http://www.ck12.org Chapter 3. Applications of Derivatives
We get a slightly better approximation for the quadratic than for the linear. If we reflect on this a bit, the finding
makes sense since the shape and properties of quadratic functions more closely approximate the shape of radical
functions.
Finally, as in the first example, we wish to determine the range ofxvalues that will ensure that our approximations
are within 0.01 of the actual value. Using the[TABLE]feature of the calculator, we find that if 4. 444 ≤×≤ 7. 87 ,
then|√x− 2 −T(x)|< 0 .01.
Lesson Summary
- We extended the Mean Value Theorem to make linear approximations.
- We analyzed errors in linear approximations.
- We extended the Mean Value Theorem to make quadratic approximations.
- We analyzed errors in quadratic approximations.
Review Questions
In problems #1–4, find the linearizationL(x)of the function atx=a.
- f(x) = 2 x^4 − 6 x^3 neara=− 2
- f(x) =x^23 neara= 27
- Find the linearization of the functionf(x) =√ 5 −xnear a = 1 and use it to approximate√ 4 .01.
- Based on using linear approximations, is the following approximation reasonable?
- 0014 = 1. 004
- Use a linear approximation to approximate the following:
16. 0834 - Verify the following linear approximation ata= 1 .Determine the values ofxfor which the linear approxima-
tion is accurate to 0. 01.
√ (^32) −x≈ 4
3 −
x
3
- Find the quadratic approximation for the function in #3,f(x) =
√
5 −xneara= 1.- Determine the values ofxfor which the quadratic approximation found in #7 is accurate to 0. 01.
- Determine the quadratic approximation forf(x) = 2 x^4 − 6 x^3 neara=− 2 .Do you expect that the quadratic
approximation is better or worse than the linear approximation? Explain your answer.
Texas Instruments Resources
In the CK-12 Texas Instruments Calculus FlexBook® resource, there are graphing calculator activities designed
to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9728.