Fundamental Ideas of Calculus – a survey
What is one central idea of calculus you would want everyone in the world to understand?
Add your ideas in comments.
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What is one central idea of calculus you would want everyone in the world to understand?
Add your ideas in comments.
If I had to choose one topic, I would choose derivatives and how to apply them.
If I could choose another, it would be integrals.
That we can understand (approximate) difficult quantities by breaking them down into much smaller, easier to understand parts. For differential calc: “How much is it changing by here” vs “how much there” OR “Can I predict how much it will be changing under different conditions?” For integral calc: slice it up and add!
** If you were to explain calculus to a non-mathematician friend through one idea, what would it be?
Compound interest.
Consider earning interest on a saving account, or paying interest on a loan.
Interest is usually quotes as percent per year.
if you left 100 in a bank at 6% pa, you would have 106 at the end of 1 yrer,
buy only 100 at the end of 364 days!
Which doesnt seem fair.
So what if the interest was earned an compuounded every month? *0.5% per month = 6% per year.
At the end of 1 month you would have 100.50
At the end of 2 month you would have 101.0025
At the end of 3 months, 101.5075125
At the end of 12 months, 106.1677812, or almost 17 cents more.
But what if we continued to divide the time into smaller and smaller units
6% per year is 0.5% per month or .01667% per day
But what about per second? per millisecond?
Would we keep making more and more money? Would we ever make $200? Would we ever even make 107?
In calculus, we imagine adding up the “compound interest” as the intervals getting smaller and samller while the number of intervals gets correspondingly larger and larger.
It turns out that, for many relationships of this kind that occur in the natural world,
we can actually determine a “limit” to the sum.
** How about that friend’s curious five-year-old kid?
Let’s consider Baby Bunnies.
If you start with two bunnies, and next month they had two babies, a boy and a girl, how many bunnies would you have? [4]
Now if the next month your two bunny-pairs each and a boy and girl bunny, how many would there be? [8]
And the monthe after that? [16]
And if every month all those bunny-pairs each had a pair of babies, how many would you have at the end of a year:
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan
2 4 8 16 32 64 128 256 512 1024 2048 4096 9192
So CALCULUS is about adding up the total, when in each unit of time,
what you have is multiplied by the same amount.
Or divided by the same amount.
Suppose on Halloween you got 64 candies.
The next day you eat half of them–how many are left (32)
And the next day you eat half of those–how many left now (16)
Next day? 8 then 4 then 2 then 1,
then you start cutting it in 1/2,
then 1/4, 1/8, 1/16, etc.
When will you candy be all gone? [never] / when it gets to small to cut anymore.
** What is one calculus idea you’d want everyone in the world to understand?
The basic concept of calculus is the DIFFERENTIAL EQUATION
If the amount something changes depends on the amount we already have, in some ratio,
as for example in population growth,
then a DIFFERENTIAL EQUATION expresses how the amount of increase is related to the amount we start with.
And calculus tells us how we can calculate the total by imagining the time intervals getting smaller and smaller while the number of intervals get larger and larger, approximating an “instantaneous” change by an infinitesimal amount.
It turns out differential equations express the way a lot of things change in nature, such as:
population growth
radioactive decay
the motion of objects under the influence of gravity or electrical or magnetic charge.
the motion of air as it propagates sound or shock waves, or water as it propagates waves,
or the flow of heat through a substance
or the behavior of sub-atomic particles in an atom
The shape (behavior) of basic functions.
Understanding why y=1/x looks the way it does provides a great foundation for grasping the concept of “limits”, ( the key that unlocks integrals and derivatives! ).
Recursion/induction – f(x+dx) = f(x) + g(x) dx