For the function \(f\), \(f'(x)=2x+2\) and \(f(2)=3\). What is the approximation for \(f(2.1)\) found by using the line tangent to the graph of \(f\) at \(x=2\) ?

Summary Submit

First, write the equation of the tangent line at \(x=2\).

$$ f'(x)=2x+2 $$ $$f'(2)=2(2)+2 = 6 $$

Given a slope of \(6\) and point \((2,3)\),

$$ y-3=6(x-2) $$

At \(x=2.1\),

$$ y-3=6(2.1-2) $$ $$ y-3=.6 $$ $$ y=\boxed{3.6} $$