If $f(x) = 3x^2+15x+5$, then find the approximate value of $f(3.02)$.

The correct answer is 77.66

To find $f(3.02)$ = $f(3 + 0.02)$

Given that $f(x) = 3x^2+15x+5$

⇒ $f(3) = 3(3)^2+15(3)+5$ = $77$


We can also derive $f'(x) = 6x+15$

⇒ $f'(3) = 6(3) + 15$ = $33$


We know that $f(x + \triangle{x})$ $\approx$ $f(x) + f'(x) \triangle x$

⇒ $f(3 + 0.02)$ $\approx$ $f(3) + f'(3) (0.02)$


Substituting the values of $f(3)$ and $f'(3)$ in the above equation, we get

$f(3 + 0.02)$ $\approx$ $77 + 33 (0.02)$ = $77.66$