3.8. Approximation Errors http://www.ck12.org
The linear approximation can be used to approximate functional values that deviate slightly from known values. The
following example illustrates this process.
Example 1:
Use the linear approximation of the functionf(x) =√x−2 ata=6 to approximate√ 3 .95.
Solution:
We know thatf( 6 ) =√ 6 − 2 =2. So we will find the linear approximation of the function and substitutexvalues
close to 6.
L(x) =f( 6 )+f′( 6 )(x− 6 ).We note thatf′(x) = 2 √x^1 − 2 ,f′( 6 ) =^14.
We also know thatf( 6 ) = 2.
By substitution, we have
f(x)≈f( 6 )+f′( 6 )(x− 6 )forxnear 6.
HenceL(x) = 2 +^14 (x− 6 ) =^12 +^14 x.
We observe that to approximate√ 3 .95 we need to evaluate the linear approximation at 5.95, and we have
L( 5. 95 ) =^12 +^14 ( 5. 95 ) = 1 .9875. If we were to compare this approximation to the actual value, √ 3. 95 ≈ 1 .9874,
we see that it is a very good approximation.
If we observe a table ofxvalues close to 6,we see how the approximations compare to the actual value.
f(x) =√x− 2 x L(x) =^12 +^14 x Actual
√
√^3.^955.^951.^98751.^9874
√^3.^995.^991.^99751.^9974
√^4622
√^4.^16.^012.^00252.^0024
4. 05 6. 05 2. 0125 2. 0124Setting Error Estimates
We would like to have confidence in the approximations we make. We therefore can choose thexvalues close to a,
to ensure that the errors are within acceptable boundaries. For the previous example, we saw that the values ofL(x)
close toa= 6 ,gave very good approximations, all within 0.0001 of the actual value.
Example 2:
Let’s suppose that for the previous example, we did not require such precision. Rather, suppose we wanted to find
the range ofxvalues close to 6 that we could choose to ensure that our approximations lie within 0.01 of the actual
value.
Solution:
The easiest way for us to find the proper range ofxvalues is to use the graphing calculator. We first note that our
precision requirement can be stated as|√x− 2 −(^12 + 4 x)|< 0. 01.
If we enter the functionsf(x) =√x−2 andL(x) =^12 +^14 xinto theY=menu asY 1 andY 2 , respectively, we will
be able to view the function values of the functions using the[TABLE]feature of the calculator. In order to view