Notes On Approximations - CBSE Class 12 Maths
Let y = f(x) be a differentiable real function defined on an interval I. From the figure, y + Δy = f(x + Δx) and y = f(x). ⇒ Δy = f(x + Δx) - f(x) Dividing both sides by Dx, we get Δy/Δx = [f(x + Δx) - f(x)]/Δx.... (1) Let us suppose that limΔx→0 Δy/Δx exists. ⇒ limΔx→0 Δy/Δx = limΔx→0 [f(x + Δx) - f(x)]/Δx Using first principles, the right hand side of the equation can be written as dy /dx ⇒ limΔx→0 Δy/Δx = f '(x) = dy/dx ⇒ Δy/Δx = dy/dt + ϵ         (e>0 and e is a very small quantity) ⇒ Δy = dy/dt . Δx + ϵΔx As ϵΔx is very small. Δy = dy/dx . Δx (Approximately) …. (2) Now, f(x + Δx) = Δy + f(x) (x + Δx) = dy/dx Δx + f(x)…..from (2) Note : (i) dy ≃ Dy          (ii) dx = Dx

#### Summary

Let y = f(x) be a differentiable real function defined on an interval I. From the figure, y + Δy = f(x + Δx) and y = f(x). ⇒ Δy = f(x + Δx) - f(x) Dividing both sides by Dx, we get Δy/Δx = [f(x + Δx) - f(x)]/Δx.... (1) Let us suppose that limΔx→0 Δy/Δx exists. ⇒ limΔx→0 Δy/Δx = limΔx→0 [f(x + Δx) - f(x)]/Δx Using first principles, the right hand side of the equation can be written as dy /dx ⇒ limΔx→0 Δy/Δx = f '(x) = dy/dx ⇒ Δy/Δx = dy/dt + ϵ         (e>0 and e is a very small quantity) ⇒ Δy = dy/dt . Δx + ϵΔx As ϵΔx is very small. Δy = dy/dx . Δx (Approximately) …. (2) Now, f(x + Δx) = Δy + f(x) (x + Δx) = dy/dx Δx + f(x)…..from (2) Note : (i) dy ≃ Dy          (ii) dx = Dx

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