The Exponential Function
and the Natural Logarithm.
Alternative development via infinitesimals
This Material will not be Assessed
in the HW/Test/Exam
Look at this point
x
The exponential function coincides
with the tangent line y=x+1 near x=0
Infinitesima

MA104 Practice 3
Course Lecturer: Dr Dan Brawn
Be sure to check integration results by differentiating back.
1. (a) Find
Z
xenx dx
where n is any non zero integer. Then use this result to find
Z
e3 sin x cos x sin x dx
(b) Integrate:
Z
x2 (ln x3 ) dx
Z
ln

Differentiation
Straight lines (Linear maths)
" y a bx"
x=0
X=a+h
For any h we have
Height of line = height at 0
+ h Tan()
Straight line
f(a)
x=a
X=a+h
For any h we have
line(h) line(a) h.gradient
Compare with the familiar
" y a bx"
The Derivative of a fu

Law of cosines
From Wikipedia, the free encyclopedia
Fig 1. A triangle
In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) is
a statement about a general triangle which relates the lengths of its sides to the cosine of
on

Example MA104 Exam question on Complex Numbers
(a) Let z = 1 + 4i and w = 2 + 3i, where i2 = 1.
(i) Express 3z w2 in the form a + bi where a and b are integers.
[20%]
(ii) Calculate |zw 1|, giving your answer in an exact form.
[20%]
(iii) Show that for an

MA104 Practice 2
Course Lecturer: Dr Dan Brawn
1. Calculate, from first principles, keeping the algebra as simple and direct as
possible, the derivative of
(a) f (x) = x4
2. (a) Using a well-known trigonometric identity and the rules of logarithms,
show t

Week 1 MA104, extra thoughts (not assessed)
The Pythagoras proof problem
A, B C, X and Y are areas. A, B and C are squares. Look at the shape C+Y, do you see two
other Similar shapes ? If area Y is k times area C then area X is k times area B , what is k