If the desired error is less than .5, the starting equation should be |sqrt(x+3) - ( 7/4 + x/4) | < 0.5. You wrote > instead of <.
For what values of x is the linear approximation sqrt(x+3) ~ 7/4 + x/4
accurate to within 0.5?
"~" stands for about
Solution: Accuracy to within 0.5 means that the function should differ by less than 0.5:
|sqrt(x+3) - ( 7/4 + x/4) | > 0.5
Equivalently we could write
sqrt(x+3) - 0.5 < ( 7/4 + x/4) < sqrt(x+3) + 0.5
I have two questions:
First, how do they get from "linear approximation sqrt(x+3) ~ 7/4 + x/4 accurate to within 0.5" to the expression "|sqrt(x+3) - ( 7/4 + x/4) | > 0.5"
Second, how do they get the equvalent term from
|sqrt(x+3) - ( 7/4 + x/4) | > 0.5
to
sqrt(x+3) - 0.5 < ( 7/4 + x/4) < sqrt(x+3) + 0.5
I thought |u| < a --> -a < u < a
So, I get -0.5 < sqrt(x+3) - ( 7/4 + x/4) < 0.5
What do I have to do to get to
sqrt(x+3) - 0.5 < ( 7/4 + x/4) < sqrt(x+3) + 0.5
Thanks for any help!
Additionally I want to add this information I have:
F(x)= sqrt(x+3)
Linear approximation f(x) ~ f(a)+f'(a)(x-a)
a=1