Mathematics 1E2: Lecture 1
Prof. Sergei Fedotov
Prof. Sergei Fedotov
Mathematics 1E2: Lecture 1
1 / 12
Useful information
Where and when to find me...
Office: Alan Turing Building, room 2.141
Office hours: Friday 1.30-3.00
Email: [email protected]
Assessment
Online tests: 4 occurrences in semester, 20% of final mark
Final exam: 80% of final mark
Resources on Blackboard
Lecture notes and example sheets
Slides after each lecture
Recording of the lectures
Prof. Sergei Fedotov
Mathematics 1E2: Lecture 1
2 / 12
Structure of the course
Program for first 11 lectures of the semester
Lectures 1-4: Integration, revision and applications
Lectures 4-6: Sequences and series
Lectures 6-7: Taylor series
Lectures 7-9: Functions of more than one variable
Lectures 9-11: Line integrals & multiple integrals
Let’s start...
Prof. Sergei Fedotov
Mathematics 1E2: Lecture 1
3 / 12
Indefinite integrals
Definition
“The indefinite integral can be thought of as the reverse operation to
differentiation”
Given a function f (x), its indefinite integral F (x) is denoted
Z x
Z
F (x) =
f (t) dt or F (x) = f (x) dx,
and satisfies
dF
(x) = f (x)
dx
Indefinite integrals are defined up to a constant C !
x5
+C
Ex 1: If f (x) = x4 then F (x) =
5
Ex 2: If f (x) = cos(x) then F (x) = sin(x)+C
Prof. Sergei Fedotov
Mathematics 1E2: Lecture 1
4 / 12
Some usual (i.e. to be known!) indefinite integrals
Z
f (x)
x−1
xn+1
+C
n+1
ln (|x|) + C
ex
ex + C
e−x
−e−x + C
xn (n 6= −1)
sin (x)
cos (x)
1
1 + x2
1
√
1 − x2
Prof. Sergei Fedotov
f (x) dx
− cos (x) + C
sin (x) + C
tan−1 (x) + C
sin−1 (x) + C
Mathematics 1E2: Lecture 1
5 / 12
Domain of definition of a function
Definition
“The domain of definition Df of a function f (x) consists of all the points
of R where the function is well defined”
Ex 1: f (x) = x2 is well defined everywhere, so Df = R = (−∞, ∞)
Ex 2: f (x) = 1/(x + 1) is well defined everywhere but at x = −1, in
this case Df = R\{−1} = (−∞, −1) ∪ (−1, ∞)
p
Ex 3: f (x) = 1/ 1 − x2 (On board)
Prof. Sergei Fedotov
Mathematics 1E2: Lecture 1
6 / 12
Definite integral
View 1: using the indefinite integral
For a given function f (x) defined on a ≤ x ≤ b and with indefinite integral
F (x), the definite integral of f (x) between a and b is defined by
Z
a
b
f (x) dx = F (b) − F (a) = [F (x)]ba
View 2: using the area under a curve
For a function f (x), the definite integral of f (x) between a and b can be
regarded as the area A under the curve y = f (x) from x = a to x = b.
y
y = f (x)
A=
x=a
Prof. Sergei Fedotov
Z b
f (x) dx
a
x
x=b
Mathematics 1E2: Lecture 1
7 / 12
Some examples
Ex 1: Calculate
Z
2
4x3 dx. (On board)
1
Ex 2: At time t = 0, a particle starts moving in straight line with a
velocity v(t) = 2t. The position of the particle is given by x(t), and
we know that x(0) = 0. Where is the particle at t = 10? (On board)
Ex 3: Calculate the area bounded by the x-axis and the curve
y = 2 + sin(x) for 0 ≤ x ≤ 2π. (On board)
Prof. Sergei Fedotov
Mathematics 1E2: Lecture 1
8 / 12
Computing an integral with rectangles... 1/2
y
y
y = f (x)
N =4
N =7
y = f (x)
f (ξ1)
f (ξ1)
=
dx
xN
x1 ξ1 x2 ξ2 x3
xN
=
ξ2
=
=
a
ξ1
x
x
x2
x1
b
a
dx
b
Split [a, b] with N points, N − 1 intervals, dx = (b − a)/(N − 1)
Choose intermediate points ξk such that xk ≤ ξk ≤ xk+1
Construct rectangle: height f (ξk ), width dx, area Ak = f (ξk )dx
Prof. Sergei Fedotov
Mathematics 1E2: Lecture 1
9 / 12
Computing an integral with rectangles... 2/2
y
y = f (x)
N =7
f (ξ1)
A1
A2
AN −1
x
xN
=
=
x1 ξ1 x2 ξ2 x3
a
Riemann (1826-1866)
dx
b
Approximate total area A ≈ A1 + ... + AN −1
Z b
N
−1
X
f (ξk )dx
Let N become large:
f (x) dx = A = lim
a
N →∞
k=1
Riemann sum
Prof. Sergei Fedotov
Mathematics 1E2: Lecture 1
10 / 12
Before next time...
Make sure you are familiar with
Integration by parts (p7 of the notes)
Method of Substitution (p8 of the notes)
Practice by reading through
Harder Integration Techniques (p10-11 of the notes)
Examples 1 (p44 of the notes): Start looking at 1(a), 1(b), 1(c)
Prof. Sergei Fedotov
Mathematics 1E2: Lecture 1
11 / 12