>>10866489Algebra as you learn it in school is the language in which you can express general facts about numbers.
Literals such as ‘x’ are used to represent arbitrary elements in a collection S.
Formulas involving an x in a collection S are general statements about elements of that collection. Proving such a formula amounts to proving that the statement holds for all members of that collection.
For example the formula (x+y)^2 = x^2 + 2xy + y^2 holding for all real numbers x and y means that I can substitute any real numbers to x and y and get a true statement.
For example, setting x = 1000, y = 1, I get 1001^2 = 1000^2 + 2000 + 1 = 1002001.
tl;dr: variables allow you to easily express general facts about numbers. There was a time when there was no algebra as we know it and people would just write really long sentences for what we write in one line. Proofs were even worse.
Calculus is a tool that allows you to analyze variations in quantities, in a very precise way. A function is a process that inputs a "thing" (usually a number, but not necessarily) and outputs another "thing" according to a fixed rule (ie. a given input will always lead to the same output).
The graph of a function is a visual representation of that process. If you input numbers and output numbers, then you will have the possible values of the input along one axis, the possible values of the output along another axis, and put a point at all points (input, corresponding output).
You can usually see on the graph if the output increases with the input, or decreases, or neither, but calculus allows you (for suitable functions) to quantify the change of a function via the derivative, and get even more precise information.
This is used in physics to formulate general laws. General principles of physics are classically formulated as differential equations, ie. equations relating physical quantities and their derivatives.
This allows you to relate quantities to the change of other quantities.
