Estimate Δf using the Linear Approximation and use a calculator to compute the error.
$f(x)= \sqrt{3+x}$
$a=6$
$Δx= 0.5$
$Δf \approx \frac{1}{12}$
^ I got this part, just take the derivative.
With these calculations, we have determined that the square root of _____
^I was thinking it was the $\sqrt9$, but that is incorrect
is approximately _____
^obviously thought this was $3$
The error in Linear Approximation is: _____
^Went through the whole process of trying to find the approximation, ended up with a $negative$ number though $-107.38\%$
[Note: This is not asking for relative error or percent error.]
My main struggle is trying to understand what they are asking for?
Any ideas?
Best Answer
$$\quad{f(X+\Delta x) \sim f(x) +\Delta x .f'(x)\\f(x)=\sqrt{3+x},f'(x)=\frac{1}{2\sqrt{3+x}}\\\to\\f(6+.5)\sim f(6) +.5 \frac{1}{2\sqrt{9}}=3+0.5*\frac{1}{6}=3+\frac{1}{12}$$ now $$f(6+\Delta x)-f(6)\sim \frac{1}{12}=0.083333333...}$$
without approximation $$\sqrt{3+6+0.5}-\sqrt{3+6}=0.0822070014844...$$