4
4.1
Integration
The definite and indefinite integrals
Definition of definite integral
The expression:
Z
b
f (x)dx = lim
Iab =
x0
a
x=b
X
f (xi )x .
x=a
is the definition of the definite integral of f (x) between the lower limit x = a and the upper limit
5
5.1
Partial Differentiation
The partial derivative [see Riley et al, Sec. 5.1]
So far we have considered functions of a single variable ie f = f (x) and the slope or gradient at x have
(x)
been given by dfdx
where
f (x + x) f (x)
df (x)
= lim
.
x0
dx
x
3
3.1
Dierentiation
Definitions
Definition of limit
Consider the function f (x). If we can make f (x) as near as we want to a given number l by choosing x suciently near to a number a, then l is said to be the limit of f (x) as x a and it is written
as
li
2
2.1
Complex Numbers
The imaginary number i [see Riley 3.1, 3.3]
Complex numbers are a generalisation of real numbers. they occur in many branches of mathematics and
have numerous applications in physics.
The imaginary number is
i=
1 i2 = 1
The obvious p
1
1.1
Vectors
Scalars and vectors [Riley 7.1]
Scalars: These are the simplest kind of physical quantity that can be completely specified by its magnitude,
a single number together with the units in which they are measured. Examples include temperature, ti
1
Williams/Working Folder One/Career Analysis Essay
Audience: Academic
Purpose: To Explore and Inform
Genre: Expository
Format: Times New Roman, 12pt font, MLA, Double Spaced, Suggested page length is 3 pages
Research and write your own Job Description of
Running head: CULTURAL DIFFERENCES CAUSE PROBLEMS
Cultural Differences Cause Problems
Name:
Course: World Literature
Institution:
1
CULTURAL DIFFERENCES CAUSE PROBLEMS
2
I agree with the above idea to some extent, given that a mans cultural habits, educat
PHAS1245Problem Solving Tutorial 2, Solutions
1. (a) We have y = x2 + 2 + x2 and
dy
2
= 2x 3 .
dx
x
1
1
(b) We have y = x 2 + x 2 and
dy
1 1 1 3
=
x 2 x 2
dx
2
2
1
1
= 1
.
2 x
x
1
1
(c) We have y = (x 2 + 1)(x 2 1)1 and
1
1
dy
1 1 1
1 1
=
x 2 (x 2 1)1 +
PHAS1245Problem Sheet 7, Solutions
1. The condition for the two lines to be parallel is that the vectors b1 and b2 are
parallel, i.e. b1 b2 = 0 .
To find the distance between them : If P and Q are the points where a line
segment perpendicular to the two l
PHAS1245Problem Sheet 3, Solutions
1. One way is to write cos2 x = (1 + cos 2x)/2 and then
Z
1Z
1
1
cos x dx =
(1 + cos 2x) dx =
x + sin 2x + c .
2
2
2
2
The other is to use integration by parts :
u = cos x,
du
= sin x,
dx
I=
Z
2
dv
= cos x
dx
v = sin x
c
PHAS1245Problem Sheet 8, Solutions
1.
e2iz = e2i(x+iy) =
=
=
2iz
Re(e ) =
e2ix2y
e2y ei 2x
e2y (cos 2x + i sin 2x)
e2y cos 2x .
2. The
student should plot an Argand
diagram and sketch 1 + i 3 on it. Then
r = 1 + 3 = 2 and arctan( 3/ 1) = /3 + , so the a
PHAS1245Problem Sheet 1, Solutions
1. We have a diagonal of 27 inches, a width of 4x and a height of 3x, where x needs
to be determined so as to find the values in inches. From Pythagoras
(4x)2 + (3x)2 = 272 25x2 = 272 x = 5.4
and therefore the width is 2
PHAS1245Problem Sheet 2, Solutions
1.
d 2 x
x e = 2xex + x2 ex = xex (x + 2) .
dx
2. (a) Since the function defined only for x > 0, f (x) = 0 is only possible if
ln x = 0 x = 1 .
(b)
1
df
= ln x + x = ln x + 1 = 0
dx
x
ln x = 1
x = e1 .
To determine whe
PHAS1245Problem Sheet 5, Solutions
1. If df = A dx + B dy, then a necessary and sufficient condition that df is exact
is that A
= B
. Apply this test in each case.
y
x
(a)
(b)
(c)
(d)
[(3x + 2)y] = 3x + 2,
y
[(x + 1)x] = 2x + 1, unequal not exact.
x
(y ta
PHAS1245Problem Sheet 4, Solutions
1. (a)
d
(sin(cos x) = cos(cos x) sin x
dx
= sin x cos(cos x)
(b)
y = xcos x ln y =
=
1 dy
=
y dx
dy
=
dx
ln xcos x
cos x ln x
cos x
sin x ln x
x
cos x cos x
x
sin x ln x
x
2.
I=
Z
2
xn ex dx =
0
Z
2
xn1 x ex dx
0
u
PHAS1245Problem Sheet 6, Solutions
1. Since a = 3i + 5j 7k and b = 2i + 7j + k,
a + b = 5i + 12j 6k
|a| =
83
|b| =
a b = i 2j 8k ,
54
a b = 34
cos = 0.508 .
2. The vector product of two vectors gives a vector perpendicular to both i.e.
c = (i + j k) (2i j