study pack ma100 inside all (1)

7/23/2019 Study Pack MA100 Inside All (1)

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MA100 Mathematical Methods

General Information for 2013/14

Calculus Lecturer: Dr. Ioannis Kouletsis office: COL 2.05C

Linear Algebra Lecturer: Dr Michele Harvey office: COL 4.15

Mathematics Departmental Office: COL 4.01

Course Materials

Study Pack

That is what you are holding now, and formed of two parts: Directed Reading and Notes,and the Exercise Sets. All this material, and more, including copies of the lecture slides,will be made available on the MA100 Moodle page as well.

The lectures will broadly follow the material as presented in the notes.

Course Texts

All students are expected to have a copy of these books. The notes contain detailedreferences to the relevant parts of the textbooks.

Calculus: K. Binmore and J. Davies, Calculus, Concepts and Methods.

Linear Algebra: M. Anthony and M. Harvey, Linear Algebra, Concepts and Methods.

Reference Texts (for additional reading and exercises; find the texts in the library):

H. Anton, Elementary Linear Algebra(or Applications version by Anton and Rorres);

S. L. Salas and E. Hille, Calculus, One and Several Variables;

Schaum Outline Series: Mathematics for Economists, Linear Algebra, Advanced Calculus,Differential and Integral Calculus.

Teaching Arrangements and What You Need To Do!

Go to the lectures.There is one Calculus lecture (Tuesday) and one Linear Algebra lecture (Friday) each week.Links to the video recordings of the lectures will be published on the MA100 Moodle page,but these videos are not a substitute for attending the lectures.

Read the texts.Each week you should do the reading as directed in the study pack for that lecture, inconjunction with any notes you have taken.

Work all exercises to the best of your ability before each class.There is one set of exercises for you to complete each week. Start working the questionsas soon as the material is covered in a lecture. (You can start on questions 1, 2, 3, 9of Exercise Set 1 immediately after the first lecture.) Starred and un-starred questionsare equally important. (The only difference is that starred ones are marked by your classteacher.) Also look at the reading and its connection to the exercises.

There are answers (but not full solutions) to the un-starred questions on the Moodle page soyou can check your work before class. Questions appearing with the symbol are optionalextras and will not be discussed in class unless time permits.


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Solutions to all questions will be put on the MA100 Moodle page after the Exercise Sethas been discussed in the classes. Use these to check your work each week, as soon as yoursolutions have been returned to you. Note that often other solutions may be possible.

Go to all classes.

Class attendance is obligatory. Each week complete the Exercise Set for that week. Prepareand hand in solutions of the starred questions to your class teacher, either before or atthe nextweeks class depending on your class teachers instructions. Your solutions will bemarked and returned to you at the following class.

The classes will be used to discuss the exercises and any questions you may have on thecourse material.

See an MA100 class teacher or lecturer in their office hours as soon as you havedifficulty doing or understanding any part of the material: lectures, reading or exercises.

You can see any MA100 class teacher or lecturer during their Office Hours. Office hour times

can be found via the MA100 Moodle page and on the Mathematics Department website:www2.lse.ac.uk/maths/Courses/Office Hours.aspx.

Attend Extra Example Sessions (optional).The Extra Examples Sessions are optional interactive sessions designed to reinforce thematerial covered in the lectures. There will be no new material presentedin these sessions.They will be held each week (starting Week 1): Thursday, 11.00-12.00, in the Old Theatre.

Maple.

Maple is a computer algebra program which will be used throughout the course for

visualising graphs and solutions to certain exercises, and to enhance your understandingof the material.

Before the end of Week 3, work through theMapletutorial found on the MA100 Moodlepage. IfMapledoesnt start immediately when clicking on the file name, save the file, andopen it inside Maple.

Classes during Week 5 (only) will be held in one of the computer rooms (the room will belisted on your LSEforYou timetable), where you will use Maple to work parts of ExerciseSet 4 using the Maple Instructions in your Study Pack.

A few Words of Advice

Each weeks material builds on the previous weeks work. Success in this course depends onworking the exercises and understanding the material covered week by week.

Do not fall behind!

If after the classes you still have questions or difficulties with any of the material,

see a class teacher or one of the lecturers immediately!


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MA100 Mathematical Methods

Background You Should Know and Exercises

This is background material for MA100 and is intended to be a review of your A-level

Mathematics course. Please work through it before term or in your spare time. (You donot need and should not use a calculator.) If you have difficulties with anything, dont

panic but do make a note of any background you are missing and work on it, or see one

of the lecturers.

It is also recommended that you look through the review booklets, An Algebra Refresher

and A Calculus Refresher. These can be found at: www.maths.lse.ac.uk/Refreshers/.

References are made to these booklets (AlgRand CalcR) for corresponding material in the

exercises that follow.

Solutions to the exercises can be found on the MA100 Moodle page at the start of term.

1. The set of real numbers, denoted R, includes the following subsets:

N, the set of natural numbers: 1, 2, 3, 4,. . .

Z, the set of integers: . . . ,3,2,1, 0, 1, 2, 3, . . .Q, the set of rational numbers: p/q with p, q Z, q= 0; such as, 2

5,9

2, 41

= 4.

the set of irrational numbers: real numbers which are not rational; for example,

2, .

The absolute value (modulus) of a real number:

|a

|=

a ifa0

a ifa

0

.

The absolute value satisfies |a|= a2, and the Triangle Inequality: |a+b| |a| + |b|.Intervals of real numbers, such as: (a, b) ={x|a < x < b}, [a, b] ={x|axb},

(, b) ={x|x < b} [a, ) ={x|ax}.

2. Polynomials of degree n: Pn(x) =a0+a1x+a2x2 +. . .+anx

n,

where the ai are real constants, an= 0, and x is a real variable.

How to graph: straight lines, P1(x) =a0+a1xquadratics, P2(x) =a0+a1x+a2x2

cubics, P3(x) =a0+a1x+a2x2 +a3x

3.

The laws of indices: xrxs =xr+s (xr)s =xrs

How to expand expressions like (1 +x)n, (x+y)n

How to factorize quadratics (such as x2 5x+ 6 and 3x2 + 14x 5)and simple cubics, (such as x3 1 and x3 2x2 5x+ 6).

The Quadratic Formula.

How to sum: arithmetic series, a+ (a+d) +...+ (a+ (n 1)d)geometric series, a+ar+ar2 +...+arn1


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Exercises. (AlgR, sections 23, 810)

(2.1) Sketch f(x) = 4 x2 and g(x) = 2x+ 1 and find their points of intersection.

(2.2) Expand (2 + 5x)4 and

x2 4/x

3

.

(2.3) Find an expression for the profit function (q) as a function ofqgiven that

(q) =pq C(q), where 2p+ 0.4q= 155 and C(q) =q2 10q+ 300 .

(2.4) Find the complete solution of each of the following systems of equations, both

algebraically (by solving the equations simultaneously) and graphically (by illustrating

them as lines on a graph).

(a)

x+ 2y= 42x

y= 3

(b)

x+ 2y = 42x+ 4y = 4

(c)

x+ 2y= 42x+ 4y= 8

Note. (x, y) is a solution of a system of equations if it satisfies all equations simultaneously.

(2.5) Factorise P(x) =x3 7x+ 6 and sketch the graph ofP(x).

3. Be able to manipulate algebraic expressions accurately and efficiently.

Be able to manipulate indices.

Exercises. (AlgR, sections 17)

(3.1) Simplify (a) 6ab ab

(b2 4bc) (b) 49x2

35y 4xy

2

(2xy)3

(3.2) Solve for s: 5

3s+ 1 2

s+ 1 = 0 .

(3.3) Rewrite the simultaneous equations

x=a c by

2b y=

a c f bx2b

as a system of linear equations in xand y. Solve the system for x and y (in terms of theconstantsa, b, c, f), and then show that

x+y = 1

3b(2a 2c f) .

4. Properties of the exponential (ex) and logarithmic (ln x = logex) functions and their

graphs:ln ex =x

eln y =y

(er)s =ers

er+s =eresln xr =r ln x

ln(xy) = ln x+ ln y


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Exercises.

(4.1) Sketch the graphs of ex, ex, ln x .

(4.2) What is the value of e0, ln 0, ln1 ?

(4.3) Show that: ln

ab

= ln a ln b (a >0, b >0).

(4.4) Simplify: ln x5 2 ln x+ ln y3 (x >0, y >0).For which values ofx and y is this expression positive?

5. Properties of the trigonometric functions (sin, tan,etc.) using radian measure of an angle:

Be able to sketch their graphs.

Know their values at multiples of, 2

, 4

, 3

, 6

.

(Know the triangles associated with these last values.)

Know the basic trigonometric identities, e.g.

sin2 x+ cos2 x= 1

sin(a+b) = sin a cos b+ cos a sin b ; cos(a+b) = cos a cos b sin a sin b ;sin x= sin(x) is an odd function; cos x= cos(x) is an even function.

Exercises.

(5.1) Sketch the graphs of sin x, cos x, 2 c o s 3x, tan x .

(5.2) Deduce the formulas for sin 2x, cos2x from the identities above, and deduce that

sec2x= tan2x+ 1 ; tan 2x= 2tan x

1 tan2x;

sin x cos y=1

2[sin(x+y) + sin(x y)] .

(5.3) Evaluate (without using a calculator): sec 3

, cot(6

), sin(54

), cos(114

).

6. Rules of differentiation: Product, quotient and chain rules

Be able to differentiate polynomials.

Know the derivatives of the exponential, logarithmic and trigonometric functions.

Exercises. (CalcR, sections 19)

(6.1) Calculate

(a) d

dx e3x

(b)

d

dxln(5 2x) (c) d

dx e3x ln(5 2x)

(d)

d

dxln

1

x2

(e) d

dx

xex

2x2 + 1

(f)

d

dx

3x2 1 (g) d

dxtan3x (h)

d

dxsin(x2 5)


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7. Know that the equation of the tangent line to the curve y = f(x) at the point where

x= a is given by:

y f(a) =f(x)(x a)

Exercise.(7.1) Find the equation of the tangent line to the curve y = 2x3 9x2 38x+ 21

at the point where x= 1.

8. Find the stationary (turning) points of graphs of functions and determine which point is

a maximum and which is a minimum.

Exercises.

(8.1) Do this for the function y = 2x3 9x2 38x+ 21.(8.2) Sketch the graph off(x) =x3 3x2.

Find the maximum and the minimum value offon the interval [0, 3],

and on the interval [0, 4].

9. Techniques of integration: partial fractions, integration by parts, substitution.

Exercise. (CalcR, sections 10-17; AlgR, section 11)(9.1) Calculate

(a)

sin3x dx (b)

4

0

sec2 x dx (c)

1

x2 5x+ 6 dx

(d)

x

x 1dx (e)

x cos2x dx (f)

x2ex dx

(g)

x(x2 2)4 dx (h) (2x+ 2)ex2+2x+3 dx

10. The technique ofCompleting the Squarein a quadratic expression.

Exercise. (AlgR, section 9)

(10.1) Complete the squares in the expressions: x2 + 6x+ 11 and 2x2 4x+ 7.

(10.2) Complete the square in ax2

+bx+c.Use this to derive the quadratic formula to solve ax2 +bx+c= 0 for x.


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MA 100Mathematical Methods

Calculus

Directed Reading and Notes

Textbook

Ken Binmore and Joan Davies, Calculus, Concepts and Methods

Cambridge University Press, 2001. ISBN 978-0-521-77541-0

All students are expected to have a copy of this book.The notes for this part of MA 100 consist mainly of detailed references to the

relevant parts of the textbook. Reading those parts is essential for understanding

the material in the course.

2013/14

c LSE 2013


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Course Content CALCULUS

1 Vectors and general geometric concepts

Lines, planes, hyperplanes; their different types of equations. Flats. Curves and conics.

2 Scalar valued functions

One-variable functions f : R R. Constrained and unconstrained optimisation.

Inverse and local inverse functions.

Two-variable functions f : R2 R. Partial derivatives, tangent planes, gradients,

derivatives, and directional derivatives. Examples including linear and quadric sur-

faces.

Functions of several variables f : Rn R. Gradients and derivatives, tangent hyper-

planes. Taylor approximation. Convex and concave functions.

The Chain Rule for differentiation.

3 Stationary points and optimisation of functions of several variables

Calculation and classification of stationary points, including using the quadratic term

in the Taylor series expansion of functions. Local and global extrema.

Lagranges method for constrained optimisation.

4 Vector-valued functions

Functions f : Rn Rm. Derivatives and tangent flats.

Inverse functions and their derivatives. Local inverses and critical points.

5 Summation and integration

Methods for summation of series.

Techniques of integration.

6 Differential and difference equations

First-order differential equations. Solution methods for differential equations of the

following types : separable, exact, linear, homogeneous. Use of substitution.

Linear differential and difference equations with constant coefficients. Nonhomoge-

neous linear differential and difference equations.

General and particular solutions. Asymptotic behaviour of solutions.

7 Applications

Applications of all the topics above.

MA 100 Mathematical Methods Calculus


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Sectionandp

agenumberreferencesrefertothecalcu

lustextbook

.

Readeachsectioncarefully,

workingthroughtheexam

ples.

Tobegin

,you

shouldlookthroughthefirstsectionsofChapter2:

Intervals(page51)

Realvalued

functionsofonerealvariable(pages515

2)

Someeleme

ntaryfunctions(pages52-5

5)

Combinatio

nsoffunctions(pages55-56)

(Ifyouhavea

nydifficultywiththereading,a

ttendofficehours.)

MA100Mathe

maticalMethods

Calculus

Lec

ture

1

Ve

ctorsan

dLines

1.1

VectorsinR

2

ForthiswefollowSection1.3ofthetextbook

,untilline8ofpage19.

1.2

VectorsinR

3


.

1.3

Lines


.

1.4

Optionalextraexercisesrelatedtothesetopics

VectorsinR

2:1.4

.1,1.4.2

(exceptpart(vi)),1.4.3and1.4.4

;

VectorsinR

3:1.8

.7and1.8

.9;

Lines:1.8

.1,1.8.2,

1.8

.4,1.8.5,1.8.6,1.8.1

1,1.8.12

,1.8.1

3,1.8

.14and1.8.15

.

(Bysomethinglike1

.4.1

,w

emeanSection1.4,Question1,o

fthetextbook

.)

MA100MathematicalMethods

Calculus


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Lec

ture

2

Planes

Curvesan

dCon

ics

2.1

Planes


,until

line12ofpage31

.Youcanskip

Example8on

pages29-30

.

2.2

Curvesandconics

Acurve

isthe

setofpoints(x,y)R

2satisfyinganequation

P(x,y)=

0.Theequation

P(x,y)=

0is

saidtorepresentthecurve.

EXAMPLES

Alineisrepre

sentedbytheequationax+

by+

c=0

,forsomerealnumbersa,b,c.

Thecirclewit

hcentre(0,0)andradius1isthesetof

pointssatisfyingx2+

y2

1=

0

(whichweus

uallywriteasx2+

y2=

1).

Forconics

havealookatthepicturesinSection2.14o

fthetextbook

.

Theequation

x2+

ay=

brepresentsaparabola

.

Theequation

x2 a2+

y2 b2=

1representsanellipse.

Anellipsewitha=

bcanberewrittenasx2+

y2=

a2,

whichistheequationofacircle

.

Anhyperbol

aisrepresentedbytheequation

x2 a2

y2 b2=

1.

Hyperbolashavetwo

asymptotes(sometimesalsocalledslantasymptotes

);seepage80ofthetextbook

.

2.3


Planes:1.8.16

,1.8.1

7,1.8.18

,1.8.1

9,1.8

.21,1

.8.2

2,1.8

.23,1.8.24

,1.8.2

5and1.8

.26;

Curvesandconics:2.15.7and2

.15.10

.

MA100Mathe

maticalMethods

Calculus

Lec

ture

3

Funct

ionso

fOne

Var

iable

Der

ivat

ives

Tay

lor

Ser

ies

Stationary

Po

intsan

dOp

tim

isat

ion

3.1

Intervals

ReadSection2.1oft

hetextbookyourself.

3.2

Functionsofoneva

riable


.

3.3

Derivatives

ForthiswefollowSections2.7,2.8and2

.10ofthetextbook.

Therulesofdifferentiation

onpage67areveryimportant.

3.4

Higherorderderiva

tives


.

3.5

Taylorseriesofone

-variablefunctions


.

3.6

Stationarypointsandoptimisationofone-variablefunctio

ns


.

3.7


Derivatives:2.11.1,2

.11.2,2.11

.9,2.1

1.11

,2.1

1.13and2

.11.15;

TaylorSeriesofone

-variablefunctions:2

.15.1,2.15

.2,2.1

5.4and2.15

.5;

Stationarypointsandoptimisationofone-variablefunctio

ns:4.5.4and4.5

.5.

MA100MathematicalMethods

Calculus


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Lec

ture

4

Globalan

dConstrained

Op

tim

isat

ion

Convexan

d

Concave

Funct

ions

Functionso

fTwo

Var

iables

4.1

Globalandconstrainedoptimisationofone-variablefunctions

Forthiswefo

llowSections4.2and4.3ofthetextbook

.

4.2

Convexandc

oncavefunctions

Forthiswefo

llowpage126and127ofSection4

.1ofthetextbook

.

4.3

Functionsof

twovariables

Forthiswefo

llowSection3

.1ofthetextbook

.

4.4


Globalandconstrainedoptimisationofone-variablefunctions:

4.5.1,4.5.2,4.5.6,4.5.8,4

.5.1

0,4.5.11and4.5.12;

Convexandc

oncavefunctions:4.5.17;

Functionsof

twovariables:3.7.1.

MA100Mathe

maticalMethods

Calculus

Lec

ture

5

Par

tial

Der

ivat

ives

Tangen

tPlanes

Homogen

eous

Funct

ions

5.1

Partialderivativesa

ndtangentplanes

ForthiswefollowSections3.2and3.3ofthetextbook

.

5.2

Homogeneousfunc

tionsandEulersformula

Afunction

f:R

2

R

iscalledhomogeneousofdegreeni

fforallx,yand

:

f(

x,

y)=

nf(x,y)

.

EXAMPLE

Thefunction

f(x,y)

=

x2+

2xyishomogeneousofdeg

ree2,since

f(

x,

y)=

(

x)2+

2(

x)(y)=

2x2+

22xy=

2(x2+

2xy)=

2f(x,y)

.

ECONOMICINTERPR

ETATION

Ifxandyarethough

tofas

inputsandf(x,y)astheoutput

,thenahomogeneousfunction

ofdegree1hasconstantreturnstoscale:ifwemultiplyallinputsby

,thentheoutput

ismultipliedby

.

Iff(x,y)ishomogeneousofdegreen>

1,then

fhasincreasingreturnstoscale

.If

f(x,y)ishomogeneousofdegreen

0:

thenforallu

=0small:f(

+u

)>

f()

,

so

isa

localminimum;

ifforallu

=0wehaveu

Tf()u

0

andfx

x

0andfx

x=

0.Butitseasytoconvince

yourselfthatiffx

x=

0,then=

0or

0;

,

ifA

0;

,

ifC x^2 + 2*y^2

The arrow is typed as the hyphen - followed by the greater than sign, >. After

youve type ^2, use the right-arrow key to make sure you go back to the baseline

when typing the +, otherwise you get x2+2y2

instead ofx2 + 2y2.

Take a minute to understand the code you have just typed. You have assigned (:=)

the name f to afunctionwhich takes a point (x, y) andmaps it (->) to x2+ 2y2.

An asterisk * is useful to indicate multiplication. (Maple knows that 2x is the

same as 2*x. But you must type a*b to multiply a and b, as Maplethinks that ab

indicates a variable with name ab.)

Now press the return key. Maplewill respond by showing that is has accepted your

definition of the function f:

(x, y) x2 + 2y2

M 1


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2. To plot the function as a surface in 3-dimensional space, type and enter the following

input. Maple interprets f(x, y) as the function expression x2 + 2y2, and plots the

graph of the surface z= x2 + 2y2.

plot3d( f(x,y), x=-5..5, y=-5..5 )

(Maplewill ignore spaces in the input; they have been put in so that you will notice

the precise way to type the commands.)

LOOK very carefully at the menu and tool bars at the top of your window. Now

CLICK on the picture of the surface with your mouse. A frame appears around the

surface and the lower toolbar changes into the plot toolbar.

PLACE your cursor (which will look like an arrow) on the surface and hold down the

left mouse button. MOVE the mouse slowly and OBSERVE what happens.

Now experiment by clicking on the different icons on the plot toolbar. Which view

shows the level curves on the surface? Which view allows you to see vertical intersec-

tions of the surface?

The plot commands can also be used with a function expression. So you could as well

have entered

plot3d( x^2 + 2*y^2, x=-5..5, y=-5..5 )

Since you need to graph several functions, the function notation is more efficient. (It

will become clear why as you continue.)

3. Before you can plot the contours in the xy plane, you need to instruct Mapleto load

the plots package, which will give us extra commands.

Place your cursor in the next Maple input line, type and enter

with(plots)

You will see a list of additional commands which are now available. The one you are

interested in is the contourplot command. To find out for yourself how to use any of

these commands, highlight the name with your mouse and then click on Help in the

menubar. The box that appears will offer help on that command as the fifth option

from the top.

M 2


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To graph some level curves f(x, y) =c in the xy plane for the function f(x, y), use

contourplot( f(x,y), x=-5..5, y=-5..5 )

To graph a specific set of level curves, say at c= 0, 1, 2, 3, use

contourplot( f(x,y), x=-5..5, y=-5..5, contours=[0,1,2,3] )

(Explain why you see only 3 contours.)

Note the use of square brackets [0, 1, 2, 3] to denote a list in Maple.

The curves in the last graph are not as smooth as the first set. If you want a smoother

set, you have to instruct Maple to plot more points. Place your cursor in the input

line that you have just typed, just after the final square bracket of the list, and addthe option as shown below (on the same line, and dont forget the comma).

contourplot( f(x,y), x=-5..5, y=-5..5,

contours=[0,1,2,3], numpoints=2000 )

Is the scaling on the two axes different, distorting the shape of the curves? To fix this,

either click on the graph and then on the [1:1] icon in the toolbar, or add a scaling

option to your command line:

contourplot( f(x,y), x=-5..5, y=-5..5,contours=[0,1,2,3],

numpoints=2000, scaling=constrained )

Experiment: CLICK on the graph and look at the effects of clicking on different icons

in the plot bar. Experiment how many points you wantMapleto compute to obtain

sufficiently smooth curves without wasting computational time and space.

4. To graph the vertical intersection with the xz plane (when y = 0) and with the

yz plane (when x= 0), use the following plot commands:

plot( f(x,0), x=-5..5)

plot( f(0,y), y=-5..5)

To compare the graphs, you might want to see them on the same set of axes, and

specify the colour of each graph.

plot( [f(t,0),f(0,t)], t=-5..5, color=[red,blue] )

Note the square brackets since you are now asking Mapleto plot a list of functions.

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You are now ready to plot the graphs, contours and vertical intersections of the surfaces

in Question 2. SCROLL back to the beginning. Your Maple Worksheet should

contain the following commands:

f:= (x,y)-> x^2 + 2*y^2 (function definition)

plot3d(f(x,y), x=-5..5, y=-5..5 ) (graph of surface)

contourplot(f(x,y), x=-5..5, y=-5..5 ) (general contours)

contourplot(f(x,y), x=-5..5, y=-5..5, contours=[0,1,2,3],

numpoints=2000, scaling=constrained ) (specific contours)

plot(f(x,0), x=-5..5 ) (xz intersection, y= 0)

plot(f(0,y), y=-5..5 ) (yz intersection, x= 0)

Instead of re-typing these lines for each surface (or using cut and paste), CHANGE the

definition of the function f and then RE-EVALUATE the input lines. For example

to do part (b), place your cursor in the input line defining fand change it to

f:= (x,y)-> sqrt(x^2 + 4*y^2)

When you press return, Maplewill replace the previous definition of the function f

with this new one.

Now place your cursor anywhere in the plot3d line and press return. Maple will

respond by graphing the new surface z=f(x, y).

EXPLORE this new surface as before. Rotate it to view it from different angles. Click

on the icons to view the surface with level curves and a set of axes through the origin.

CONTINUE to re-evaluate the other input lines. COMPARE the contours on the

surface with the graph of the general contourplot. ALTER the list in the specific

contours plot to match the set of contours required in the question.

Do the above for all functions in Question 2.

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Question 1

First define the cost function:

C := x -> x^3 - 24*x^2 + 117*x + 784

Define the marginal cost function (using the differential operator D which maps the

cost function into its derivative function),

mc := D(C)

Note Maples feedback.

Finally define the average cost function,

ac := x->C(x)/x

Each time that you enter an input line, verify that Maplehas the correct definition.

To graph the functions on the same axes use a list,

plot( [C(x), mc(x), ac(x)], x=0.5..20 )

ENLARGE the graph: Click on it and a frame appears. Position the mouse over the

bottom right corner on the frame until it turns into a double arrow. Hold down the

left mouse button as you drag the frame down and right to enlarge it.

You can alter the input to specify the colours yourself, so that you can tell which graph

corresponds to which function, as you did for the vertical intersections of Question 2.

Note that the lowest x value graphed was 0.5 and not 0. What happens if you change

the lower x limit to 0? Why?

Were you able to find the minimum of the average cost function, the point at which

the marginal cost and the average cost are equal? If not, try

solve( mc(x)=ac(x) ,x )

Or look for the minimum value ofac(x):

solve( diff(ac(x),x)=0, x )

Note that Maplereturns the real and the complex roots.

(Note the use ofdiff for the derivative of an expression. For more on this, see the

final section More Maple.)

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Question 3

You should be able to plot this function without any difficulty:

plot( x^2-cos(x), x=-3..3 )

Linear Algebra

To learn how to use Mapleto solve systems of linear equations, Ax= b, we will work

examples from Question 5 of Exercise Set 2. The equations are

(i)

x1+ x2+x3 = 2

2x2+x3 = 0

x1+ x2 x3 = 4

(ii)

x1+ x2+ 2x3 = 2

2x2+x3 = 0

x1+x2 x3 = 0

(iii)

x1+ x2+ 2x3= 2

2x2+x3= 0

x1+x2 x3= 2

First you need to load the linear algebra package,

with(LinearAlgebra)

Then input the vector b and the coefficient matrix Afor (i):

b := Vector([2,0,-4])

A := Matrix( 3, 3, [1,1,1, 0,2,1, -1,1,-1] )

The first two numbers indicate the size of the matrix (3 3). Determine yourself how

the list of elements must be entered.

Obtain the augmented matrix and assign it the name Ab, then find the reduced row

echelon form:

Ab :=

ReducedRowEchelonForm(Ab)

from which you easily can obtain the solution.

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The solution x can also be found directly by

x := LinearSolve(A,b)

You can now modify the Mapledefinitions ofAand b to work parts (ii) and (iii).

DO this to see how Maple handles systems which are inconsistent or which have

infinitely many solutions.

Questions 4, 5, 6

The inverse of a matrix is found as follows:

MatrixInverse(A)

Try to find out what happens if you ask for the inverse of a matrix which is not

invertible.

Matrix multiplication is denoted in Mapleby a dot. Once you have as input a matrix

assigned the name A and a vector assigned the name b, the product matrix Ab is

given by

A.b

Use this method to find x= A1band to verify that Ax= b.

For Question 6 you need to create a 3 3 identity matrix; assign it the name Id.

Id := Matrix( 3, 3, [1,0,0, 0,1,0, 0,0,1] )

INPUT the consumption matrix C and the demand vector d, and then solve the

system:

LinearSolve(Id-C,d)

More Maple

Whatever question you ask in Maple, if you intend to use the result again, assign a

name to the commandline, using :=. For instance

sola := LinearSolve(A,b); A.sola - b

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functions vs. expressions

To understand the difference, enter and OBSERVE the output of the following lines:

f := x -> x^3 + 3*x (a function)

g := x^3 + 3*x (an expression)

f(1); g(1)

(The semi-colon is there to separate the two questions, otherwise Maple thinks you

want to see the product off(1) and g(1).)

If f is defined as afunction, then f(x) is the correspondingexpression.

f(x); g

diff( f(x),x ); diff( g,x )

Use diff( expression, x ) to differentiate anexpressionwith respect to the vari-

able x; its output is an expression.

To have the derivative as a function, you need to use the differential operator D; it

is applied to a functionand its output is a function.

D(f)

Maplehas excellent Help facilities. You can search for topics. If you know the com-

mand or topic, but want to know more how to use it, type a question mark followed

by the topic,

?plot, color

or highlight a word with your mouse, click on Help in the menubar, and click on the

fifth option Help on . . . .

IfMapledoes something unexpected or gives error messages, it is often because it is

using old definitions you have made. For instance, in the above part on Linear Algebra

you used x to denote the solution ofAx= b. This gives surprises if you later try to

do something with a completely different x; say want to solve x2 +x= 0.

To clear all definitions, assignments, etc., enter

restart

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Exercise Set 5

1. For each of the three functions below, compute

z

x and

z

y . Show that all the functionsare homogeneous, and verify Eulers formula in each case.

(a) z =x3 +y3

x+y(b) z=

1

5x1/5 +

4

5y1/5

5

(c) ex/y sinx

y+ey/x cos

y

x

2. *(i) Find the equation of the tangent plane to the surface

z = f(x, y) = 4x2 +y2,

at each of the points (1, 1) and (1,1).

Find also a normal vector to the surface at each of these points.

At what points, if any, is the tangent plane to the surface z =f(x, y) horizontal?

(The next part of this question will be covered in the next calculus lecture, but you should

be able to answer it before then by thinking carefully about what is being asked.)

Find and sketch the contour 4x2 +y2 = f(1, 1) = f(1,1), together with the tangent

line and a normal vector to the contour at each of the points (1 , 1) and (1,1).

(ii) Do all of the above (part (i)) for the function z = g(x, y) = x2 y2, and the points(2, 1) and (2,1).

3. Determine whether the following permutations are even or odd, and find the total number

of inversions: (i) (2 4 3 1) (ii) (1 3 4 5 2).

4. Evaluate the following determinants using the cofactor expansion along an appropriate

row or column.

(i)

2 6 11 0 53 1 4

(ii)

7 5 2 32 0 0 0

11 2 0 023 57 1 1

(iii)

1 2 1 03 2 1 00 1 6 50 1 1 1

(iv)

0 1 0 01 0 0 00 0 1 00 0 0 1

(v)

0 0 0 0 0 10 0 0 0 3 20 0 0 2 9 30 0 1 0 7 40 6 9 8 7 51 3 4 2 9 6

(vi)

3 t 21 5 32 1 1

*For what values oft is the determinant (vi) equal to zero?


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* 5. Use Mapleto plot the functions f(x) and g(x) in question 2 of Exercise Set 3 about the

= respective points x = 3 and x = 4, together with the Taylor approximations you found.

Plot the functions in red and the Taylor approximations in blue. Hand in a printout of

the graphs (you can colour it by hand).

6. Let a=

a1a2

a3

, b=

b1b2

b3

; e1=

10

0

, e2=

01

0

, e3=

00

1

The vector product or cross product, a b of the two vectors is the vectordefined by

a b=

e1 e2 e3a1 a2 a3b1 b2 b3

= (a2b3 a3b2)e1 (a1b3 a3b1)e2+ (a1b2 a2b1)e3

The vector a b is perpendicular to both a and b (see part (b)).

(a) Calculate w= u v for the vectors u=

12

3

and v=

25

4

.

Check that w is perpendicular to both uand v

(b) Show that for general vectors a, b, c R3 the scalar triple product, a, b c

is given by

a, b c=a1 a2 a3b1 b2 b3c1 c2 c3

()Use this and properties of the determinant to show that the vector bc is perpendicular

to both band c.

(c) Show that the vectors a, b, c are coplanar (lie in the same plane)

the determinant () is equal to zero.

Obtain the constant t if the vectors

31

2

,

t

5

1

,

23

1

are coplanar.

Objectives.

(i) Master partial differentiation

(ii) Be able to check whether a function is homogeneous, know Eulers theorem.

(iii) Draw contours of surfaces, tangent lines and normals.

Find equations of tangent planes, contours and tangent lines, find normal vectors.

(iv) Evaluate a determinant by cofactor expansion.

Make sure you have done the reading for this week.


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Exercise Set 6

* 1. Roughly sketch a contour map of the function h: R2 R defined by

h(x, y) = 4000 0.001x2

0.004y2

which represents the surface of a mountain. Suppose a mountain climber is at the point

(500, 300, 3390). In what direction should he move in order to ascend at the greatest rate?

What is this rate of ascent?

There is a path through the point in the direction (3, 4)T. Find its rate of ascent.

For each case, sketch a vector along the path of the climber.

2. A firm producing two goods x and y has the profit function

(x, y) = 32x 6x2 + 8xy+ 16y 3y2 20.

Find the level of output for each of the two goods for which x(x, y) =y(x, y) = 0.

* 3. The production function P(k, l) = 3k1/3l2/3 represents the output of a firm if it uses k

units of capital and l units of labour. Sketch some contours P(k, l) =c.

At present, k= 8 and l= 1. Find a normal vector to the contour P(k, l) = 6 at the point

(8, 1). At what rate would production increase if the firm increases k and l at the same

rate? In what ratio should it increasek and l in order to increase production most rapidly?

Find Pfor a general Cobb-Douglas function, P(k.l) =Akl (for constants A, , ).

4. Evaluate the following determinants (use row operations to simplify the calculation):

*(a)

1 4 3 22 7 5 11 2 6 02 10 14 4

(b)

1 4 1 3 01 7 4 3 82 8 2 6 02 0 5 5 71 9 0 9 2

(c)

3 3a 3a2

2 2b 2b2

1 c c2

5. For which values of is the matrix A=

2 3

2 1

not invertible?

* 6. A is 3 3, det(A) = 5. Find det(3A), det(A2), det(2A1), det((2A)1)).

7. Use the method of the adjoint matrix to find the inverse of

A= 2 0 30 3 11 4 2

B= 1 0 2

2 1 30 1 1

C= 1 2 0

0 1 12 1 1


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8. Use Cramers rule to find the value of x, y, z for system (a) and to find the value of z

for system (b) (wherea, b are constants, a =b, a = 2b).

(a)

x+y+z = 8

2x+y z= 3

x+ 2y+z = 3

(b)

ax ay+bz = a+b

bx by+az = 0

ax+ 2by+ 3z = a b

9. A portfolio is a row vector Y = ( y1 . . . ym) in which yi is the number of units

of asset i held by the investor. After a year, say, the value of the assets will increase

(or decrease) by a certain percentage. The change in each asset depends on states the

economy will assume, predicted by economists as a returns matrix, R = (rij) where

rij is the factor by which investment i changes in one year if state j occurs.

Suppose an investor has assets in y1= land, y2= bonds and y3= stocks, and that

R=

1.05 0.95 1.01.05 1.05 1.05

1.20 1.26 1.23

Then the totalvalues of the portfolio in one years time are given by YR where (YR)jis the total value of the portfolio if state j occurs.

(a) Find the total values of the portfolio W = ( 5000 2000 0 ) in one year for each of

the possible states.

(b) Show that U= ( 600 8000 1000 ) is a riskless portfolio, that is, it has the samevalue in all states j.

An arbitrage portfolio Y= ( y1 . . . ym) is one which costs nothing (y1+ +ym= 0),

cannot lose ((YR)j 0), and in at least one state makes a profit ((YR)j >0 some j).

(c) Show that Z= (1000 2000 1000 ) is an arbitrage portfolio. (The bond asset of

2000 indicates that this sum was borrowed from the bank).

Objectives.

(i) Understand the idea of the gradient and the derivative of a function of two variables,and be able to evaluate these.

(ii) Understand the concept ofdirectional derivativefor a two variable function, know its

formula,f, u, and the direction and value of maximum increase/decrease.

(iii) Understand the effect of row operations on a determinant and use them to simplify

its evaluation.

(iv) Use |A| to determine ifAis invertible or what solutions Ax= b might have.

(v) FindA1 by cofactor method and use Cramers rule for solving systems ofn equations

in n unknowns.



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Exercise Set 7

* 1. Let f(x,y ,z) = 3x2 + 2y2 z2. Find f

x,

f

y and

f

z.

(a) What is the tangent hyperplane to the hypersurface u= 3x2 + 2y2 z2

where x= y = z = 1?

(b) Find the equation of the plane that is tangent to the surface

3x2 + 2y2 z2 = 13 at the point (2,1,1)

Write down its normal vector.(Have a look at this surface on Maple. It is called a hyperboloid of one sheet.)

* 2. Find all the stationary points of the functions, f : R2 R defined by:

(a) f(x, y) =xy2 +x2y xy

(b) f(x, y) =x3 +y3 3xy 8

3. Find all the stationary points of the function, f : R3 R given by

f(x,y ,z) = ln

(x3 2y3 3z3 6xy+ 4)1

2

* 4. Which of the following are subspaces ofR3?

S1=

xy

z

x+y+z = 0

S2=

xy

z

x2 +y2 +z2 = 1

S3=

xy

z

x= 0

S4=

xy

z

xy= 0

S5=

xy

z

x= 0 and y= 0

=

00

z

z R

Provide proofs or counterexamples to justify your answers. Describe the sets geometrically.

5. IfAis an m nmatrix, show that the null space, N(A), is a subspace ofRn.


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* 6. Express each of the vectors below as a linear combination of u=

21

1

and v=

11

3

if possible:

(a) 324

(b)

000

(c)

757

7. (a) Solve the matrix equation to find the coefficients and by finding A1:

25

=

12

+

11

=

1 12 1

=Ax

(b) Show that lin{w1, w2} = lin

12

,

11

= R2

. That is, show any vector b R2

can be expressed as a linear combination ofw1 and w2 by solving b= Ax for x:

b1b2

=

12

+

11

=

1 12 1

(c) Show, in general, that ifv and w are non-zero vectors in R2, v=

a

c

, w=

b

d

lin{v, w} = R2 v =tw for any t R a bc d = 0

Objectives

(i) Extend concepts of three dimensional vectors to n dimensions.

(ii) Understand concept of a function of three or more variables; Find its

derivative, tangent hyperplanes to the hypersurface and contour, gradient and directionalderivatives. Find tangent plane to a contourof the graph of a 3-variable function.

(iii) Find stationary points of functions of two or more variables. (Systematically solve the

equations simultaneously; factorise expressions where possible and delete non-zero factors

before solving.)

(iv) Understand concept of a linear subspace: Decide whether a set is a subspace and proveit

is or use a counterexample to show that it isnt.

(v) Express a vector as a linear combination of given vectors, or show that it is not possible.



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Exercise Set 8

1. The function f : R3 R2 with component functions f1 and f2 is defined by

u= f1(x,y,z) =x2 +y2 +z2; v= f2(x,y,z) =x y.

Find all the points x = (x,y,z)T such thatf(x) = (8, 0)T, and describe the curve consisting

of these points. = Use Mapleto verify your answer.

* 2. Let A=

3 01 1

), c=

10

).

The affine function f : R2 R2 is defined by f(x) =Ax + c.

Write down the equations of two hyperplanes whose intersection is the graph off.

3. A function f : R R2 with component functions f1 and f2 is defined by

(a)x= f1(t) = 2t

y=f2(t) = 4t2

*(b)

x= f1(t) = sin t

y = f2(t) = ln t

For the function in (a) find the set of (x, y) which correspond to the set of all real values of

t. Find the derivative att= 1, then write down the equation of the tangent line at t= 1.

For the function in (b) find the set of (x, y) which correspond to the set of all positive realvalues oft. Find the derivative at t = /2, and the equation of the tangent line at t = /2.

4. Find the derivatives at the given point of the functions defined below. Write down the

equations for the corresponding tangent flats in each case.

(a) f : R2 R2 with component functions f1 and f2 defined by

u= f1(x, y) =xy

v= f2(x, y) =x y

at (x, y) = (1, 1).

*(b) f : R2 R3 with component functions f1, f2 and f3 defined by

u= f1(x, y) =x

v=f2(x, y) =x+y

w= f3(x, y) =xy

at (x, y) = (1, 1).

(c) f : R3 R2 with component functionsf1 and f2 defined by

u= f1(x,y,z) =xy

v= f2(x,y,z) =yz

at (x,y,z) = (1, 1, 1).


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* 5. Show that the setS1spans R3, but any vectorv R3 can be written as a linear combination

of the vectors in S1 in infinitely many ways. Show thatS2 and S3 do not span R3.

S1=

12

3

10

1

01

1

11

0

S2=

10

1

21

3

12

9

S3=

11

1

20

1

6. Determine which of the following sets of vectors are linearly independent.

L1=

12

)13

) L2=

12

)13

)25

) L3=

00

)12

)

L4=

12

0

27

0

35

0

L5=

121

3

201

2

111

2

L6=

1213

2012

4418

L7=

1213

2012

4418

1112

* 7. Determine which of the following sets of vectors spans R3, and which of the sets is linearly

independent.

S4=

12

3

214

101

S5= 12

5

214

101

* 8. Let A be any matrix. Let v1 and v2 be 2 non-zero vectors and suppose that Av1 = 2v1and Av2= 5v2. Prove that {v1, v2} is linearly independent.

(Hint: Assume 1v1+2v2 = 0; multiply this equation through by A to get a second

equation for v1 and v2; then solve the two equations simultaneously).

Can you generalize this result?

Objectives.

(i) Understand concept of avector-valued function, f : Rn Rm, find derivatives and

tangent flats. Know that a tangent flat of a function f : R Rm is a line.

(ii) Understand the concept of an affine function.

(iii) Determine whether a set {v1, , vr} is linearly independentor spansa given space.

Understand these concepts in terms of linear combinations of the vectors.



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Exercise Set 9

1. The length and width of a rectangle decrease at the rate of 2 cm per minute and 3 cm per

minute respectively. When the length is 6m and the width is 3m how fast are the followingchanging:

(a) the area and *(b) the diagonal?

2. Suppose that z= f(x, y) is a function of two variables x and y, and that y depends on x

viathe function y=g(x). Write down an expression for the derivative dz

dx .

Set z=xy4 + 3y2 + ln y and y=x2 + 3x+ 10. Find

dz

dx .

(Your answer can involve y.)

*3. Find the second derivative of the real-valued function f : R2 R defined by

f(x, y) =ex sin y

Write down the quadratic Taylor approximation at the point (x, y) = (1, 0).

4.Recall that a function

f is

homogeneous of degreenif

f(x, y) =nf(x, y).

Fix x and y and think offas a function of. Use the Chain Rule to differentiate the above

equation with respect to . Put = 1 to obtain Eulers Theorem:

xfx(x, y) +yfy(x, y) =nf(x, y).

*5. For each of the sets of vectors given below, find a basis of the vector space Lin(Si) and state

its dimension. Describe any propersubspaces ofR2 or R3 geometrically (give Cartesian

equations for any lines or planes).

S1=

12

23

S2=

11

00

223

3

S3=

10

1

21

3

12

9

S4=

121

3

20

1

2

441

8

111

2


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*6. Which of the following sets are a basis for R3? (State reasons for your answers.)

S1=

123

210

410

721

S2=

101

111

S3=

21

1

121

33

0

S4=

21

1

121

33

1

*7. Find the coordinates of the vector

121

with respect to each of the bases for R3:

B1=

10

0

01

0

00

1

B2=

11

1

11

0

233

8. Find a basis for each of the following subspaces ofR3.

(a) The plane x 2y+z = 0 (b) The yz plane

9. Ifv1,v2 Rn, explain the difference between the sets A= {v1,v2} and B= Lin{v1,v2}.

Objectives.

(i) Know howchain ruleformula applies to functions of several variables and use it to find

derivatives of composite functions.

(ii) Find the Taylor approximation at a point for a function of several variables.(iii) Show that a set of vectors is a basisof a vector space V Rn.(iv) Find a basis of any subspaceV ofRn.

(v) Know that any vector inV is a unique linear combination of the basis vectors, and be

able to find the coordinates of any vector in the basis.



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Exercise Set 10

1. The total cost of producing x units of a product and y units of another product is given by

the functionC(x, y) = 5x2 + 7y2 2xy46x86y+ 600

Find the production level that minimizes this cost and the corresponding cost ofproduction.

*2. For each of the following functions, find all the stationary points and classify them as localmaxima, local minima, or saddle points.

(a) f(x, y) = 1y3 3yx2 3y2 3x2 (b) f(x, y) = 4xyx4 y4

(c) f(x, y) =ex sin y (d) f(x, y) = 4x2ey 2x4 e4y

=

Use Mapleto plot the contours of the functions near their stationary points.

Print out the contours graph of (a) and mark the stationary points in red.


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3. Find the general solution of each of the following systems of equations Ax=b in the formx= p +1s1+ . . .+rsr where p is a particular solution of the system and {s1, . . . , sr} is abasis for the null space ofA.

*(a) x1+ 2x2+ 3x4+ x5= 4, x3+ 2x4+ x6= 1, x7= 2

*(b)

x1+ x2+x3+ x4 = 42x1+ x3x4 = 2

2x2+ x3+ 3x4 = 6(c)

x1+ 2x2 x3 x4 = 3x1x22x3 x4 = 1

2x1+ x2 x3 = 3

*4. For each of the following systems of equations Ax= b, determine the set of all vectors bforwhich the system is consistent. In each case the set will be a subspace ofR3; describe what

that subspace is.

(b)

x1+x2+ x3+x4 = b12x1+x3x4 = b2

2x2+x3+ 3x4 = b3

(c)

x1+ 2x2x3x4 = b1x1x22x3x4 = b2

2x1+ x2x3 = b3

5. (a) Convince yourself that F(R) is a vector space. Which of these sets are subspaces:

S1={f F(R)| f(0) = 1} S2={f F(R)| f(1) = 0}

(b) Show S3= {f C1(R)| f f= 0} is a subspace.

6. Which of the following subsets are subspaces of M2(R)?

W1 =

a 00 b

|a, b R

W2 =

a 11 b

|a, b R

W3 =

a2 00 b2

|a, b R

Objectives.(i) Classify stationary points of functions of two or more variables.

(ii) Recognise the subspaces associated with a system of equationsAx= b:the null space ofA, N(A) ={x| Ax = 0} Rn,the range ofA, R(A) ={Ax| x Rn}= {b| Ax = b is consistent }= C S(A) Rm.

(iii) Know Ax= b consistent general solution is x= p +1s1+2s2+ +kskwhere Ap= b and {s1, s2 . . . , sk} is a basis of the null space ofA.

(iv) Understand functions can be treated as vectors, e.g. C(R) is a vector space.

Make sure you have done the readingfor this week.


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Exercise Set 11

*1. Show that the function

f(x,y,z) =x2 + 5y2 + z2 + 2xy+ Ayz

is convex if the positive constant A is less than 4, but not convex ifA >4.

Identify one stationary point off, and show that this is the global minimum off ifA 4, evaluate f(t,t, 2t) for any t, and deduce that fhas no global minimum.

2. (a) Find the production levels x and y which minimize the cost function C : R2 R

defined by

C(x, y) = 4x2

+ 4y2

2xy 40x 140y+ 1800

for a firm producing two goods x and y.

(b) Find the production levels x and y which minimize this cost function subject to each

of the following production requirements.

(i) x + y 25 (ii) x + y 35

For the following threequestions consider the matrices

A=

1 0 11 1 10 1 0

B=

1 2 1 3 00 1 1 1 11 3 2 0 1

3. (a) Find a basis of the row space ofA,RS(A), a basis for the column space ofA, C S(A),

and find the rank ofA, (A).

(b) Show that the RS(A) and CS(A) are each planes in R3. Find Cartesian equations

for these planes and hence show that they are two different subspaces.

(c) Find the null space of A, N(A), and verify the dimension theorem. Show that the

basis vectors of the null space are orthogonal to the basis vectors of the row space ofA.


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*4. (a) Find a basis of the row space ofB.

(b) Find a basis of the column space ofB.

(c) Find a basis of the null space ofB.

(d) Find the rank ofB and verify the dimension theorem for matrices. Show that the basis

vectors of the null space are orthogonal to the basis vectors of the row space ofB.

*5. (a) Using the information you obtained in questions 3 and 4, and without solving the

system, determine for each of the matrices A and B above if the system of equations

Ax= b or Bx= b is consistent, for each of the given values ofb:

b1 =

112

b2 =

213

(b) If the system is consistent, then find the general solution in each case. Hence or

otherwise express b as a linear combination of the column vectors of the given matrices.

6. A matrix Ahas full column rank def the columns ofA are linearly independent.

IfAis an m k matrix with full column rank, show that:

(1) ATA is a symmetric k k matrix,

(2) ATA is invertible.

Then verify the above results for the matrix M=

1 00 2

1 1

.

Objectives.

(i) Optimise a several variable function constrained either within, or on the boundary of

a region.

(ii) Understand convex and concavefunction and investigate if a function is one of these.

Understand the connection with global optimisation.(iii) Know coefficients xi in b= x1v1+ + xnvn= components ofx in Ax= b,

where columns ofA are the vectors vi.

(iv) Know rank(A) = number of leading ones in RREF(A) = dim(RS(A)) = dim(CS(A)).

Know CS(A) =R(A) (Column space and range are the same subspace.)

(v) Find a basis ofRS(A), CS(A) and N(A) from RREF(A).

(vi) Understand the rank-nullity theorem: rank(A)+nullity(A) =n, where n is the

number of columns ofA, rank(A) = dim(R(A)) and nullity(A) = dim(N(A)).



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Exercise Set 12

*1. The Cobb-Douglas production function for a particular manufacturer is given by

P(x, y) = 100x3/5y2/5

where x represents the units of labour (at 150 per unit) and y represents the units of

capital (at 250 per unit).

Using the method of Lagrange multipliers,

(i) find the maximum production level if the total cost of labour and capital is limited to

100,000, and

(ii) find the minimum cost of producing 20,000 product units.

Roughly sketch appropriate contours P and Cto justify your use of Lagranges method

and to establish whether you have minimised or maximised the function.

In each case find and interpret the value of the Lagrangian multiplier, .

2. Plot z =

3i and w = 1 +i as points in the complex plane. Express them inexponential form and evaluate q=

(

3 i)6(1 +i)10

. Express your answer in the forma + ib.

*3. Write each of the following complex numbers in the form a+ib

ei

2 ei3

2 ei3

4 ei11

3 e1+i e1 e3+i 25 4ei

7

6

4. Show that the functions y(x) =eix and y(x) =eix each satisfy the differential equation

d2y

dx2 +y = 0.

*5. Find the roots w and w of the equation x2 4x+ 7 = 0For these values of w and w, find the real and imaginary parts of the following functions,

f(t) =ewt t R, g(t) =wt t Z+


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*6. The function y(t) can be written in the form

y(t) =Aet +Bet

where A and B are non-zero complex constants and = a+ib.

Show that y(t) can be written in the alternative forms

y(t) =eat(Aeibt +Beibt)

=eat(A cos bt+Bsin bt)where A= A+B and B =i(A B).How can A and B be chosen so thatA andB are real?ForA andB real, show that y(t) can be written as

y(t) =eat(A cos bt+Bsin bt) =C eat cos(bt )where C= A2 +B2 = 2AB and satisfies tan = (B/A).

7. Show that for any z C, the expressions ezt +ezt , t R and zt + zt , t Z+are both real.

8. (a) Find the four complex roots of the equation z4 =1. Illustrate the roots as pointsin the complex plane.

Find the roots of z3

=1 and illustrate them on another graph of the complex plane.Without actually solving the equations, illustrate the roots of

x5 =1 and x6 = 64as points in the complex plane.

(b) Use the roots you found in part (a) to write z4 + 1 as a product of quadratic factors

with real coefficients.

Objectives.(i) Sketch and use the contours of a function of several variables and its constraint to

define a Lagrangian function.

(ii) Deduce constrained maximum and minimum points from the Lagrangian function.

Interpret the value of the Lagrangian multiplier, .

(iii) Know: a complex number is z=a+ib where i2 =1.All roots of a polynomial with real coefficients exist as complex numbers.

(iv) Know Eulers formula: ei = cos +i sin .

(v) Change between forms of a complex number: a+ib= rei =r cos +ir sin .



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Exercise Set 13

1. For the following, you may use the laws of exponents, but do not assume any property of

the logarithm other than its definition as the inverse of the exponential function.

(a) Fora >0, show that logay = ln y/ ln a

(b) Find the derivative of f(x) =ax .

2. Use the chain rule to show that, for 0< x < , d

dx(arcsin(cos x)) =1 .

What can you deduce about h(x) = arcsin(cos x) for x in this range? Can you explain

this?

*3. Letp be the price per unit required to sell x units of a commodity. Verify that each of the

following demand functions:

(i) x= 85e6p and (ii) x= 90 2p

is invertible and sketch the graphs of the functions and their inverses. Find the revenue

functions and the outputs x and pricespwhich maximize them.

For (i) express the price elasticity of demand given by

= dx/xdp/p

= p/xdp/dx

as a function ofxand verify that ||= 1 when the revenue function is maximised.

If the average cost function for (ii) is

AC(x) =x2 8x+ 157 +2

x

find the output x which minimises marginalcosts.

4. Two linear transformations T and S : R2 R2 are defined by

T

xy

=

xy

and S

xy

=

yx

(a) Sketch the effects ofT andSon the standard basis, and hence on the unit square with

sides e1, e2. Describe T and S in words.

(b) Find the matrices for T and S.

(c) What are the linear transformations ST and S2T? First do this directly and describe

them in words, then check by multiplying matrices. IsST =T S?


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*5. T and Sare linear transformations with respective matrices:

AT =

12

12

12

12

AS=

1 00 1

(a) Describe T and S in words.

(b) Illustrate ST and T Susing the unit square. Then calculate their matrices to check

that ST =T S.

*6. v1=

10

1

, v2=

11

3

, v3=

00

1

w1=

11

1

, w2=

11

0

, w3=

01

1

(a) Show that {v1, v2, v3} is a basis ofR3

. Show that {w1, w2, w3} is a basis ofR3

.

(b) Write down the matrix AT of the linear transformation T given by

T(e1) =v1, T(e2) =v2, T(e3) =v3 where {e1, e2, e3} R3 is the standard basis.

ExpressT

xy

z

as a vector in R3 (in terms ofx, y and z).

(c) Write down the matrix AS of the linear transformation S given by

S(v1) = e1, S(v2) = e2, S(v3) = e3 (Hint: What is the relationship between S

and T?)

(d) Write down the matrix AR of the linear transformation R given by

R(e1) =w1, R(e2) =w2, R(e3) =w3.

(e) Is RSdefined? What does this linear transformation do to v1, v2 and v3? Find the

matrixARSand use it to check your answer.

Objectives.

(i) Understand concept of an inverse of a one variable function and know when it exists.(ii) Find a formula for an inverse function and its derivative. Find inverses of elementary

functions.

(iii) Know and apply the definition of a linear transformation.

(iv) Understand why and how a linear transformation T : Rn Rm can be given as

multiplication by a matrix AT. Columns ofATare the images of a basis ofRn.

(v) Recognise the order of composed mappings. ST is first T and then S.



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Exercise Set 14

1. = Use Maple to plot the function f: R R defined by y=x4 6x2 + 12 .

Find a formula for a local inverse to the function at each of the points x= 1, 1, 2. Sketch

the graph of each local inverse function.Calculate the derivative of the local inverse at x = 2 directly from your formula. Evaluate

it at the point y= 4 and verify that at this point

dx

dy =

dy

dx

1

Find any points at which no local inverse exists.

*2. Functions f : R2

R2

are defined by

(a)

u

v

=

2x yx + y

(b)

u

v

=

y x2

x + y

Find the derivatives of these functions. Find where the functions have a local

inverse, and find the derivative of each local inverse at points where it exists.

Find also a formula for each local inverse, when it exists, at each of the points (x, y) = (1, 1).

and (x, y) = ( 12, 0).

*3. For each of the following linear transformations, find a basis for the null space ofT, N(T),

and the range ofT, R(T). Verify the rank-nullity theorem (dimension theorem) in each

case. If any of the linear transformations are invertible, find the inverse, T1.

(a) T : R2 R3 by T

x

y

=

x + 2y0

0

(b) T : R3

R3

by Txyz

= x + y+ z

y+ zz

(c) T : R3 R3 by T(x) =

1 1 00 1 1

1 0 1

xyz

x=

xyz


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4. Is it possible to construct a linear transformation T : R3 R3 where

N(T) is the line with Cartesian equation x= y = z ,

and R(T) is the plane 2x + y z = 0?

If not, explain why. If yes, then write down a matrix for such a linear transformation.

5. F = D2 2D+ 1 is a linear operator on C(R), the vector space of functions on R

with continuous derivatives of all orders, where D is the differential operator, D(f) =f.

Find F(sinx) and F(3xex).

6. S is the set of sequences, yt : {0, 1, 2, 3, . . .} R. Convince yourself thatS is a vector

space. (Are the axioms satisfied? What is the 0sequence? What is the additive inverse?)

E is the shift operator, E(yt) =yt+1. Show that Eis a linear operator on S.

(a) Let yt be the sequence yt= (1

2)t. Then E(yt) =yt+1=zt, where zt= (

1

2)t+1.

Write out the first four terms of the sequences yt and zt to see the effect of E.

(b) Is 2E 1 also a linear operator on S? Justify your answer.

(c) The null space of the linear transformation 2E1 is therefore a subspace of S.

Show that yt = (1

2)t is in the null space by showing it is a solution of the difference

equation

2yt+1 yt= 0

Given that the null space has dimension 1, explain why anysolution can be written as

yt=(12)t for some R.

Objectives.

(i) Understand concept of a local inverse for a function at a point and find it.

(ii) Understand the concept of a local inverse of a vector-valued function of several variables

at a point. Be able to find critical points, and to find formulae for a local inverse and

its derivative when these exist.

(iii) Understandnull space (kernel) and range (image) of a linear transformation, and

their correspondence to null space and range of an associated matrix.(iv) Know therank-nullitytheorem (the dimension theorem) for linear transformations.



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Exercise Set 15

1.*(a) Find the effective annual rate of interest on 1000 at 8% compounded

(i) quarterly and (ii) continuously.

*(b) Determine the interest rate needed to have money double in 8 years when compounded

semiannually.

(c) Find the present value of 10,000 in 5 years (that is, find how much to invest now in

order to obtain 10,000 in 5 years time) at 6% interest, when interest is compounded

(i) annually and (ii) continuously.

*(d) The estimated value of a collection of antiques bought for investment is increasing

according to the formula

V= 325, 000 (1.95)t2/5

The discount rate under continuous compounding is 6.8%. How long should the collection

be held to maximize the present value?

2. *(a) A deposit ofP is made each month for t years in an account that is compounded

monthly at an interest rate r. Find the sum accumulated after t years.

(b) A deposit of50 is made each month in an account that is compounded monthly at

12% interest. Find the balance in the account after 10 years.

*3. In question 6 of exercise set 13, you showed that each of the sets is a basis ofR3:

B=

10

1

11

3

00

1

and B =

11

1

11

0

011

(a) Write down the transition matrix P from B coordinates to standard.

Write down the transition matrix Q from

B coordinates to standard.

(b) If v=

312

, fi n d [v]B

(c) Given [x]B = 21

3

B

find [x]B

(d) Write down the transition matrix fromB coordinates to B coordinates.(Noteyou will need to change coordinates fromB to standard to B).


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4. Change the basis in R2 by a rotation of the axes through an angle of

6 clockwise:

(a) Write down the matrix of the linear transformation which accomplishes this rotation.

(b) Write down the new basis vectors, v1 and v2 (which are the images ofe1 and e2).

Denote the new basis by B, B ={

v1,v2}

.

Write down the transition matrix P from B coordinates to standard coordinates.

(c) A curve Cis given in standard coordinates by the equation 3x2 + 2

3xy+ 5y2 = 6.

Find the equation of the curve in the new B coordinates, (X, Y).

(d) Hence sketch the curve in the x, y plane.

*5. Let M=

21

,

11

v=

12

T

x

y

=

7x 2yx + 8y

(a) Show that M is a basis of R2

. Write down the transition matrix from Mcoordinates to standard coordinates.

(b) Find [v]M, the M coordinates of the vector v.

On one graph, sketch the vector v and show how to obtain it geometrically:

(1) as a linear combination of the standard basis vectors and

(2) as a linear combination of the M basis vectors.

(c) Write down the matrix of the linear transformation T, T : R2 R2 with respect tothe standard basis. Call it A.

(d) Find the matrix of T in M coordinates. Call it B.

Describe geometrically the effect of the transformation T as a map from R2 R2.(e) Find the image of [v]M using B.

On the same graph as in part (b), use the M basis vectors (and the information you

obtained about the linear transformation in part (d)) to sketch T(v) as a linear combi-

nation of the M basis vectors.

Check your answer using standard coordinates.

Objectives.

(i) Know formulae for interest compounding and discounting and their uses.

(ii) Find the transition matrix P from coordinates with respect to a new basis B to

standard coordinates and use P to change coordinates. Know v= P[v]B.

(iii) Understand changing coordinates does not change the position of the vectorv.



80/92

Exercise Set 16

*1. Given the inverse demand function yd = 116 + 2x x2 and inverse supply function

ys = (x+ 2)2, where x represents quantity and yd and ys each represents price, find theconsumers surplus and the producers surplus.

2. Use partial fractions to find the integrals

(a)

dx

(1 x2)2 (b)

x2 dx

x2 +x 2

3. (i) Find

(a)

x3 x2 +x+ 1

(x2 + 1)2 dx (b)

cos8 x sin3 x dx

(ii) Find, using appropriate changes of variable

(a)

(x2 + 2)6x3 dx (b)

sinx cosx

(1 + sinx)3dx

4. Use integration by parts to evaluate

1

0

arcsinx dx

5. A firm has fixed costs of 125. Its marginal revenue function is R(x) = 170 0.4x, and its

marginal cost function isC(x) = 10+2x, wherex represents quantity. Find the revenue

function, R(x) and the cost function C(x). Find also the value ofx which maximises profit.

*6. Let A=

1 43 2

.

(a) Find the eigenvalues and the corresponding eigenvectors for A.

(b) Hence, find an invertible matrix P such that P1AP is diagonal.

(c) Calculate P1AP directly to check your answer.


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7. Diagonalize the matrix A=

7 21 8

.

*8. Let B= 3 1 2

5 3 51 1 2

.

(a) Find the characteristic equation of B. Find the eigenvalues and the corresponding

eigenvectors for B.

(b) Hence, find an invertible matrix P such that P1BP is diagonal, and check your

answer for P by multiplying out P1BP.

Objectives.

(i) Know the concepts of consumer and producer surplus and how to calculate them.

(ii) Know the method of partial fractions.

(iii) Convert integrals to standard form by techniques of completing the square,

substitution and integration by parts.

(iv) Know definition of an eigenvectorand an eigenvalueof a matrix.

Find eigenvalues and corresponding eigenvectors for a matrix.

(v) Diagonalizea matrix A: use eigenvalues and corresponding eigenvectors to create (if

possible) an invertible matrix P and a diagonal matrix D such that P1AP= D.

Know that eigenvectors and eigenvalues are in corresponding columns, and that the

eigenvectors form a basis ofRn.

(vi) Check eigenvectors and eigenvalues by computingAP.



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Exercise Set 17

1. Classify each of the differential equations below in one or more of the following categories:

(i) linear equation (ii) separable equation (iii) exact equation

Find the general solutions of the equations by any appropriate method.

*(a) (x2 + 6x) dy= y2 dx 12 dy

(b) 2xdy

dx= 2

x+y+ 2

*(c) (cos y+y cos x) dx+ (sin x x sin y) dy= 0

(d) dydx

=yx2 + 2x2

(e) xy= e(3x2+5) dy

dx

*(f) x dy 4y dx= (x+ 1) dx

*2. Find the general solution of 2f

xy

=ex sin y+1

x

ln(xey)

*3. Find the eigenvalues of the matrices

A=

1 1 10 1 1

1 0 2

B=

2 1 21 0 1

2 1 2

and show that neither matrix can be diagonalized over R. Why must each of these matrices

have at least one real eigenvalue?

*4. (a) Find the general solution of the following system of linear differential equations:y1(t) =y1(t) + 4y2(t);

y2(t) = 3y1(t) + 2y2(t).

(b) Then find the unique solution satisfying the initial conditions y1(0) = 1 and y2(0) = 0.

For this solution find the values ofy1(0) and y

2(0).

5. Find A5

if A=( 1 4

3 2

.


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*6. (a) Diagonalize the matrix

A=

1 3 00 2 03 3 2

(b) Find the general solution of the system of linear differential equations y

=Ay.

Objectives.

(i) Understand concepts of a differential equationand its solution.

(ii) Solve a differential equation of order one by separating variables.

Know: solution of equation (D )y= 0 is y= cex.(iii) Recognise and solve exact and linear differential equations of order one.

(iv) When possible, solve a partial differential equation by partial integration.

(v) Recognise if a matrix can be diagonalized or know why it cannot.

(vi) Calculate An (n N) using diagonalization.(vii) Solve systems of linear differential equations y =Ay using diagonalization

(by changing to a basis of eigenvectors).



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Exercise Set 18

1. Find the general solution of each of the equations by any appropriate method.

(a) xydydx y2 = 3x2e2y/x (b) x2 dy

dx+ 2xy= 6x2

2. Solve the following differential equations with the help of the given substitutions:

(i) xdydx

+y = x3y6 (Let y5 =u)

(ii) (1 +y2) = (arctan y x) dydx

(Let arctany= u)

*3. Write down a linear differential equationP(D)y= 0 whose solutions include the function

e2x and another such equation whose solutions include the function xex.

Hence write down a homogeneous linear differential equation whose solutions include both

e2x and xex together with its generalsolution.

*4. Find the general solutions of the following differential equations:

(i) d4y

dx4+ 2

d3y

dx3+

d2y

dx2 = 0 (ii)

d2y

dx2 2dy

dx+ 10y= 0

Find and sketch the particular solution of equation (ii) which satisfies the conditions

y(0) = 1 and y

6

= 0

Record your answers to this question for later use.

*5. (a) Orthogonally diagonalize the matrix A=

1 22 1

.

(b) Use this to sketch the curve xTAx= 3 in the xy plane.

Find the points of intersection of the curve with the old and new axes.


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*6. Let C be the curve defined by 3x2 + 2

3xy+ 5y2 = 6.

(a) Find a symmetric matrix A such that C is given by xTAx= 6.

(b) Orthogonally diagonalize the matrixA. (Find an orthogonal matrixP and a diagonal

matrix D such that P1

AP= PT

AP= D).(c) Use Pas a change of coordinates to sketch the curve in the xy plane, showing the old

and new axes on your diagram.

(d) How does this relate to question 4 of Exercise Set 15? Why would a rotation

anti-clockwise by

3also have been an appropriate choice there?

7. (a) Show that ifP is an orthogonal matrix, then det(P) = 1.(b) Show that ifA is orthogonally diagonalizable, then A is symmetric.

(c) IfP is an orthogonal matrix and x= Pz, show thatx = z.

Objectives.

(i) Recognise and solve homogeneous differential equations.

(ii) Recognise when a change of variable can transform a differential equation of order one

to one of the types: separable, homogeneous, exact or linear.(iii) Solve linear constant coefficient higher order differential equations P(D)y= 0 when

the roots of P(D) are distinct, repeated or complex.

(iv) Know matrix P is orthogonal PT =P1 its columns are an orthonormalbasis ofRn.

(v) Orthogonally diagonalize a symmetric matrixA, PTAP= D.

In the 2 2 case create Pfrom eigenvectors so P is a rotation anticlockwise and usethis to change coordinates to sketch a curve xTAx= k, where k is a constant.

Find points of intersection of the curve with new andold axes.

(vi) Know A is orthogonally diagonalizable A is symmetric.



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Exercise Set 19

*1. Describe the behaviour as x of the general solutions of the equations in question 4

of Exercise Set 18.

2. Find the general solutions of the following difference equations; describing their behaviour

as x

(i) yx+2 yx+1 2yx = 0

(ii) yx+3 3yx+2+ 9yx+1+ 13yx = 0

*Find and sketch the particular solution of equation (i) which satisfies the conditions

y1 = 1 and y2 = 5

Record your answers to question 2 for use next week.

*3. Let A=

1 4 24 1 22 2 2

(a) Find the eigenvalues of A and for each eigenvalue find an orthonormal basis for thecorresponding eigenspace.

(b) Hence find an orthogonal matrixP such that PTAP= P1AP= D.

(c) Write downD and check that PTAP= D.

(d) Express f(x,y ,z) =xTAx as a function of the variablesx, y and z. Write down a

matrix Q and an expression zTDz = 1X2 +2Y

2 +3Z2 where 1 2 3 such

that f(x,y ,z) =xTAx= zTDz if x= Qz.

Is the quadratic form f(x,y ,z) positive definite, negative definite or indefinite?

Evaluate f(x,y ,z) at one unit eigenvector corresponding to each eigenvalue.

*4. (a) Find an orthonormal basis for the subspace ofR3 given by

V =

x

y

z

5x y+ 2z = 0

.

(b) Extend this to an orthonormal basis ofR3.


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5. Beginning with the vector v1, use Gram-Schmidt to obtain an orthonormal basis for the

subspace ofR4 spanned by the vectors,

v1 =

10

10

v2 =

30

20

v3 =

21

13

*6. (See Calculus Lecture 10 and Algebra Lecture 19)

(a) Let q= (x )Tf()(x ). For x near , , an eigenvalue of the matrix f(),

q >0 all s> 0 = is a local minimum.

q 0, F(a2, b2)< 0,

G(a3, b3)> 0, G(a4, b4)< 0.

8. IfAis an m k matrix of rank k, show that the matrix ATA is positive definite.

Objectives.

(i) Describe the behaviour of solutions of differential equations as x .

(ii) Know the difference equation (E )yx = 0 has solution yx =c()x. Solve linear,

constant coefficient, higher order difference equations. Describe behaviour as x .

(iii) Sketch particular solutions of difference equations.

(iv) Use Gram-Schmidt orthogonalization to obtain an orthonormal basis of a vector space.

(v) Obtain information about a quadratic form xT

Ax from the eigenvalues ofA.



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Exercise Set 20

*1. Find the general solutions of the following differential equations where x R+:

(i) d4y

dx4+ 2

d3y

dx3+

d2y

dx2= sin x (ii)

d3y

dx3 3

d2y

dx2+ 9

dy

dx+ 13y= x+ex

2. Find the general solutions of the following difference equations where x N:

(i)

yx+2

yx+1 2

yx= (1)

x

(ii

) yx

+2+ 2yx

+1+ 4yx

= 12

(iii) yx+3 3yx+2+ 9yx+1+ 13yx = 2x

*3. Find the general solution of the differential equation

2f

x2

2f

y2 =x

by making the change of variable x= u+v, y= u v.

*4. A Markov process satisfies the difference equation:

xk =Axk1 where A=

0.7 0.60.3 0.4

x0 =

0.60.4

Solve the equation and find an expression for xk as a linear combination of the eigenvectors

ofA. Use this to predict the long term distribution; that is, find limxk as k .

5. The population of osprey eagles at a certain lake is dying out. Each year the new population

is only 60% of the previous years population. Conservationists introduce a new species of

trout into the lake and find that the populations satisfy the following system of difference

equations, where xt is the number of osprey in year t and yt is the number of trout in

year t.

xt =Axt1 where xt =

xtyt

A=

.6 .2.25 1.2

and x0 =

20100


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(a) Give a reason why this system isnota Markov process.

(b) Describe in words how each of the populations depends on the previous years popu-

lations.

(c) Solve the system (as you would a Markov process). Show that the solution xt

satisfies xt =Atx0. Diagonalize the matrixA. Hence find an expression for xt as alinear combination of the eigenvectors ofA.

(d) Show that the situation is not stable, that according to this model both osprey and

trout will increase without bound as t .

What will be the eventual ratio of osprey to trout?

In order to have the populations of osprey and trout achieve a steady state, they decide to

allow an amount of fishing each year, based on the number of osprey in the previous year.

The new equations are

xt =Bxt1 where xt =

xtyt

B=

.6 .2 1.2

and x0=

20100

where >0 is a constant to be determined.

(e) What property of the transition matrix of a Markov process determines that there is a

(finite, non-zero) long-term distribution?

Deduce a condition on the eigenvalues of the matrix Bto produce the same effect.

Then find the value of which satisfies this condition.

(f) Show that for this value of the population now reaches a steady state as t and

determine what this stable population of osprey and trout will be.

Objectives.

(i) Find a particular solution of a nonhomogeneous linear differential/difference equation

with RHS itself the solution of a linear differential/difference equation.

(ii) Recognise defining properties of a Markov process, xk =Axk1:

A has non-negative entries and the entries of each column sum to 1.

Know that A always has an eigenvalue = 1 with other eigenvalues |i| 1.

(iii) Find the solution of a Markov process xk =Axk1. Express it as a linear combination

of eigenvectors ofA. Use it to predict lim xk as k .

Apply the same method to solve any system of linear difference equations,xk =Axk1.



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Some Review Exercises

(optional)

1. Consider a matrix, for example the 3 3 matrix A given below, and think about . . .

what this matrix can represent . . . what you can ask about the matrix (e.g. invertible?diagonalizable?) . . . what subspaces ofRn are associated with the matrix?

A=

7 6 95 4 5

2 2 4

(i) Solve Ax= 0.

(ii) Describe the linear transformation T(x) =Ax by finding the images ofe1, e2, e3

(iii) Can this matrix be used as a change of coordinates in R3

?(iv) Find A1 if it exists.

(v) Diagonalise the matrix A. (Find an invertible matrix P and a diagonal matrix D such

that P1AP= D, and check your work.)

(vi) Use the diagonalisation to give a description of the effect of the linear transformation

T(x) =Ax on R3.

(vii) Use the diagonalisation to solve the system of differential equations y =Ay

(viii) Find the row space, column space and nullspace of A. Find the kernel and the

range of the linear transformation T(x) = Ax. Describe these spaces geometrically,giving Cartesian equations for any planes and vector equations for any lines.

2. Show that the function f(x,y,z) = 5x2+5y2+9z26xz12yz has a line of stationary

points and determine their nature.

Find a symmetric matrix A such that f(x,y,z) =xTAx.

Is A positive (or negative) definite (or semidefinite)? What information does this give

about the function f? About the stationary points off?

How is A related to the Hessian off?

3. Let A be an m k matrix with full column rank (i.e. the columns of A are linearly

independent). Show that:

(1) ATA is a symmetric k k matrix,

(2) ATA is invertible,

(3) ATA is positive definite.


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4. Functions f, g: R3 R are defined by

f(x,y,z) =x2 + 4y2 + 2z2

g(x,y,z) =x2 + y2 2z2

(a) Show that the point (3,

2, 1) lies on both of the following level surfaces and henceon their curve of intersection:

f(x,y,z) = 27 and g(x,y,z) = 11

Describe these surfaces, sketching their intersections with the coordinate planes.

(b) Find normal vectors and the tangent planes to both surfaces at (3,2, 1). Do the

surfaces touch tangentially at this point?

(c) Using the normal vectors found in (b) write down Cartesian equations for the tangent

line to the curve of intersection of the level surfaces at (3,2, 1)

5. The function f(x, y) is

study pack ma100 inside all (1)

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