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
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
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
A Quick Tour of Game
Theory
Paul Dutting
1
Non-cooperative game theory
Players motivated by individual incentives
Interactions resulting in payoffs
Explains:
Selfish but collectively damaging behavior
How to think strategically
More than one possible
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
MA107: Quantitative Methods (Mathematics) 201415
Answers for Extra Exercises 6: Stationary points and Lagrangeans
Extra Exercise 1. The managers expect 60 days of good weather with a profit of 2 y y on these
days, and 40 days of poor weather with a profit
MA107: Quantitative Methods (Mathematics) 201415
Answers for Extra Exercises 7: Consumer choice
1/4 1/2
Extra Exercise 1. The utility function for two goods is u(x1 , x2 ) = x1 x2 and the budget
constraint is p1 x1 + p2 x2 = M . We are asked to find the L
MA107: Quantitative Methods (Mathematics) 201415
Exercises 4: Optimisation
For all homeworks in this course please note that no marks will be given for answers only. Always
show the method used to obtain your answers.
Please write clearly. Dont just use f
MA107: Quantitative Methods (Mathematics) 201415
Answers for Exercises 1: Sets, functions, graphs, equations
Exercise 1. This should be standard. Note that R2+ is the non-negative quadrant which is the set
of points (x, y) where x 0 and y 0. The answers a