I have a homework question that is asking to use the linear approximation to estimate $\Delta f=f(3.02)-f(3)$ for $f(x)=x^2$.
I know that the linear approximation is $L(x) = f'(a)(x-a)+f(a)$.
I however do not quit understand what the problem is asking in the first place. Couldn't I just estimate $\Delta f=f(3.02)-f(3)$, by plugging both 3.02 and 3 into the original function of $f(x) = x^2$ and subtracting them. How do I do this with linear approximation?
Best Answer
You can indeed compute: $$f(3.02)-f(3) = (3.02)^2-3^2 = (6.02)\cdot 0.02 = 0.1204,$$ the exact value. The point of the exercise is avoiding this computations. Using linear approximation gives: $$L(3.02) = f(3) + f'(3)\cdot0.02,$$ so that $\Delta f \approx f'(3)\cdot 0.02 = 6 \cdot 0.02 = 0.12$. Good, no?