For y=4x+x^3, express Δx and dy as functions of x and Δx, and for Δx infinitesimal find an infinitesimal ε such that Δy=dy+εΔx.

Question

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Answer 1

y' = 4 + 3x^2

dy/dx = 4 + 3x^2

dy = 4 dx + 3 x ^2 dx

Mean value theorem says:

There exists E such that

f'(E) = deltaY/ deltaX

4 + 3E^2 = delatY / delta X

delta X * (4 + 3E^2) = delta Y

4 *delta X + 3 * E^2 * delta X = delta Y