Question 1

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

Question 2

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

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

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

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