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Question: If \(f(x) = 3{x^2} - x\) where x=1 and \(\delta x = 0.02\)then \(\delta f = \) A) 0.1012 B) ...

If f(x)=3x2−xf(x) = 3{x^2} - x where x=1 and δx=0.02\delta x = 0.02then δf=\delta f =

A) 0.1012
B) 1.012
C) 0.101
D) 0.1

Explanation

Solution

We know that δx\delta xmeans small difference in x , which also can be solved using differentiation. It is indirectly differentiation with respect to x. So, apply delta on both sides which means partially differentiate the whole equation and we should consider that δ(x2)=2(δx)\delta ({x^2}) = 2(\delta x).

Complete step by step answer:
The given equation is f(x)=3x2−xf(x) = 3{x^2} - x;
By applying delta on both sides we get,
δf(x)=δ(3x2−x)\delta f(x) = \delta (3{x^2} - x) ………….(1)
Now let us see what does this delta exactly mean:
These are used to represent infinitesimal changes in a variable. On a graph this is equivalent to a tangent to a curve. Physically it is used when applying derivative on a variable ‘y’ w.r.t another variable ‘x’ giving infinitesimal change in ‘y’ for infinitesimal change in ‘x’, that is derivative.
Now we get a doubt that how is derivative and delta is related?
So, here is the explanation:
Delta is used for demonstrating a large and finite change . The partial derivative symbol is used when a multivariable function is to be differentiated with respect to only a particular variable , while treating the other variables as constants . Small delta is used to represent an improper (or discontinuous) derivative.
This implies δ(ax)\delta (ax)will be d(ax)d(ax)which is a.d(x)=a.δ(x)a.d(x) = a.\delta (x)where a is any constant.
And also δ(x2)\delta ({x^2}) will be d(x2)d({x^2})which is 2xd(x)=2xδ(x)2xd(x) = 2x\delta (x)
This implies for equation (1), that can be written as
δf(x)=δ(3x2)−δ(x)\delta f(x) = \delta (3{x^2}) - \delta (x)
δf(x)=6x.(δx)−(δx)\delta f(x) = 6x.(\delta x) - (\delta x)
Given that x=1 and δ\delta x=0.02
By substituting these above values in the above equation , we get it as
δf(x)=6×1×(0.02)−(0.02)\delta f(x) = 6 \times 1 \times (0.02) - (0.02)
δf(x)=0.12−0.02\delta f(x) = 0.12 - 0.02
That is δf(x)=0.1\delta f(x) = 0.1
Therefore the correct option is D.

Note:
Learn the concepts and differences between delta function and derivative functions. Also go through the properties of derivatives. Do not make calculation mistakes while solving the problems. Delta functions and derivatives are not the same everywhere.