Welcome to MA 212
Further Mathematical Methods
Jozef Skokan
Calculus
Office: COL.3.04
Arne Lokka
Adam Ostaszewski
Linear Algebra
Offices: COL.4.08/COL.4.06
Department of Mathematics
London School of Economics and Political Science
Lectures
Calculus: weeks
Theorem 10.2.1 Let be an eigenvalue of the matrix
A, with algebraic multiplicity m. Then, for some rm,
the matrix (A-I)r has a nullspace U of dimension m.
Moreover, for each u in U, Au is also in U
Let A be an n x n matrix with characteristic polynomial
(
MA212: Further Mathematical Methods (Calculus) 201617
Exercises 4: The fundamental theorem of calculus and double integrals
For these and all other exercises on this course, you must show all your working.
1.
Let f be a continuous function taking positive
MA212: Further Mathematical Methods (Calculus) 201617
Exercises 5: FTC (again), transformations and more double integrals
For these and all other exercises on this course, you must show all your working.
1.
The function, p(x, y), of two variables is defin
MA212: Further Mathematical Methods (Calculus) 201617
Solutions to Exercises 3
1. (a) This requires evaluating successive derivatives of f (x) = ln x at x = 1. We see that
f (n) (x) = (1)n+1
(n 1)!
,
xn
for n 1. This means that the Taylor series is
X
f (j
MA212: Further Mathematical Methods (Calculus) 201617
Solutions to Exercises 4
1. This question is mostly about the Fundamental Theorem of Calculus, which says that
Z x
d
f (z) dz = f (x),
dx z=c
for any constant c, provided f is continuous at x.
This que
MA212: Further Mathematical Methods (Calculus) 201617
Solutions to Exercises 1
I think these answers are an important part of the course: do have a look through them even if you
and your class teacher agree that your answers were perfect.
The answer given
MA212: Further Mathematical Methods (Calculus) 201617
Solutions to Exercises 5
1. The first part of this question, which tests your understanding of the idea that an integral is an
antiderivative, is more-or-less exactly Question 2 from Exercises 4.
2
Wha
MA212: Further Mathematical Methods (Calculus) 201617
Solutions to Exercises 2
1. Plugging in a few values should suggest to you that the limit is likely to be 1. Sure, the evidence
could equally support the suggestion that the limit is 0.999999999567, bu
MA212: Further Mathematical Methods (Calculus) 201617
Exercises 6: Improper integrals
For these and all other exercises on this course, you must show all your working.
1. Sketch a graph of f (x) = sin2 ( x) for x 0. (Or use Maple.)
Z
f (x) dx converges.
MA212: Further Mathematical Methods (Calculus) 201617
Exercises 7: More improper integrals and the manipulation of proper integrals
For these and all other exercises on this course, you must show all your working.
1.
Determine whether each of the followin
MA212: Further Mathematical Methods (Calculus) 201617
Exercises 1: Assumed background
The lectures do not cover practical techniques for integration of functions of a single variable as
students on this course are supposed to be skilled in this already. H
1 Revision of Integration
Compute the following inde nite integrals
1. Quickies:
Z
Z
Z p
2 + ln x
x
cos x
p
dx;
dx;
dx;
x
(1 + x2 )2
1 + sin x
Z
Z
Z
sin x
sin2 x cos xdx;
tan xdx =
dx;
cos x
Z 0
Z
f (x)
f 0 (x)
p
dx;
dx
f (x)
f (x)
Note these are all of t
MA212: Further Mathematical Methods (Calculus) 201617
Exercises 2: Approximate behaviour and convergence
For these and all other exercises on this course, you must show all your working.
1
1. For x > 0, let f (x) = x ln 1 +
.
x
Evaluate f (x) for some lar
MA212: Further Mathematical Methods (Calculus) 201617
Exercises 3: Taylor series, more limits and the definition of the Riemann integral
For these and all other exercises on this course, you must show all your working.
1.
(a) Find the Taylor series expans
Theorem 20.3.1: For every function f in V, we
have |f|2 = 2k-(-,)|ak(f)|2 Moreover, for
each t in (-,), we have f(t)=k-(-,)ak(f)ekit
Complete Approximation
Fourier Series
Theorem 20.2.1: If f is a bounded continuous real-valued
function dened on (-,), the
Theorem 18.1.2 If A is an n x m matrix and both
Ax and Ay are double sided generalised
inverses of A such that N(Ax) = N(Ay) and R(Ax)
= R(Ay), then Ax = Ay
Theorem 18.1.1 If A is an n x m matrix of rank k,
and Ag is a double-sided generalised inverse of
Then d=dt and:
In particular, taking (t)=t we get
(t)=1 and so:
Let f(t) be a bounded function
dened on [a,b]
1) Smoothness Rule: Suppose is
a smooth function on [a,b]
(derivative exists and is
continuous).
1) Additivity over Intervals: If
a<c<b then: Pro
Homogeneous the derivative of
each function is in the linear span of
itself and other functions
Homogeneous Linear Dierential
Equations
Application to Differential
Equations
The Method
Which can be written as y=Ay
must solve
An Example with Jordan Normal
Theorem 14.1.1: If a vector space
V is equal to U1+U2 for subspace
U1 and U2 then dim(V) +
dim(U1nU2) = dim(U1)+dim(U2)
Theorem 14.1.2: If U1 and U2 are
Denition: For U a subspace of an
inner product space V, the
orthogonal complement U is the
set
cfw_v i