Verify the given linear approximation at a=0. Then determine the values of x for which the linear approximation is accurate to within 0.1
(i) 3root(1-x) = 1-(1/3)x
(ii) tan x = x
(iii) 1/(1+2x)^4 = 1-8x
(iv) e^x = 1 + x
I understand verifying the linear approximation, I am just confused on how to determine the values for x thats accurate within 0.1. Any help would be greatly appreciated!!!!
Once you have verified the linear approximation, to find the values for x that make the linear approximation within .1 of the actual value:
Let f(x) be the sactual value and A(x) be the approximation. Then we want `|f(x)-A(x)|<=.1` or `-.1<f(x)-A(x)<.1`
You can solve either with a graphing utility. For example find the values of x so that `1-1/3x` is within .1 of the actual value of `root(3)(1-x)` :
You can graph both the absolute value and the line y=.1 and find the intersections:
The approximate values are x=-1.204336 and x=.706649
Alternatively you can work algebraically:
Working on the right side first:
Again using a algebra utility we find `x~~-1.204336,.706649`
You would use a similar procedure for the rest of the problems:
Find x so that |tanx-x|<.1 etc...