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Evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. f(x)

User Reformy
by
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1 Answer

1 vote

Answer:


(f(x + \triangle x) - f(x))/(\triangle x) = 3x^2+ 3x \cdot \triangle x + (\triangle x)^2

Explanation:

Given


f(x) = x^3

Required

Evaluate


(f(x + \triangle x) - f(x))/(\triangle x)


(f(x + \triangle x) - f(x))/(\triangle x) becomes


(f(x + \triangle x) - f(x))/(\triangle x) = ((x + \triangle x)^3 - x^3)/(\triangle x)

Expand


(f(x + \triangle x) - f(x))/(\triangle x) = (x^3 + 3x^2 \cdot \triangle x+ 3x \cdot (\triangle x)^2 + (\triangle x)^3 - x^3)/(\triangle x)

Collect like terms


(f(x + \triangle x) - f(x))/(\triangle x) = (x^3 - x^3+ 3x^2 \cdot \triangle x+ 3x \cdot (\triangle x)^2 + (\triangle x)^3 )/(\triangle x)


(f(x + \triangle x) - f(x))/(\triangle x) = (3x^2 \cdot \triangle x+ 3x \cdot (\triangle x)^2 + (\triangle x)^3 )/(\triangle x)

Factorize


(f(x + \triangle x) - f(x))/(\triangle x) = (\triangle x(3x^2+ 3x \cdot \triangle x + (\triangle x)^2) )/(\triangle x)


(f(x + \triangle x) - f(x))/(\triangle x) = 3x^2+ 3x \cdot \triangle x + (\triangle x)^2

User Raj Chaurasia
by
6.5k points