CK12 Calculus - Single Variable

for near
Hence

We observethat to approximate we needto evaluatethe linearapproximationat , and we
have

. If we wereto comparethis approximationto the actualvalue,
, we see that it is a very goodapproximation.

If we observea tableof valuescloseto we see how the approximationscompareto the actualvalue.

Actual

5.951.9875 1.9874

5.991.9975 1.9974

6 2 2

6.012.0025 2.0024

6.052.0125 2.0124

SettingErrorEstimates
We wouldlike to haveconfidencein the approximationswe make.We thereforecan choosethe values
closeto a to ensurethat the errorsare withinacceptableboundaries.For the previousexample,we saw
that the valuesof closeto gavevery goodapproximations,all within of the actual
value.
Example2:
Let’s supposethat for the previousexample,we did not requiresuchprecision.Rather, supposewe wanted
to find the rangeof valuescloseto that we couldchooseto ensurethat our approximationslie within
of the actualvalue.
Solution:
The easiestway for us to find the properrangeof valuesis to use the graphingcalculator. We first note

that our precisionrequirementcan be statedas

If we enterthe functions and into theY =menuasY 1 andY 2 , respec-
tively, we will be able to view the functionvaluesof the functionsusingtheTABLEfeatureof the calculator.