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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 need more space there are extra pages at the
end of the ex
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
of the limit of a sequence. Sketching a curve given a f
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 vector which emanates from A and
terminates at B and has len
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 ja2 jxn jan
:
Xjb
For what nonnegative values of a1 , a2
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 /, b D fx .x0 ; y0 /, and c D fy .x0 ; y0/.
5.4 THE CHAI
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
x
so
f 0 .x0 / D lim
x!x0
n
X1
xn
n 1
x0 X n
x
x0
k 1 k
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, and linear transformations, which are integral parts of
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 /.xj
f .cj /.xj
j D1
xj
xj
1/ C
D f C g ;
n
X
1/
g.cj /.
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 sequence in .C a; b; k k/.
(c) Show that .C a; b; k k/ i
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; y; / D 4 x sin y cos 5 ; X0 D .1; =2; /
w
x sin
If F
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 need more space there are extra pages at the
end of the e
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, cos x,
ex . The simple properties of denite integrals.
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 functions
such as xn (including 1/x), sin x, cos x, ex .
Obje
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 elementary matrices that were not covered in 2007 and
2
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 functions
such as xn (including 1/x), sin x, cos x, ex .
Obje
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 . We call the values of the function the terms of the
sequ