Think of these two things as your cheesecake ingredients. Like cheesecake is made up of cream cheese and eggs, Calculus is built upon these two concepts.
One of the more applicable situations that these can be used in is the relationships between position, velocity, and acceleration:
A nice graphical representation
The order goes like this derivative wise:
The derivative of the position gives you velocity, and the derivative of the velocity gives you acceleration.
And the opposite is true of integrals:
The integral of acceleration gives velocity, and the integral of velocity gives position.
These don't have to exclusively apply to position - velocity - acceleration either. This concept can be universally applied to any function of "something".
If you take the derivative of "something", then you'll get the rate of "something", and if you take the derivative of that then you'll get the change in rate of the rate of "something".
If this is confusing, applying some Cheesecake to this will clear this up:
At the Cheesecake Factory, they make a lot of cheesecake. As a matter of fact, they make such a great amount of cheesecake that the actual amount of cheesecake cannot be counted. Luckily, the rate at which they make cheesecake per hour can be measured:
Where r(t) is the rate of cheesecake created and t is time in hours. Everyday has 24 hours, which is why time is bounded from 0 hours to 24 hours.
(a) Calculate the amount of cheesecake produced per day.
(b) The rate of cheesecake production starts at one point and then begins falling from there, but at some point in the day cheesecake production begins increasing and increases until the end of the day. Find when in the day this happens.
(c) At what time is a total of 234.85 cheesecakes made? Yes, the reason why there's a fraction of a cheesecake is so that your answer is nice and even.
Solutions:
Will be posted later due to my laziness.
If you crazy people out there did not think this problem is up to AP problem standards, don't worry. The next problem will be BASED upon a REAL AP problem, but, there will definitely be cheesecake.
The derivative of the position gives you velocity, and the derivative of the velocity gives you acceleration.
And the opposite is true of integrals:
The integral of acceleration gives velocity, and the integral of velocity gives position.
These don't have to exclusively apply to position - velocity - acceleration either. This concept can be universally applied to any function of "something".
If you take the derivative of "something", then you'll get the rate of "something", and if you take the derivative of that then you'll get the change in rate of the rate of "something".
If this is confusing, applying some Cheesecake to this will clear this up:
At the Cheesecake Factory, they make a lot of cheesecake. As a matter of fact, they make such a great amount of cheesecake that the actual amount of cheesecake cannot be counted. Luckily, the rate at which they make cheesecake per hour can be measured:
Where r(t) is the rate of cheesecake created and t is time in hours. Everyday has 24 hours, which is why time is bounded from 0 hours to 24 hours.
(a) Calculate the amount of cheesecake produced per day.
(b) The rate of cheesecake production starts at one point and then begins falling from there, but at some point in the day cheesecake production begins increasing and increases until the end of the day. Find when in the day this happens.
(c) At what time is a total of 234.85 cheesecakes made? Yes, the reason why there's a fraction of a cheesecake is so that your answer is nice and even.
Solutions:
Will be posted later due to my laziness.
If you crazy people out there did not think this problem is up to AP problem standards, don't worry. The next problem will be BASED upon a REAL AP problem, but, there will definitely be cheesecake.
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