Exam
III
Practice
Problems:
Calculus
II
–
Dr.
Staples
Scope
of
Exam:
8.5‐8.6,
9.1‐9.3,
and
10.1‐10.2
Section
8.5
and
8.6
Ratio,
Root
and
Comparison
Tests,
Alternating
Series
Test
For
the
following
series,
determine
whether
they
diverge
or
converge.
Clearly
STATE
the
REASON/TEST
that
you
employed.
Show
the
values
of
all
computed
limits.
To
save
time,
you
may
assume
the
function
is
decreasing
when
applying
the
Alternating
Series
Test.
!
1.
3.
! !
!!! ! ! 2.
!
!
!!! ! ! !"# ! !!
!!! ! ! 4.
!
!
!!! ! !! 5.
! ! !!
!!! ! ! 6.
7.
9.
! !
!!! !! 8.
!
!
!
!!! ! ! !!
!
! !
!!! !! !
!
!!! (! !"#)! 10.
11.
! ! !" !
!!! ! 12.
!
!
!!! !
!
!!!( !! ) !
!
!!
!!!(! +3 )
!
!
!
!
13.
!
!!! ! ! !!
14.
!!!(1 + ! ) !
!!
! !
15.
!
!!!(−1) ! 16.
!!!(−1) !" ! !
! !
! !!
17.
!
!!!(−1) ! ! 18.
!!!(−1) !" ! ! !"# !
! !!!!
19.
!
!!!(−1) !!!!
20.
!!! ! ! For
#15
to
#20,
now
determine
if
the
series
converge
absolutely,
converge
conditionally,
or
diverge.
Section
9.1:
Taylor
Polynomials
For
the
following
functions
find
the
5th
degree
Taylor
Polynomial
centered
at
0.
21.
𝑓 𝑥 = sin 𝑥
22.
𝑓 𝑥 = 𝑒 !! 23.
𝑓 𝑥 = ln(1 − 𝑥)
24.
For
the
function
𝑓 𝑥 = cos 𝑥 ,
find
the
4th
degree
Taylor
Polynomial
centered
at
π/6.
25.
For
the
function
𝑓 𝑥 = 𝑥,
find
the
4th
degree
Taylor
Polynomial
centered
at
4.
Section
9.2:
Power
Series,
Radius
of
Convergence,
and
Interval
of
Convergence
For
the
following
power
series,
find
the
radius
and
interval
of
convergence
26. 28. !
!
! (!!) (!!!)
27. !!!
!!!
!
!
! (!!) (!!!)
!!!
!!
!!!!
!
!!! !!!! !
29. ! (!!!)
!!! !!!!
!!!
(!)!!
30. !
!!! ! ! !! Using
information
for
geometric
series
and
rules
of
manipulation
of
power
series,
find
the
power
series
and
interval
of
convergence
for
the
following
functions.
!!
31.
𝑓 𝑥 = !!!
!
32.
𝑓 𝑥 = !!!!
Using
the
power
series
for
𝑓 𝑥 find
the
power
series
and
interval
of
convergence
for
the
function
𝑔 𝑥 .
!
33.
𝑓 𝑥 = !!!
and
𝑔 𝑥 = ln 1 − 𝑥 !
!
34.
𝑓 𝑥 = !!!!
and
𝑔 𝑥 = (!!!!)! Section
9.3:
Taylor
Series
For
the
following
functions
find
the
Taylor
Series
35.
𝑓 𝑥 = sin 𝑥
about
𝑥 = 0. 36.
𝑓 𝑥 = ln 𝑥, about
𝑥
=
4.
37.
𝑓 𝑥 = 𝑒 !! about
𝑥 = 1. Section
10.1:
Parametric
Equations
For
the
following
problems,
a)
eliminate
the
parameter
to
obtain
an
equation
in
x
and
y
b)
graph
the
curve
and
indicate
the
positive
orientation
c)
find
the
requested
derivative
and
tangent
line
38.
𝑥 = 𝑒 !! , 𝑦 = 𝑒 ! + 1, 𝑓𝑜𝑟 0 ≤ 𝑡 ≤ 25.
Find
the
derivate
dy/dx
when
t=1
and
the
equation
of
the
tangent
line
to
the
curve
there.
39.
𝑥 = 4 cos 𝑡 , 𝑦 = 3 sin 𝑡 , 𝑓𝑜𝑟 0 ≤ 𝑡 ≤ 2𝜋
Find
the
derivate
dy/dx
when
t=𝜋/4
and
the
equation
of
the
tangent
line
to
the
curve
there.
40.
Find
a
set
of
parametric
equations
for
the
circle
centered
at
the
origin
with
radius
6,
generated
counterclockwise.
Section
10.2:
Polar
Coordinates
!
41.
Change
the
polar
coordinates
(2, ! )
to
rectangular
coordinates.
42.
Express
the
point
(‐4,
4 3)
in
polar
coordinates
with
the
angle
𝜃
between
and
2π.
43.
Graph
the
polar
curve
𝑟 = 4 sin 𝜃.
44.
Graph
the
polar
curve
r=2.
45.
Graph
the
polar
curve
r=
2(1‐cos
𝜃).
46.
Graph
the
polar
curve
𝜃 = 2𝜋/3.
Convert
the
following
equations
to
Rectangular
(x‐y)
form.
47.
𝑟 sin 𝜃 = 3.
48.
𝑟 = 6 cos 𝜃.
49.
𝑟 = 4.
50.
𝑟 = cot 𝜃 csc 𝜃.