High Level View

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One way to think about calculus is by thinking, about the area of a circle.
Area of a circle, $ Area = \pi R^2 $
In this model, we can learn about 3 ideas in calculus: Integrals, Derivatives, and the fact that they are opposites.

Think of dividing the circle into many concetric rings:
Area of the circle, will be the sum of the areas of all the rings.
If we flat out the ring:
We can approximate this shape as Rectangle-ish.
NOTE: This is not exactly a rectangle, but we can think of it as one for approximation.
Area of this rectangle is approximately: $ Area \approx 2 \pi r \ dr $ where $ dr $ is the thickness of the ring.
The smaller the value of $ dr $, the better the approximation.
A way to think about it is to put all these flattened rectangles side by side along the axis of the radius:
Now, as the $dr$ becomes smaller and smaller, the area becomes more accurate and since its a triangle we can calculate it using: $$ Area = \frac{1}{2} \cdot base \cdot height = \frac{1}{2} (R)(2 \pi R) = \pi R^2 $$ Example: If $ R = 3 $ then, which is the formula for the area of a circle.
The way we transition from something approximate to something precise, cuts deep to what calculus is all about.

A lot of hard problems in math and science can be broken down and approximated as a sum of many small quantities.
Example: Finding out how far a car has travelled based on its velocity at each point in time:

We were lucky in case of a circle because after putting all the rectangles side by side we got a triangle which is easy to calculate area for.
But, what if we had a different curve? Like a parabola?
Whats the area under this curve from $ x=0 $ to $ x=3 $?
If we fix the left endpoint in place at $ x=0 $ and let the right endpoint vary:
A function like this which gives us the area under the curve from a fixed point to a varying point is called an Integral of $ x^2 $.
Calculus has tools to figure out what an integral like this is.

If I slightly increase $ x $ by a small amount $ dx $, how much change in area, $ dA $ will I get?
This gives us a way to think about how $ A(x) $ is related to $ x^2 $: $ dA \approx x^2 dx $
Also, we can say: $$ \frac{dA}{dx} = x^2 $$ So, the ratio of a tiny change in A to a tiny change in x is whatever $ x^2 $ is at that point. And it gets better as $ dx $ becomes smaller.
Example 1: If $ x = 3 $ then, for a tiny change $ dx = 0.001 $: $$ \frac{A(3.001) - A(3)}{0.001} \approx 3^2 $$
Example 2: If $ x = 2 $ then, for a tiny change $ dx = 0.001 $: $$ \frac{A(2.001) - A(2)}{0.001} \approx 2^2 $$
So, it works for any value of $ x $.
And there is nothing special about $ x^2 $. This works for any curve.
The ratio of a tiny change in the area under the curve to a tiny change in x, when $ dx $ approaches 0, is called the Derivative of the area function.
$$ \frac{dA}{dx} \approx f(x) \\ \\ dx \to 0 $$
Derivative is a measure of how sensitive a function is to small changes in its input.
There are many ways to visualize derivatives, as we’ll see later.
After gaining enough practice computing derivatives, we will be able to look at a situation like this, where we don’t know what the function is, but we do know that its derivative will be $ x^2 $ and from that reverse engineer what the function must be:

The back and forth between integrals and derivatives, where the derivative of a function for the area under a curve gives us back the function defining the curve is called the Fundamental Theorem of Calculus.
It shows in some sense that integrals and derivatives are opposites of each other.