notes

Basics of Calculus

Calculus is the study of changing variables.

Important Symbols

  1. d The differential symbol. Means a “little bit of”. So, d𝑥 means a “little bit of 𝑥“. Could be infinitely small.
  2. ∫ The integral symbol. Long s means “sum of”. So, ∫d𝑥 means the “sum of the little bits of 𝑥“

On smallness

At a certain point, small values become negligible. Consider a square where 𝑥 represents the length of one side. The area of the square, therefore, is 𝑥2. Now, consider that you were to add d𝑥 to 𝑥, meaning increasing each side of the square by d𝑥. How much would the area grow by? This can be represented by the following diagram:

By taking the area with 𝑥+d𝑥, we get (𝑥+d𝑥)2, which can be expanded to: 𝑥2+2d𝑥+d𝑥2. Now, note that the d𝑥2 term is an order of magnitude smaller than even 2d𝑥. Because it is so small, we are free to ignore it and consider it to be effectively zero. This is even more true for smaller and smaller values of d𝑥.

On relative growings

Calculus is largely about how different values can grow and change in relation to one another. Consider the following example: Let 𝑥 represent the horizontal distance, from a wall, of the bottom end of a ladder AB. Let 𝑦 be the distance at which the top of the ladder rests on the wall. The question arises: If we were to pull the ladder further away from the wall, thereby increasing 𝑥, how much would 𝑦 change? The following figure represents the situation:

We will solve this with example data first. Suppose the ladder was so long such that where 𝐴 is 19inches from the wall, 𝐵 meets the wall 15ft above. If 𝑥 changes by 1inch, then, how much does 𝑦 change by?

𝑥=19in𝑦=180ind𝑥=1ind𝑦=?Because 𝑦 must decrease, we know that ℎ, the new height, is:ℎ=𝑦−d𝑦And, conversely, the distance from the wall, 𝑙, is:𝑙=𝑥+d𝑥By pythagorean theorem, we know the length of the ladder is:√(180)2+(19)2=181Squaring both sides gets us:1802+192=1812Becuase the length of the ladder must remain constant, the newheight must be:(𝑦−d𝑦)2=(181)2−(20)2=32761−400=32361Taking the square root,𝑦−d𝑦=179.892So, d𝑦 is:0.108This makes the ratio at which y changes in relation to x, or:d𝑦d𝑥=−0.1081

The above example also delivers a key intuition: in order for d𝑦d𝑥 to make any sense, y and x must have some relation. In this example, the relation was the following:

𝐿=√𝑦2+𝑥2𝑦=√𝐿2−𝑥2

So, because there is a relation, a change in x affecting y by some amount makes sense.

Simplest Cases

Power Rule

Power rule is defined as the following:

𝑦=𝑥𝑛d𝑦d𝑥=𝑛𝑥𝑛−1 Consider𝑦=𝑥2We know that if 𝑥 grows, 𝑥2 grows, so 𝑦 grows𝑦+d𝑦=(𝑥+d𝑥)2By expansion,𝑦+d𝑦=𝑥2+(2𝑥d𝑥)+(d𝑥)2By the discussion on smallness, (d𝑥)2 is simply too small to matterMoreover, we said before 𝑦=𝑥2. Let us substitute:𝑥2+d𝑦=𝑥2+(2𝑥d𝑥)Simplifying,d𝑦=2𝑥d𝑥So,d𝑦d𝑥=2𝑥

This works for any 𝑛, and this can be viewed intuitively from the following:

(𝑥+d𝑥)𝑛=(𝑥+d𝑥)(𝑥+d𝑥)(𝑥+d𝑥)…(𝑥+d𝑥)|𝑛times|(𝑥+d𝑥)𝑛=𝑥𝑛+𝑛𝑥𝑛−1+rest is negligible!

Sources

Much of this note comes from one source: Calculus Made Easy by Silvanus P. Thompson This is an attempt to intuitively explain fundamental concepts of calculus rather than going through rigorous math. The other source considered is: 3Blue1Brown’s Essence of Calculus Series

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