MA212
CALCULUS
NOTES
Adapted from the 2015 course pack and lecture slides
Prepared by Luke Milsom
MARCH 2015
Chapter 1
Pre-integration
1.1
Limits
In this section we give an informal overview of what a limit is and what it means
to say that a function f (x
MA212: Further Mathematical Methods (Calculus) 201415
Exercises 2: Approximate behaviour and convergence
For these and all other exercises on this course, you must show all your working.
1.
For x > 0, let f (x) = x ln 1 +
1
.
x
Evaluate f (x) for some lar
MA212: Further Mathematical Methods (Calculus) 201415
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
MA212: Further Mathematical Methods (Calculus) 201415
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) 201415
Exercises 3: Taylor series, more limits and the denition 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 expansio
MA212: Further Mathematical Methods (Calculus) 201415
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) 201415
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
j=0
f
MA212: Further Mathematical Methods (Calculus) 201415
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) 201415
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 dened
Further Mathematical Methods (Linear Algebra)
Solutions For The 2001 Examination
Question 1
(a) For a non-empty subset W of V to be a subspace of V we require that, for all vectors x, y W
and all scalars R:
i. Closure under vector addition: x + y W .
ii.
MA200 Examination 2004.
Answers
Note: All questions are worth 25pts. The detailed marking scheme is indicative only.
1 (a) [11pts]
The region R is indicated in the diagram below.
y
y=x
2
4
R
2
2
x
Its not possible to integrate the function with respect to
Further Mathematical Methods (Linear Algebra)
Solutions For The 2002 Examination
Question 1
(a) To be an inner product on the real vector space V , a function hx, yi which maps vectors x, y V
to R must be such that:
i. Positivity: hx, xi 0 and, hx, xi = 0
MA212
LINEAR ALGEBRA
NOTES
Adapted from the 2015 course pack and lecture slides
Prepared by Luke Milsom
MARCH 2015
Chapter 1
Background
1.1
Vector spaces
A vector space is a set V of vectors with two operations
1. u, v V , u + v V
2. v V and a R (for real
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Chapter 6
Integration
Reading There is no specic essential reading for this chapter. It is
essential that you do some reading, but the topics discussed in this
chapter are adequately covered in so many texts on the
applications of
calculusthat it would b
Chapter 8
Improper Integrals
Reading There is no specic essential reading for this chapter. It is
essential that you do some reading, but the topics discussed in this
chapter are adequately covered in so many texts on the
applications of
calculusthat it
Chapter 3
Approximate behaviour and
convergence
Reading There is no specic essential reading for this chapter. It is
essential that you do some reading, but the topics discussed in this
chapter are adequately covered in so many texts on the
applications
Chapter 9
Manipulation of Integrals
Reading There is no specic essential reading for this chapter. It is
essential that you do some reading, but the topics discussed in this
chapter are adequately covered in so many texts on the
applications of
calculust
Chapter 7
Double Integrals
Reading There is no specic essential reading for this chapter. It is
essential that you do some reading, but the topics discussed in this
chapter are adequately covered in so many texts on the
applications of
calculusthat it wo
Chapter 10
Manipulation of Improper
Integrals
Reading There is no specic essential reading for this chapter. It is
essential that you do some reading, but the topics discussed in this
chapter are adequately covered in so many texts on the
applications of
Chapter 11
Laplace Transforms
Reading There is no specic essential reading for this chapter. It is
essential that you do some reading, but the topics discussed in this
chapter are adequately covered in so many texts on the
applications of
calculusand in
Chapter 12
The Riemann-Stieltjes
Integral
Reading There is no specic essential reading for this chapter. It is
essential that you do some reading, but the topics discussed in this
chapter are adequately covered in so many texts on the
applications of
cal