Let f be a differentiable function such that f(3) = 2 and f'(3) = 5. if the tangent line to the graph of f at x = 3 is used to find an approximation to a zero of f, that approximation is?
so confused

The linear approximation to centered at is



What this means is that you can use the tangent line to at to get a decent approximation of at some other value of to within a certain degree of accuracy depending on how close this value is close to . (Note that when , the approximation is exact; .)

So what you're asked to do is find an approximate value of a zero of near . That is to say, you're looking for some value such that , but all you have at your disposal is the linear approximation to the function.





You know that if is a zero of , then , so you get



Solving for , you find



This means that an approximate zero of is .