Increments and Differentials

When a changing quantity passes from an initial value to another value, it is said that it had an increment.

To denote an increment we use upper case delta: Δ

Used to show a change in the value of a variable.

Δp: price

Δy: change in the volume of an object

ΔT: change in Temperature

Δt: change in time

Δy: change on the variable y

To calculate an increment you need to obtain the difference between the final and the initial value:

Δy = y2 – y1
Δx = x2 – x1

Can be positive or negative (decrement).

Increments

Give the real change between two quantities.

The information comes from the actual situation (f(x)).

y2 = Δy + y1 x2 = Δx + x1
Δy = y2 – y1 Δx = x2 – x1

Differentials

Give an estimate of the real change among two values.

The information comes from the tangent line.

dy ≈ Δy dx = Δx
dy = f ’(x) * dx dx = x2 – x1
f ’(x) * dx ≈ y2 – y1

Infinitely small quantities.

They give an estimate of the real change on the value of a quantity (or variable).

dy ≈ Δy

dy = f ’ (x) * dx