Final Examination
MATH 262
December 12, 2005.
1. Find the interval of convergence of the series
∞
X
n
=0
(
x

2)
n
2
2
n
+ 1
and determine whether the series converges absolutely or conditionally at the
endpoints.
2.
(a) Find the Maclaurin expansion for the function
F
(
x
) =
Z
x
0
sin(
t
2
)
dt
and compute
F
(1) to 2 decimal places.
(b) Find the Taylor series expansion of
f
(
x
) =
x

1
1 + (
x

1)
2
about
x
= 1 and use this to compute
f
(9)
(1) and
f
(10)
(1).
3. Find the solution of the differential equation
(1

x
2
)
y
00

xy
0
+ 9
y
= 0
satisfying
y
(0) = 0,
y
0
(0) = 1.
4. Given the curve
r
=
t
2
i
+
t
2
j
+
t
3
k
,
(a) find the arc length of the curve from
t
= 0 to
t
= 1;
(b) find the unit tangent vector
ˆ
T
, principal normal
ˆ
N
, and binormal
ˆ
B
of the
curve at the point
t
= 1;
(c) find the curvature
κ
and torsion
τ
of the curve at point
t
= 1.
1
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Final Examination
MATH 262
December 12, 2005.
5. Show that the function
f
(
x, y
) =
(
ax
+
by
2
√
x
2
+
y
2
if (
x, y
)
6
= (0
,
0)
,
0
if (
x, y
) = (0
,
0)
is continuous at (0
,
0) if
a
= 0 but is not continuous at (0
,
0) if
a
6
= 0.
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 Spring '09
 GREGRIX
 Math, Calculus, Taylor Series, McGill University Faculty of Engineering, Professor J. Labute

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