MATH 015 Foundations Mathematics A
Lecture 5 Ex. 0.6 and 1.3
0.6 FACTORING
Question 1
Factor:
a.
b.
c.
x 2 36
25 x 2 36 y 2
4 x 2 12 x 9
Page 1 of 8
Question 2
Factor completely:
a.
b.
c.
2 x 2 50
3x
Section 4.1 Sequences of Real Numbers
179
4.1 SEQUENCES OF REAL NUMBERS
An infinite sequence (morebriefly,
a sequence) of real numbers is a realvalued function
defined on a set of integers n n k . W
Section 6.4 The Implicit Function Theorem
x 2 y C xy
; X0 D .1; 1/
2xy C xy 2
2
3
2
3
u
2x 2y C x 3 C
5 ; X D .0; 1; 1/
(c) 4 v 5 D F.x; y; / D 4
x 3 C y
w
xCyC
2
3
2
3
u
x cos y cos
(d) 4 v 5 D F.x
Section 8.2 Compact Sets in a Metric Space
25.
(a) Show that
kf k D
Z
535
b
a
jf .x/j dx
is a norm on C a; b,
(b) Show that the sequence ffn g defined by
fn .x/ D
26.
27.
28.
29.
30.
x
b
is a Cauchy s
136 Chapter 3 Integral Calculus of Functions of One Variable
Proof Any Riemann sum of f C g over a partition P D fx0 ; x1; : : : ; xn g of a; b can
be written as
f Cg D
D
n
X
j D1
n
X
f .cj / C g.cj /
CHAPTER 6
VectorValued Functions
of Several Variables
IN THIS CHAPTER we study the differential calculus of vectorvalued functions of several
variables.
SECTION 6.1 reviews matrices, determinants, a
74 Chapter 2 Differential Calculus of Functions of One Variable
and so on, and
d nf
D f .n/ :
dx n
Example 2.3.1 If n is a positive integer and
f .x/ D x n ;
then
f .x/
x
f .x0 /
xn
D
x0
x
x0n
x
D
x0
Section 5.4 The Chain Rule and Taylors Theorem
25.
339
Prove: If f is differentiable at .x0 ; y0 / and
f .x; y/ a b.x x0 / c.y
p
.x;y/!.x0 ;y0 /
.x x0 /2 C .y y0 /2
lim
y0 /
D 0;
then a D f .x0 ; y0 /
316 Chapter 5 RealValued Functions of Several Variables
6.
7.
8
< sin.x 2 y 2 /
;
(e) f .X/ D
x2 y2
:
1;
x y;
x D y
Define (a) limjXj!1 f .X/ D 1 and (b) limjXj!1 f .X/ D 1.
Let
f .X/ D
jx1 ja1 jx2 j
Vectors summary
Quantities which have only magnitude are called scalars. Quantities which have magnitude and
direction are called vectors.
AB is the position vector of B relative to A and is the vecto
The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises and Solutions for Week 2
2014
Assumed Knowledge: Sigma notation for sums. The ideas of a sequence of numbers and
8002A SEMESTER 1 2022 PAGE 16 OF 30
Extended Answer Section
There are three questions in this section, each with a number of ports, Write your answers
in the space provided below each part. If you nee
8002A SEMESTER 1 2009 PAGE 2.8 0F 30
Extended Answer Section
There arethree questions in this section, each with a number ofperts. Write your answers
in the space provided below each part. If you ne
The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises and Solutions for Week 3
2014
Assumed Knowledge: Integrals of simple functions such as xn (including 1/x), sin x,
The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises and Solutions for Week 4
2014
Assumed Knowledge: Sketching curves of simple functions. Integrals of simple functi
Attached are questions selected from the MATH1002 examinations for 2005, 2006 and
2007, together with brief solutions. Question 3 in the 2005 and 2006 examinations has
been omitted, since it involves
The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises and Solutions for Week 5
2014
Assumed Knowledge: Sketching curves of simple functions. Integrals of simple functi
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Summer School, 2012 Page 4 of 26
1. [14 marks total]
{a} [2 marks] Find the general solution ydr) of the diffmonrial equation
11; + 9y 2 U. Summer School, 2012 Page 5 01 26
{h} [3 marks] Find a pa