242 Math Section 014: Test #3
N AME:
'
Please circle your lab/discussion: 060 (12:00) 061 (11:00) 062 (2:30)
There are 6 questions on 6 pages. Points per page: 5, 5, 5, C, 5, 6.
The 7th page has some f ormulas you eari refer to if n eeded.
[5 p oints] Question 1. D etermine w hether the infinite series
is absolutely convergent, conditionally convergent, or divergent.
/? +2.
boes 'it
/
'
[5 pointy] Question 2. Find the interval of convergence of t he power
seres
oo
X^
1)^
i < - y-2
<3
,
1
->
Z
or
[5 p oints] Question 3. Find a power series r epresentation of t he f unction
1 + 2.7;
1 - x- '
Then use the first t hree terms of t hat series ( the constant t erm, the x term.
,
?
,
*,
^O - .
and, the x t erm) t o h ud a n approximation t o 02
1
l
0.999
x
fl /.S0
X
-
/
f ^a f -y -f -X
1 + IK -0-001
0.0(93 +
i +0.000003
= 1.003003
[6 points] Question 4. Find the first t hree terms of the Taylor series for the
function
and use those f irst t hree terms to get an approximation for ( 1.04) 1 / 2 .
60 =
1
X
-f- 0.02 - 0.0002.
' i.om
[5 points] Question 5. Find the first three terms of the Taylor series for t he
function
f ( x ) = c~x sin.x.
-2L f *! _:*! + . . .
|>
-^
-?'
*
3!
>
^
3)
-
l_^
-
5!
.
V- - -
'7!
3!
3.
5
.
i-x
1'
I-'X
z~ *
'Y
A
^
L
/v
A
3
A/3
^/v3
X -if" ^^
3
O^
- /X
-
[6 p oints] Question 6. Consider the p arametric curve :/; t + lnt, y ~ t hit,
where 0 < t < oo. F ind dy/dx and d2y/dx'2. Find w here the curve is
increasing, decreasing, concave up. or concave down. Sketch t he c urve.
i
+
1 - J=1 -b
<L
<!
dt
-t +1
,,i
1
up