Final Exam 2014 - Math 121A Final exam 1(a Calculate the...

Math 121A: Final exam
1. (a) Calculate the eigenvalues and eigenvectors of the matrix
A =


1 0 1
0 1 1
1 1 0


.
(b) Solve the linear system
A


x
y
z


=


0
4
2


.
2. (a) Calculate the radius of convergence R of the power series
∞
∑
n = 1
x
n
n ( - 3 ) n
.
By considering the series for x = ± R and using appropriate series tests, deter-
mine the exact range of x for which it converges.
(b) For the function f
(
t
) = -
5
+
2 t
2
, calculate f
(
0
)
, f
(
1
)
, f
(
2
)
, and f
(
3
)
. Use the
results to sketch f over the range
-
3
≤
t
≤
3.
(c) Determine the precise set of values of t for which the series
∞
∑
n = 1
[
f
(
t
)] n
n
( -
3
)
n
will converge.
3. (a) Let a function f ( x ) have the Fourier transform
˜
f ( α ) . Let g ( x ) = f ( - x ) and
h
(
x
) =
x f
(
x
)
. Show that the Fourier transforms of g and h are given by
˜
g
(
α
) =
˜
f
( -
α
)
and
˜
h
(
α
) =
i
˜
f
0
(
α
)
.
(b) Determine the Fourier transform
˜
f
(
α
)
of the function
f ( x ) =
e
- x
for x
>
0,
0 for x
≤
0.
(c) By using the above results, or otherwise, determine the Fourier transform ˜ s ( α )
of
s
(
x
) =
x
2
e
-| x |
.
Show that ˜ s ( α ) is real.