9.7 Taylor Polynomials and Approximations
574
9.7 Taylor Polynomials and Approximations
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1.
Find a first-degree polynomial function P1 whose value and slope agree with the
value and slope of
at
.
a.
b.
c.
d.
e.
____ 2.
Use a graphing utility to graph
and P1, a first-degree polynomial
.
a.
d.
b. e.
575 Chapter 9: Infinite Series
c.
____ 3.
Find a first-degree polynomial function P
1
whose value and slope agree with the
value and slope of f at
. What is P1 called?
a.
tangent line to
at
b.
secant line to
at
c.
secant line to
at
d.
secant line to
at
e.
tangent line to
at
9.7 Taylor Polynomials and Approximations
576
____ 4.
Find a first-degree polynomial function P1 whose value and slope agree with the
value and slope of f at
. What is P1 called?
a.
; tangent line to
at
b.
; secant line to
at
c.
; tangent line to
at
d.
; differential of
at
e.
; tangent line to
at
____
5.
What is
a first-degree polynomial function whose value and slope agree with the
value and slope of
at
?
a.
is the tangent line to the curve
at the point
b.
is the tangent line to the curve
at the point
c.
is the tangent line to the curve
at the point
d.
is the tangent line to the curve
at the point
e.
is the tangent line to the curve
at the point
____
6.
Consider the function
and its second-degree polynomial
at
Compute the value of
and
Round
your answer to four decimal places.
a.
b.
c.
d.
e.
577 Chapter 9: Infinite Series
____ 7. Find the Maclaurin polynomial of degree 4 for the function.
a.
b.
c.
d.
e.
____ 8. Find the Maclaurin polynomial of degree 3 for the function.
a.
b.
c.
d.
e.
____ 9. Find the Maclaurin polynomial of degree 5 for the function.
a.
b.
c.
d.
e.
9.7 Taylor Polynomials and Approximations
578
____ 10.
Find the Maclaurin polynomial of degree 4 for the function.
a.
b.
c.
d.
e.
____ 11. Find the fourth degree Maclaurin polynomial for the function.
a.
b.
c.
d.
e.
____ 12. Find the Maclaurin polynomial of degree two for the function .
a.
b.
c.
d.
e.
579
Chapter 9: Infinite Series
____
13.
Find the third Taylor polynomial for
expanded about
.
a.
b.
c.
d.
e.
____
14.
Find the third degree Taylor polynomial centered at
for the function.
a.
b.
c.
d.
e.
____
15.
Find the fourth degree Taylor polynomial centered at
for the function.
a.
b.
c.
d.
e.
____ 16. Determine the degree of the Maclaurin polynomial required for the error in the
approximation of the function to be less than 0.001.
2
5
1
3
9.7 Taylor Polynomials and Approximations
580
____ 17.
Determine the values of x for which the function
can be replaced by the
Taylor polynomial
if the error cannot exceed 0.008. Round your answer to four
decimal places.
a.
b.
c.
d.
e.
581 Chapter 9: Infinite Series
9.7 Taylor Polynomials and Approximations
Answer Section
9.8 Power Series
582
9.8 Power Series
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. State where the power series is centered.
0
–3
2
3
–2
____ 2. Find the radius of convergence of the power series.
a.
b.
5
25
____ 3. Find the radius of convergence of the power series.
16
64
0
8
583
Chapter 9: Infinite Series
____
4.
Find the interval of convergence of the power series. (Be sure to include a check for
convergence at the endpoints of the interval.)
a.
b.
c.
d.
e.
____ 5. Find the interval of convergence of the power series. (Be sure to include a check for
convergence at the endpoints of the interval.)
a.
b.
c.
d.
e.
____ 6. Find the interval of convergence of the power series. (Be sure to include a check for
convergence at the endpoints of the interval.)
a.
b.
c.
d.
e.
9.8 Power Series
584
____ 7.
Find the interval of convergence of the power series. (Be sure to include a check for
convergence at the endpoints of the interval.)
a.
b.
c.
d.
e.
____ 8. Find the interval of convergence of the power series . (Be sure to
include a check for convergence at the endpoints of the interval.)
a.
b.
c.
d.
e.
____ 9. Find the interval of convergence of the power series . (Be sure to
include a check for convergence at the endpoints of the interval.)
a.
b.
c.
d.
e.
585
Chapter 9: Infinite Series
____
10.
Write an equivalent series of the series
with the index of summation
beginning at
.
a.
b.
c.
d.
e.
____ 11.
Write an equivalent series of the series
with the index of
summation beginning at
.
a.
b.
c.
d.
e.
9.8 Power Series
586
____ 12. Consider the function given by . Find the interval of convergence
for .
a.
b.
c.
d.
e.
____ 13. Consider the function given by . Find the interval of convergence
for .
a.
b.
c.
d.
e.
____ 14. Consider the function given by . Find the interval of
convergence for .
a.
b.
c.
d.
e.
587 Chapter 9: Infinite Series
____ 15. Consider the function given by . Find the interval of
convergence for .
a.
b.
c.
d.
e.
____ 16. Identify the graph of the first 10 terms of the sequence of partial sum of the series
for .
a. d.
b. e.
9.8 Power Series
588
c.
____ 17. Find the differential equation having the solution .
a.
b.
c.
d.
e.
589 Chapter 9: Infinite Series
9.8 Power Series
Answer Section
9.9 Representation of Functions by Power Series
590
9.9 Representation of Functions by Power Series
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find a geometric power series for the function centered at 0.
a.
b.
c.
d.
e.
____ 2. Find a geometric power series for the function centered at 0.
a.
b.
c.
d.
e.