The definite and indefinite integrals
Definition of definite integral
f (x)dx = lim
f (xi )x .
is the definition of the definite integral of f (x) between the lower limit x = a and the upper limit
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
been given by dfdx
f (x + x) f (x)
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
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
1 i2 = 1
The obvious p
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
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CULTURAL DIFFERENCES CAUSE PROBLEMS
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PHAS1245Problem Solving Tutorial 2, Solutions
1. (a) We have y = x2 + 2 + x2 and
= 2x 3 .
(b) We have y = x 2 + x 2 and
1 1 1 3
x 2 x 2
(c) We have y = (x 2 + 1)(x 2 1)1 and
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
cos x dx =
(1 + cos 2x) dx =
x + sin 2x + c .
The other is to use integration by parts :
u = cos x,
= sin x,
= cos x
v = sin x
PHAS1245Problem Sheet 8, Solutions
e2iz = e2i(x+iy) =
Re(e ) =
e2y ei 2x
e2y (cos 2x + i sin 2x)
e2y cos 2x .
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
d 2 x
x e = 2xex + x2 ex = xex (x + 2) .
2. (a) Since the function defined only for x > 0, f (x) = 0 is only possible if
ln x = 0 x = 1 .
= ln x + x = ln x + 1 = 0
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
. Apply this test in each case.
[(3x + 2)y] = 3x + 2,
[(x + 1)x] = 2x + 1, unequal not exact.
PHAS1245Problem Sheet 4, Solutions
(sin(cos x) = cos(cos x) sin x
= sin x cos(cos x)
y = xcos x ln y =
ln xcos x
cos x ln x
sin x ln x
cos x cos x
sin x ln x
xn ex dx =
xn1 x ex dx
PHAS1245Problem Sheet 6, Solutions
1. Since a = 3i + 5j 7k and b = 2i + 7j + k,
a + b = 5i + 12j 6k
a b = i 2j 8k ,
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