MATH2060A Elementary Analysis II (Spring 2012-13)
Exercise 8
Deadline: Mar 25, 2013.
Hand in: Supplementary Exercises 1, 3, 4 and 7.
Supplementary Exercises
1. Study the uniform convergence for the following sequences in the indicated domains:
(a) xenx ,
MATH2060A Elementary Analysis II (Spring 2012-13)
Exercise 7
Deadline: March 18, 2013.
Hand in: Supplementary Exercises 1, 3, 6 and 7.
Supplementary Exercises
Problems 9 and 10 are optional.
1
1. In Example 1 in 5.3, Notes 2, we evaluate the integral 0 1
MATH2060A Elementary Analysis II (Spring 2012-13)
Exercise 6
Deadline: March 11, 2013.
Hand in: No 18, and Supplementary Ex no 3 and 4.
Section 7.2: No 18, 19.
Supplementary Exercises
1. Deduce from the Cauchy-Schwarz inequality
b
b
b
f2
|f g |
g2,
a
a
a
MAT2060A Elementary Analysis II (Spring 2012-13)
Exercise 4
Deadline: Feb 18, 2013.
Hand in: Section 6.4 no. 12, 16; Supplementary Exercises (1) and (2)
Section 6.4 3, 6, 10, 12, 14, 16, 18
Supplementary Exercises
1. (a) For any convex function f on (a, b
MATH2060A Elementary Analysis II (Spring 2013)
Exercise 5
Deadline: March 4, 2013.
Hand in: Section 7.1 Q.14; Supplementary Exercises (1) and (3).
Section 7.1: Q.8-11, 14, 15.
Supplementary Exercises (Use the knowledge in Section 1 in Notes 2.)
1. Let P b
MATH2060A Elementary Analysis II (Spring 2012-13)
Exercise 3
Deadline: Feb 15, 2013.
Hand in: Section 6.3 no 9(a), 10(d), 11 (c), 14, Supplementary Exercises (1) and (2)
Section 6.2: no 20; Section 6.3: no 4-12, 14.
Supplementary Exercises
1. Let f : [0,
MATH2060A Assignment 2
Deadline: Feb 4, 2013.
Hand in: Section 6.2 2(a)(c), 3(a)(d), 10, 19; Supplementary problem (1)
Section 6.2: Q.2, 3, 5, 9-13, 18, 19
Supplementary Problems
1. Study the optimization problem for the function
f (x) = |x(x 1)(x + 6)|,
MATH2060A Exercise 1
Deadline: Jan 24, 2013.
Hand in: Section 6.1 no 6,8,9,13 and supplementary exercise no (2)
Section 6.1 no 5-10, 13, 14, 17.
Supplementary Exercises
1. A function f : (a, b) R has a symmetric derivative at c (a, b) if
f (c + h) f (c h)
2012-13 Second Term MAT2060A
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Solution 7
Supplementary Exercises
1. If we change [0, /2] to [0, 5/2]. Then cos t will varies from [1, 1] which exceeds the range
of [0, 1]. It violates the requirements of change of variables theorem. Also we noticed that
2012-13 Second Term MAT2060A
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Solution 6
Section 7.2
18. If f 0, then result is trivial. Otherwise, since f is continuous on [a, b], [a, b] s.t.
f ( ) = sup f > 0. By continuity, , 0 < < f ( )/2, > 0 s.t.
f (x) > f ( ) x ( , + ) [a, b].
Hence
b
b
a
a
( ,
2012-13 Second Term MAT2060A
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Solution 4
Section 6.4
3. It is trivial for n = 2. Assume it is true for n = k . Then for n = k + 1, we have
(f g )(k+1) (x)
k
d
k
= (f g )(k) (x) =
f (kj ) (x) g (j ) (x), by induction hypothesis
j
dx
j =0
k
=
j =0
k
=
j =0
2012-13 Second Term MAT2060A
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Solution 3
Section 6.2
20. (a) By Mean Value Theorem, there exists a c1 (0, 1) such that f (c1 )(1 0) = f (1)
f (0). i.e.f (c1 ) = 1.
(b) By Mean Value Theorem, there exists a c2 (0, 1) such that f (c2 )(2 1) = f (2)
f (1)