used in the equations is mostly correct, but some suggestions are made for improvement.
IronPlate
Aside from learning all the tools to solving differential equations and taking derivatives, I've been trying to practice actually applying this knowledge to the creation of my own equations to find out certain things. Where I'm uncertain is, have I achieved the actual expression?

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Assume mass is constant with time.
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(1) $F=ma$
(2) $F=m\frac{dv}{dt}$ ; F(t,v)
(3) $F=m\frac{d^{2}u}{dt^{2}}$ ; F(t,u,$\frac{du}{dt}$)

Rate at which v changes with time (acceleration)
(4) $\frac{dv}{dt}=\frac{F}{m}$

Rate at which F changes with time
(5) $\frac{dF}{dt}=m\frac{d^{3}u}{dt^{3}}$

Rate at which u changes with time
(6) $\frac{du}{dt}=\frac{1}{m}∫Fdt$

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Assume m(t), no longer constant.
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Rate at which v changes with time
(7) Same as (4)

Rate at which F changes with time
(8) $\frac{dF}{dt}=m\frac{d^{3}u}{dt^{3}}+\frac{d^{2}u}{dt^{2}}\frac{dm}{dt}$

Rate at which u changes with time
(9)Absolutely no clue, still working on it.

I'll leave it there.Is there anything wrong with my notation? Does each expression capture the descriptions?

(6) needs an additional integration constant, which corresponds to the initial velocity.

"u" for a position can be a bit confusing. In (6), I would use a different variable in the integral (like t'). Apart from that, it looks fine.

Hint for (9):
##\frac{du}{dt}=c+\int_0^{t} \frac{dv}{dt} dt'##

1. What is calculus?

Calculus is a branch of mathematics that deals with the study of continuous change. It is used to solve problems involving rates of change, optimization, and motion.

2. Why is calculus important?

Calculus is important because it provides a powerful set of tools for understanding and solving real-world problems. It is used extensively in science, engineering, economics, and many other fields.

3. How can calculus be applied in the real world?

Calculus can be applied in various ways in the real world, such as in physics to study motion and forces, in economics to optimize production and profit, in medicine to model the spread of diseases, and in engineering to design and analyze structures and systems.

4. What are the two main branches of calculus?

The two main branches of calculus are differential calculus and integral calculus. Differential calculus deals with the study of rates of change and slopes of curves, while integral calculus deals with the accumulation of quantities and the area under curves.

5. How can I improve my understanding of calculus?

To improve your understanding of calculus, it is important to practice solving problems and to have a strong foundation in algebra and trigonometry. It can also be helpful to seek out additional resources, such as textbooks, online tutorials, and study groups.

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