#calculus

[edoassets]

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[z-boson]

this is what integrating dy feels like btw

[incounting]

a good comparison

[mypeakisnowandalways]

a good comparison

[eightballidiot]

Inequalities are Weird (the math ones)

(the equations work here)

Inequalities are at their core in contrast to equality equations (its in the name). The algebra that can be applied to both sides seems pretty restrictive in comparison to regular algebraic equations.

The most striking property is that multiplying (or dividing) both sides by a negative number reverse the direction of the alligator mouth.

The reason why this happens is actually more simple than one might think. The equations below can be used to understand why this rule is true.

$$x<y$$

$$-x>-y$$

(click here if the equations arent working)

On a purely conceptual basis, the equations above make since. If y is indeed greater than x, than the negative of y must be less than the negative of x. Multiplying (or dividing) by -1 is just the same as taking the entire real number line and flipping it 180º. y is greater than x so y will be to the right of x, and -y is less than -x because it is now to the left of -x since the line was turned over.

Multiplying by a negative number "-p" is essentially the same as multiplying by -1 and then multiplying by "p" (division acts the same way).

More generally, in order to apply a function to both sides of the inequality, the function must either be "increasing*" (the alligator mouth is in the same direction) or "decreasing*" (the alligator mouth is in the opposite direction).

*- it should specifically be strictly increasing or strictly decreasing where no two outputs are equal unless their input is equal.

The reason why one reverses the inequality sign for the function f(x)=-x is because said function is always decreasing meaning that the order of the real numbers is reversed. For an equation f(x)=2x, the function is always increasing meaning the order of the real numbers is preserved.

Determining whether a function is increasing or decreasing requires two properties for each; the function must be injective meaning that f(x) and f(y) are equal if only if x and y are equal themselves, and the function must be either always greater than or equal to 0 (for increasing function) or always less than or equal to 0 (for decreasing functions).

For instance, the exponential function e^x is an increasing function as its derivative is always positive and the function is injective (meaning e^x only equals e^y if x and y are the same):

$$x<y$$

$$e^x<e^y$$

The function -x^3 is decreasing as the derivative is always zero or negative and there are no (real) numbers that have the same output. Therefore, the alligator mouth goes in the opposite direction:

$$x<y$$

$$-x^3>-y^3$$

[lalalipuyofgulg]

I don't think they have ever experienced actual racism

[flowlingual]

[bugbeep]

goodbye everyone, my calculus class starts today….