DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE NAME: Binshihoun Taha Student Id: XXXXXXXXXX Tutorial Group: B MATH136 S212 Mathematics IB Assignment 5 Due 11:30 12/11 2012 Please sign the declaration...

DEPARTMENT OF
MATHEMATICS
FACULTY OF SCIENCE
NAME: Binshihoun Taha
Student Id: XXXXXXXXXX
Tutorial Group: B
MATH136 S212
Mathematics IB
Assignment 5
Due 11:30 12/11 2012
Please sign the declaration below, and staple this sheet to the front of your solutions. Your assignment must be
submitted at the Science Centre, E7A Level 1.
Your assignment must be STAPLED, please do not put it in a plastic sleeve.
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assignment, the following acts constitute plagiarism:
a) Copying or summarizing another person’s work.
) Where there was collaborative preparatory work, submitting substantially the same final version of any
material as another student.
Encouraging or assisting another person to commit plagiarism is a form of improper collusion and may attract
the same penalties.
STATEMENT TO BE SIGNED BY STUDENT
1. I have read the definition of plagiarism that appears above.
2. In my assignment I have carefully acknowledged the source of any material which is not my own work.
3. I am aware that the penalties for plagiarism can be very severe.
4. If I have discussed the assignment with another student, I have written the solutions independently.
SIGNATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Note that marks may be deducted for late assignments. If you need an extension and have a good
eason, then you should ask your lecturers, or apply for Special Consideration. You cannot just
assume that it is OK to submit your work a day or two late.
Have you ready Guidelines on assignment presentation, on the MATH136 assignment download page?
1. Find the general solution of the following differential equations:
(a)
d2y
dx2
− 2dy
dx
+ 17y = 8 cos 3x+ 6 sin 3x, (b)
d2y
dx2
− y = 2e−x.
2. Find the solution of the differential equation
d2y
dx2
+ 9y = 3 cos 3x
that satisfies y (0) = y′ (0) = 0.
3. Determine whether the following series is convergent, explaining your reason:
∞∑
n=1
sin
nπ
2
.
First downloaded: 6/11/2012 at 0:21::20
4. (a) Use the comparison test to determine the convergence of the series
∞∑
n=1
n2
n4 + 3n2 + 4
and
∞∑
n=1
1
n!
.
(You may assume that
∑∞
n=1 n
−p converges when p > 1 and diverges when p = 1.)
(b) Is the series
∞∑
n=1
(−1)n+1√
n+ 1 log (n+ 1)
=
1√
2 log 2
− 1√
3 log 3
+
1√
4 log 4
− 1√
5 log 5
+ . . .
convergent? Explain your reasons, clearly stating any test that you apply.
5. Find the Taylor series about x = 0 of the function
f (x) = log(1 + x)
Use the obtained result to determine the Taylor series about x = 0 of the function
g(x) = log
1 + x
1− x
.
6. Suppose u =

4
3
2
1
 andW is a subspace of R4, and thatB1 =


1
1
−1
1
 ,

−2
2
1
1

 andB2 =


−1
3
0
2
 ,

0
4
−1
3


are both bases for W .
(a) Which basis for W is orthogonal?
(b) Use the orthogonal basis to find projWu.
(c) Use the other basis and the least squares method to find projWu.
7. Find the least squares line that best fits the points (1, 2), (2, 3), (3, 3), (4, 6), (5, 7).
8. [Harder] A formula for the least squares line y = β0 +β1x that best fits n data points (xi, yi) is often given
as
β1 =
n
n∑
i=1
xiyi −
(
n∑
i=1
xi
)(
n∑
i=1
yi
)
n
n∑
i=1
x2i −
(
n∑
i=1
xi
)2 , β0 = 1n
(
n∑
i=1
yi − β1
n∑
i=1
xi
)
.
(a) Derive the formula for β1 from the least squares solution of y = Xβ, where y =
y1...
yn
, X =
1 x11 ...
1 xn

and β =
(
β0
β1
)
. [Hint: start by writing the entries in XTX and XTy as sums.]
(b) Show that β0 =
1
n
(
n∑
i=1
yi − β1
n∑
i=1
xi
)
matches the expression you obtained for β0 in part (a).
(c) Why might it make sense to express β0 as given, rather than as the expression you found in part (a)?
9. Consider the quadratic form Q(x) , given in the form xtAx :
Q(x) = xtAx = 3x21 + 2x
2
2 + 2x
2
3 + 2x1x2 + 2x1x3 + 4x2x3 .
(a) Find the maximum and minimum values of Q(x) subject to the constraint ‖x‖ = 1 .
(b) For which vectors x are these values attained? (Note: they are unit vectors.)
(c) Using a suitable change of variables, write Q(x) as a sum of squares. Describe this change of variables.