For what values of x is the linear approximation $\sqrt{x + 3} \simeq \frac74 + \frac x4 $accurate to within 0.5?

Question

For what values of x is the linear approximation $\sqrt{x + 3} \simeq \frac74 + \frac x4 $accurate to within 0.5?

My Try

I found this question under llinearization lesson Since this problem does not mention about a point around which it is approximated, I took it as 'a'

$\therefore$ $f(x)=f(a)+(x-a) f'(a)$

$f(x)=\sqrt{a+3} + \frac{x-a}{2\sqrt{x+3}}$

How do I proceed to find 'a' such that the error is within 0.5? Please Help! Thanks in advance.

Answer 1

Hint:

You could try to solve $$\sqrt{x + 3} = \frac74 + \frac x4 +\frac12$$ and $$\sqrt{x + 3} = \frac74 + \frac x4 -\frac12$$ which if you square both sides will give you two quadratics and possibly some spurious solutions to check. Then spot the relevant intervals