MATHEMATICS T (1).docx

8/19/2019 MATHEMATICS T (1).docx

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STPM MATHEMATICS T SYLLABUS

Chapter 1 Functions

1.1 Functions

• (a) state the domain and range of a function, and find composite functions;

The domain of a unction is all the possible input values, and the ran!e is all possibleoutput values."omain# set of all possible input values (usually x)$an!e# set of all possible output values (usually y)

a) State the domain and range of the following relation%&'( )*+( &,( -+( &1( )1+( &*( +/

The domain is all the x-values, and the range is all the y-values "omain# %'( ,( 1( */ $an!e# %)*( -( 01( /

a) State the domain and range of the following relation%&'( ,+( &,( ,+( &1( ,+( &*( ,+/

"omain# %'( ,( 1( */

$an!e# %,/

hat is a Function2 A unction( &3+ re4ates an input to an output. Each input is

re4ated to e3act45 one output.

This is a unction. There is only one y for each x

This is a unction. There is only one arrow coming from each x; there is only one y foreach x

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This is not a unction. is in the domain, but it has no range element that corresponds toit

This is not a unction. ! is associated with two different range elements

Functions# "omain and $an!e

The unction composition of two functions ta"es the output of one function as the input

of a second one

#n in6erse unction for f, denoted by f -$, is a function in the opposite direction.- See more at% http%&&coolmathsolutions.blogspot.com&'$!&'&what-is-function.htmlsthash.t!*ri+g.dpuf

(b) determine whether a function is one-to-one, and find the inverse of a one-to-onefunction;

# one-to-one function is a function in which every element in the range of the function

corresponds with one and only one element in the domain.

# function, f(x), has an inverse function if f(x) is one-to-one.

The oriontal /ine Test% 0f you can draw a horiontal line so that it hits the graph inmore than one spot, then it is 12T one-to-one.

a) 0s below function one-to-one3

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f(x)=x3

f(x)4x! is one-to-one function and has inverse function. (The horiontal line cuts thegraph of function f at $ point, therefore f is a one-to-one function)

b) 0s below function one-to-one3

f(x)=x2

f(x)4x' is 12T one-to-one function and does 12T has an inverse function. (Thehoriontal line cuts the graph of function f at ' points, therefore f is 12T a one-to-onefunction)

0f we restricted x greater than or e5ual to . The horiontal line cuts the graph of functionf once and f(x)4x' is one-to-one function and has an inverse function.


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(c) s"etch the graphs of simple functions, including piecewise-defined functions;

ow to find the inverse of one-to-one function below3

f(x)=3x−46raw the graph of f(x)4!x-7

The horizontal line cuts the graph of function f once, therefore f is a one-to-onefunction and it has has inverse function.

8ind the inverse function

f(y)=x3y−4=x

y=x+43

f−1(x)=x+43

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The graph of an inverse relation is the reflection of the original graph over the identityline, y 4 x.

9iecewise-defined function is a function which is defined by multiple sub-

functions, each sub-function applying to a certain interval of the mainfunction:s domain.

E3amp4e 1

f(x)={2x−1,x


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f(x)=4x+1

f(x)=2x3−4x2+5x+11

f(x)=x7−6x4+3x2

*ational function is division of two polynomial functions.

f(x)=P(x)Q(x) where 9 and are polynomial functions in x.

y=2x+5x−1f(x) is not defined at x4$.


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• (d) use the factor theorem and the remainder theorem;

The Factor Theorem

0f the remainder of a polynomial, f(x), when divide by (x-a) is ero, then (x-a) is a factor.Show that (x>!) is a factor of

x3+5x2+5x−3

f(−3)=(−3)3+5(−3)2+5(−3)−3

f(−3)=0

?y factor theorem (x>!) is factor of x!>x'>x-!*emainder Theorem

0f a polynomial f(x) is divided by (x @ r) and a remainder * is obtained, then f(r) 4 *.f(x)=(x−r)⋅q(x)+R

Axample% &3+73-8'3'0*3*8309 di6ide :5 &301+

#fter dividing by x-$, there is a remainder of -7.


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(e) solve polynomial and rational e5uations and ine5ualities;

So46in! $ationa4 Ine;ua4ities

# rational function is a 5uotient of two polynomials.

x2+3x+2x2−16≥0

x2+3x+2x2−16=(x+2)(x+1)(x−4)(x+4)

as an example for an ine5uality involving a rational function.

This polynomial fraction will be ero wherever numerator is ero

(x+2)(x+1)=0

x=−2,x=−1

The fraction will be undefined wherever the denominator is ero

(x−4)(x+4)=0

x=4,x=−4

Bonse5uently, the set

(−∞,−4),[−2,−1],(4,+∞)

Solution in Cine5ualityC notation

x4

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Find the inter6a4

*ational e5uations and ine5ualities '


A rationa4 unction is a ;uotient o t


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Find the inter6a4

(f) solve e5uations and ine5ualities involving modulus signs in simple cases;

So46e the ine;ua4it5 3 * 3 8 1

• S5uare both sides of each e5uation to omit the modulus signs.

• *earrange the ine5uality into an e5uivalent form.

So the solution set is

(g) decompose a rational expression into partial fractions in cases where the denominator

has two distinct linear factors, or a linear factor and a prime 5uadratic factor;A rationa4 unction P&3+D&3+ can :e re


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rite the partia4 raction decomposition o

11x+21(2x−3)(x+6)=A2x−3+Bx+6

"istri:ute A and B

11x+21=(x+6)A+(2x−3)B0f x4-D

11(−6)+21=[2(−6)−3]B

B=30f x4,

21=6A−3B

A=5

The partia4 raction

11x+21(2x−3)(x+6)=52x−3+3x+6

This is the E3ponentia4 Function%

f(x)=ax a is any value greater than


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For > G a G 1#

• #s x increases, f(x) heads to

• #s x decreases, f(x) heads to infinity

• 0t is a Strictly 6ecreasing function

• 0t has a oriontal #symptote along the x-axis (y4).

For a = 1#

• #s x increases, f(x) heads to infinity

• #s x decreases, f(x) heads to

• it is a Strictly 0ncreasing function


The atura4 E3ponentia4 Function is the function

f(x)=exwhere e is the number (approximately '.E$+'+$+'+)

1.' E3ponentia4 and 4o!arithmic unctions

• (h) relate exponential and logarithmic functions, algebraically and graphically;

This is the E3ponentia4 Function%

f(x)=ax a is any value greater than

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For > G a G 1#

• #s x increases, f(x) heads to

• #s x decreases, f(x) heads to infinity

• 0t is a Strictly 6ecreasing function


For a = 1#

• #s x increases, f(x) heads to infinity

• #s x decreases, f(x) heads to

• it is a Strictly 0ncreasing function


The atura4 E3ponentia4 Function is the function

f(x)=exwhere e is the number (approximately '.E$+'+$+'+)

The logarithmic function is the function y 4 logax, where a is any number such that a F ,a G $, and x F .

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y 4 logax is e5uivalent to x 4 ay

The inverse of an exponential function is a logarithmic function.

Since y 4 log'x is a one-to-one function, we "now that its inverse will also be a function.


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(i) use the properties of exponents and logarithms;

$. 0ndices

an,a≠0

a0=1,a≠0a−n=1/an

'. /aw of 0ndices

am×an=am+n

am÷an=am−n

(am)n=am×n

am×bm=(a×b)m

am÷bm=(ab)m

!. /ogarithms

Ifa=bc,then logba=c

loga1=0

logaa=1

alogab=1

7. /aw of /ogarithms

logaxy=logax+logay

loga(xy)=logax−logay

logaxn=nlogax

Bhange of base of /ogarithms

logab=1logba

logab=lognblogna

(H) solve e5uations and ine5ualities involving exponential or logarithmic expressions;


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A rationa4 unction is a ;uotient o t


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1.- Tri!onometric unctions

• (") relate the periodicity and symmetries of the sine, cosine and tangent functions to their

graphs, and identify the inverse sine, inverse cosine and inverse tangent functions andtheir graphs;

• (l) use basic trigonometric identities and the formulae for sin ( A I B), cos ( A I B) and tan( A I B), including sin 2A, cos 2A and tan 2A;

• (m) express a sin θ > b cos θ in the forms r sin ( θ I J) and r cos ( θ I J);

• (n) find the solutions, within specified intervals, of trigonometric e5uations andine5ualities.

Trigonometric Identities

sin(A+B)=sinAcosB+cosAsinB

sin(A−B)=sinAcosB−cosAsinB

cos(A+B)=cosAcosB−sinAsinB

cos(A−B)=cosAcosB+sinAsinB

tan(A+B)=tanA+tanB1−tanAtanB

tan(A−B)=tanA−tanB1+tanAtanB

sin(2A)=2sinAcosA

cos(2A)=cos2(A)−sin2(A)

=2cos2(A)−1

=1−2sin2

(A)

tan2A=2tanA1−tan2A

E3press a sin J : cos in the orm $ sin & J K+

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E3press a sin J : cos in the orm $ sin& J K+(


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AxampleAxpress - sin 8' cos in the form $ sin& 8 K+

R=42+32−−−−−−√ =25−−√=5

α=tan−1 34=36.87∘So

4sinθ+3cosα=5sin(θ+36.87∘)

Summar5 o the e3pressions and conditions

E3press a sin J : cos in the orm $ cos & J K+

E3press a sin J : cos in the orm $ cos& J K+(


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=tanαSo

α=tan−1ab

R=a2+b2−−−−−−√

Then we have expressed a sin K > b cos K in the form re5uired

asinθ+bcosθ≡Rcos(θ−α)

Summar5 o the e3pressions and conditions


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Chapter *

Se;uences and Series

*.1 Se;uences

• (a) use an explicit formula and a recursive formula for a se5uence;

• (b) find the limit of a convergent se5uence;

Con6er!ent Se;uence

# se5uence is said to be convergent if it approaches some limit. 8ormally, a se5uenceconverges to the limit

limn→∞Sn=S

E3amp4e

• The se5uence '.$, '.$, '.$, '.$, . . . has limit ', so the se5uence converges to '.

• The se5uence $, ', !, 7, , D, . . . has a limit of infinity (L). This is not a real number, so

the se5uence does not converge. 0t is a divergent se5uence.

Con6er!ent se;uences ha6e a inite 4imit.

1,12, 13,14,15,16....,1n

Limit=0

1,12, 23,34,45,56....,nn+1

Limit=1

*.* Series

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(c) use the formulae for the nth term and for the sum of the first n terms of anarithmeticseries and of a geometric series;

An arithmetic progression or arithmetic sequence is a sequence of numberssuch that the dierence between the consecutive terms is constant.

The nth term of the sequence an! is given b"

an=a1+(n−1)d

The sum of the sequence of the #rst n terms is then given b"

Sn=n2[2a+(n−1)d]or

Sn=n2(a1+a2)

eometric Series is a series


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a+ar+ar2+ar3+ar4.....=∑ k=0∞ark=a1−r


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(x+y)2=x2+2xy+y2

(x+y)3=x3+3x3y+3xy2+y3

(x+y)4=x4+4x3y+6x2y2+4xy3+y4

It is possib&e to e'pand an" power of ' ( " into a sum of the form

(x+y)n=(n0)xny0+(n1)xn−1y1+(n2)xn−2y2+...+

(nn−1)x1yn−1+(nn)x0yn It can be written as

(x+y)n=∑ k=0n(nk)xn−kyk%inomia& formu&a ) invo&ves on&" a sing&e variab&e

(1+x)n=(n0)x0+(n1)x1+(n2)x2+...+(nn−1)xn−1+(nn)xn(1+x)n=∑ k=0n(nk)xk

According to the ratio test for series convergence a series converges when)

if limk→∞(ak+1ak)1,series diverges

iflimk→∞(ak+1ak)=1,result is indeterminate The %inomia& Theorem converges when *'*+,


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(g) expand ($>x) n , wheren$Q, and identify the condition M x M N $ for the validity of thisexpansion;

(h) use binomial expansions in approximations.

Chapter '

Matrices

'.1 Matrices

(a) identify null, identity, diagonal, triangular and symmetric matrices;

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In6erse o "ia!ona4 Matri3

6iagonal matrix is a matrix in which the entries outside the main diagonal are all ero.

A=- a11000a22 0 00a33/0

A−1=-11111111a11000 1a22 0 00 1a33/02222222

Axample

A=- 5000 3 0 00 1/0

A−1=-111115000 13 0 001/02222

(b) use the conditions for the e5uality of two matrices;

(c) perform scalar multiplication, addition, subtraction and multiplication of matriceswith at most three rows and three columns;

hat is Trian!u4ar Matri3

# matrix is a trian!u4ar matri3 if all the elements either above or below the diagonal areero. # matrix that is both upper and lower triangular is a diagonal matrix.

Properties o Trian!u4ar Matri3

• The determinant of a triangular matrix is the product of the diagonal elements.

• The inverse of a triangular matrix is a triangular matrix.

• The product of two (upper or lower) triangular matrices is a triangular matrix.

Upper Trian!u4ar Matri3

A=33333130250013333=3.2.1=6

A−1=33331/3−1/6−1/601/2−5/250013333

Lo


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B−1=33331/2003/2−10−11613333

(d) use the properties of matrix operations;

Properties o Matrices

AB≠BA

A+B=B+A

A(B+C)=AB+AC

(A+B)C=AC+BC

A(BC)=(AB)C

AI=IA=A

A0=I

Scalar Oultiplication

λ(AB)=(λA)B

(AB)λ=A(Bλ)

Transpose

(AB)T=BTAT

9roperties of 0nverse Oatrix

AA−1=A−1A=I

(A−1)−1=A

(AT)−1=(A−1)T

(An)−1=(A−1)n

(cA)−1=c−1A−1=1cA−1

(e) find the inverse of a non-singular matrix using elementary row operations;

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8ind the inverse of a non-singular matrix using elementary row operations

4shirts +2 pants +2 pairofshoes=$190



/et x 4 price of shirt , y4 price of pant, 4 price for a pair of shoe

-4322 4 4 232/0-xyz/0=-190295250/0/et

A=-4322 4 4 232/0,B=-190295250/0

A-xyz/0=-190295250/0

AA−1=A−1A=I

A−1A-xyz/0=A−1-190295250/0

-xyz/0=-0.50−0.5−0.5 −0.5 1.50.250.75−1.25/0-190295250/0

-xyz/0=-104035/0

A4ternati6e(

a1x+b1y+c1z=d1

a2x+b2y+c2z=d2

a3x+b3y+c3z=d3

Oode F A1 F Pn"nowns4!, input data below

a1=4,b1=2,c1=2,d1=190

a2=3,b2=4,c2=3,d2=295

a3=2,b3=4,c3=2,d3=250

x=10,y=40,z=35

(f) evaluate the determinant of a matrix;

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In6erse o a Matrices ''

A=-adgbeh cfi/0

$. 6eterminant of !x! Oatrices

4A4=4444adgbehcfi4444=a444 e hfi444@b444dgfi444+c444dgeh444

=a(ei−fh)−b(di−fg)+c(dh−eg)

'. 0nverse of !x! Oatrices

A−1=-adgbe h cfi/0−1=14A4-A D GBEHCFI/0T=14A4-A BC

DEFGHI/0A=444ehfk444,B=−444dgfk444,C=444dgeh444

D=−444bhck444,E=444agck444,F=−444agbh444

G=444becf444,H=−444adcf444,K=444adbe444

-+−+− +−+−+/0

(g) use the properties of determinants;

Properties o "eterminant

$. M#M 4 if it has two e5ual line

33333132 223133333=0

'. M#M 4 if all elements of a line are ero

33333032023033333=0

!. M#M 4 if the elements of a line are a linear combination of the others. (row ! 4 row $ >row ')

33332133252463333=0

7. The determinant of matrix # and its transpose # are e5ual. M#TM4M#M

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A=33332131213423333, |AT|=33332131243123333

|A|=|AT|=−5. # triangular determinant is the product of the diagonal elements.

A=33333130 250013333=3.2.1=6D. The determinant of a product e5uals the product of the determinants.

|A.B|=|A|.|B|

E. 0f a determinant switches two parallel lines its determinant changes sign.

33332131213423333=@5,33331232114323333=5

'.* S5stems o 4inear e;uations

(h) reduce an augmented matrix to row-echelon form, and determine whether a system oflinear e5uations has a uni5ue solution, infinitely many solutions or no solution;

(i) apply the =aussian elimination to solve a system of linear e5uations;

aussian e4imination is a method for solving matrix e5uations of the form

Ax=b

=aussian elimination starting with the system of e5uations

Bompose the Caugmented matrix e5uationC

9erform elementary row operations to put the augmented matrix into the uppertriangular form

# matrix that has undergone =aussian elimination is said to be in echelon form=iven below matrix e5uation

0n au!mented orm, this becomes

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Switching the first and second rows (without switching the elements in the right-hand column vector) gives

8irst row times ! and minus third row gives

8irst row times ' and minus second row gives

8inally, second row times -E&! and minus third row gives (augmented matrix has been reduced to ro


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Oode F A1 F Pn"nowns4!, input data below

a1=20,b1=0,c1=−10,d1=−100

a2=0,b2=20,c2=10,d2=300

a3=−10,b3=10,c3=20,d3=200

x=−5,y=15,z=0

Chapter -

Comp4e3 um:ers

- Comp4e3 um:ers


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• (a) identify the real and imaginary parts of a complex number ;

The rea4 num:ers include all inte!er, rationa4 num:er (number that can be expressed asthe 5uotient or fraction p&5 of two integers, with the denominator 5 not e5ual to ero) and irrationa4 num:ers (numbers cannot be represented as a simple fraction)

0ntergers

...−4,−3,−2−1,0,1,2,3,4....*ational 1umber

12,−12,1537,....0rrational 1umber

2√,3√3,π,e,log23,....

Ima!inar5 num:er is a number than can be written as a real number multiplied by the

imaginary unit i

i=−1−−−√

i2=−1

# complex number is a number that can be put in the form

a+biwhere a and : are real numbers and i is the s5uare root of -$, the ima!inar5 unit

i=−1−−−√

i2=−1

The a:so4ute 6a4ue or modu4us of a complex number 4 a > bi is

|z|=a2+b2−−−−−−√

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(b) use the conditions for the e5uality of two complex numbers;

(c) find the modulus and argument of a complex number in cartesian form and expressthe complex number in polar form;

hat is Ar!ument o Comp4e3 um:er

Bomplex 1umber , 4a>biOodulus is the length of the line segment, that is 29 (modulus of can Qnd using9ythagorasR theorem)

|z|=a2+b2−−−−−−√

|z|=42+32−−−−−−√

|z|=25−−√=5

The angle theta is called the argument of the complex number,

tanΘ=ba

tanΘ=34

Θ=tan−134

arg z=0.644rad

ow to change theta4!D.+E to radian

rad=Θ6π1800

rad=Θ6π1800

36.8706π1800=0.644rad

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(d) represent a complex number geometrically by means of an #rgand diagram;

Ar!and "ia!ram

The #rgand diagram is used to represent complex numbers.#rgand diagram form by y-axis which represents the ima!inar5 part and x-axisrepresent the rea4 part.

Axample%

Z1=3+2i

Z2=−4+i

BonHugate of

Z61=3−2i

Z62=−4−i

Z1+Z2=−1+3i

(e) find the complex roots of a polynomial e5uation with real coefficients;

(f) perform elementary operations on two complex numbers expressed in cartesian form;

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(g) perform multiplication and division of two complex numbers expressed in polar form;

(h) use de OoivreRs theorem to find the powers and roots of a complex number

"e Moi6reNs ormu4a 0 STPM Mathematics

"e Moi6reNs

6e Ooivre:s formula states that for any real number x and any integer n,

(cosx+isinx)n=cos(nx)+isin(nx)

From Eu4erNs ormu4a

eix=cosx+isinx

(eix)n=einx

ei(nx)=cos(nx)+isin(nx)

"e Moi6reNs Theorem in Comp4e3 um:er

0f 4 x > yi 4 reiK, and n is a natural number. Then

zn=(x+yi)n=(reiθ)n

E3amp4e 1# Pse 6e OoivreRs theorem to Qnd ($>i)$.


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=32(cos450∘+isin450∘)

=32(0+i)

=32i

E3amp4e *# Pse 6e OoivreRs theorem to Qnd (! > i ) E .


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Chapter , Ana45tic eometr5

, Ana45tic eometr5

(a) transform a given e5uation of a conic into the standard form; Transform 7onic 8ection into 8tandard 9orm

A conic section is the intersection of a p&ane and a doub&e right circu&arcone. %" changing the ang&e and &ocation of the intersection: we can producedierent t"pes of conics: such as circles, ellipses, hyperbolas and parabolas.

Equation of a Circle in Standard Form

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(x−h)2+(y−k)2=r2Equation of an Ellipse in Standard Form

(x−h)2a2+(y−k)2b2=1Equation of a Hyperbolas in Standard Form

(x−h)2a2−(y−k)2b2=1Equation of a Parabolas in Standard Form

a(x−h)2+k

(b) find the vertex, focus and directrix of a parabola;

Oerte3( Focus and "irectri3 o a Para:o4a

# para:o4a is the set of all points 9 in the plane that are e5uidistant from a fixed point 8&ocus+ and a fixed line d &directri3+.The 6erte3 of the parabola is at e5ual distance between focus and the directrix.

9arabolas are fre5uently encountered as graphs of ;uadratic unctions, such as

y=x2

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(c) find the vertices, centre and foci of an ellipse;

Oertices( Centre and Foci o an E44ipse

E44ipse is a planar curve which in some Bartesian system of coordinates is described bythe e5uation%

(x−h)2a2+(y−k)2b2=1

Oerte3 o an e44ipse. The points at which an ellipse ma"es its sharpest turns. The verticesare on the maHor axis.Centre o an e44ipse # point inside the ellipse which is the midpoint of the line segmentlin"ing the two foci.Foci o an e44ipse 0 The foci, c is always lie on the maHor (longest) axis, spaced e5uallyeach side of the centre.

c2=a2−b2

E3amp4e

16x2+25y2=400

16x2400+25y2400=1

x225+y216=1

x252+y242=1

(x2−0)252+(y2−0)242=1

Oerte3 is &0,(>+ and &,(>+( the maor &4on!est+ a3is.

Co06erte3 is &>(0-+( &>( -+ minor &shortest+ a3is

The centre is at &h(+7&>(>+

Foci o an e44ipse. The foci are three unit to either side of the centre, at &0'(>+ and &'(>+

c2=a2−b2

c2=52−42

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c2=±3

• 8ind ertices, Bentre, and 8oci of Allipse $

• 8ind ertices, Bentre, and 8oci of Allipse '

• 8ind ertices, Bentre, and 8oci of Allipse !

(d) find the vertices, centre, foci and asymptotes of a hyperbola;

(e) find the e5uations of parabolas, ellipses and hyperbolas satisfying prescribed

conditions (excluding eccentricity);

(f) s"etch conics;

(g) find the cartesian e5uation of a conic defined by parametric e5uations;

(h) use the parametric e5uations of conics.

Chapter Oectors

.1 Oectors in t


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A unit vector, or direction vector is a vector which has &ength of , ormagnitude of ,.

b!u =̂u4u4

c!d!

=osition ! is ca&&ed a position vector.

=oint A has the position vector a, if

A=(35) =oint % has the position vector b, if

B=

(62

)

e!

(b) perform scalar multiplication, addition and subtraction of vectors;

8ca&ar ?u&tip&ication of


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c[a1a2]=[ca1ca2]E3amp4e# Oultiply the vector u 4 N ', ! F of by the scalars ', U!, and $&'

2u @ =2[23]=[46]12u @ =12[23]=[13/2]−3u @ =@3[23]=[−6−9]

(c) find the scalar product of two vectors, and determine the angle between two vectors;

The scalar product( or dot product is an a4!e:raic operation that taes t


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• u(v>w)4uv>uw

• u.u4MMuMMVW'X

• c(u.v)4cu.v4u.cv

Sca4ar product can :e o:tained :5 ormu4a

a.b=|a||b|cosθMaM means the magnitude (length) of vector a. and

a.b=axbx+ayby

Balculate the scalar product of vectors a and b, given K 4 Y.

a=[−68],b=[512]

|a|=(−6)2+82−−−−−−−−−√ =10|b|=52+122−−−−−−−√ =13

a.b=|a||b|cosθ

a.b=10612cos59.5∘or

a.b=axbx+ayby

a.b=(−6)(5)+(8)12)=66

8ind the angle between the two vectors.

a=-234/0,b=-1−23/0


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a.b=axbx+ayby

a.b=|a||b|cosθ

a.b=(2)(1)+(3)(−2)+(4)(3)=8

|a|=22+32+42−−−−−−−−−−√ =29−−√

|a|=12+(−2)2+32−−−−−−−−−−−−√ =14−−√8=29−−√14−−√cosθ

cosθ=0.397

θ=66.6∘

(d) find the vector product of two vectors, and determine the area a parallelogram and ofa triangle;

Oector Product o T


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=−3i+6 j−3k

u @ =@3,6,−3B

ector product, c 4 a x b 4 (-!, D, -!)

.* Oector !eometr5

(e) find and use the vector and cartesian e5uations of lines;


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a=EFaxayazGH,b=EFbxbybzGHJ,r=EFxyzGH7artesian Cquation wi&& be

x−axbx−ax=y−ayby−ay=z−azbz−az

C'amp&e) 7ompute the cartesian equation of a straight &ine through point Aand % with position vector)

a=EF1−32GH,b=EF3−15GH,r=EFxyzGH

x−13−1=y−(−3)−1−(−3)=z−25−2

x−12=y+32=z−23

(f) find and use the vector and cartesian e5uations of planes;

Oector E;uations o P4anes

The vector e5uation of the plane

r=a+λu+µv

Axample% Bompute the vector e5uation of a plane through point #, ? and B with position

vector%a=EF2−13GH,b=EF14−1GH,c=EF0−21GH

u=b−a=EF14−1GH−EF2−13GH=EF−15−4GH

v=c−a=EF0−21GH−EF2−13GH=EF−2−1−2GH

r=a+λu+µv

r=EF2−13GH+λEF−15−4GH+µEF−2−1−2GH

Cartesian E;uations o P4anes

8ormula for Bartesian A5uations of 9lanes

n=(b−a)6(c−a)

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r.n=a.n

a=EF2−13GH,b=EF14−1GH,c=EF0−21GH,r=EFxyzGH

n=(b−a)6(c−a)

n=EF−15−4GHEF−2−1−2GH=EF−14611GH

r.n=a.n

EFxyzGH.EF−14611GH=EF2−13GH.EF−14611GH

−14x+6y+11z=(2)(−14)+(−1)(6)+(3)(11)Bartesian A5uations of 9lanes

−14x+6y+11z=−1

(g) calculate the angle between two lines, between a line and a plane, and between two planes;

Ang&e %etween Two Dines

Ang&e %etween Two Dines

θ=β−α

tanθ=tan(β−α)=tanβ−tanα1+tanβtanα

=m1−m21+m1m29or two &ine of gradient m,: mK te acute ang&e between them is a&wa"spositive

tanθ=333m1−m21+m1m2333

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m1m ! "1: this formu&a doesnLt worM for perpendicu&ar &ines.

E#ample 1) 9ind the acute ang&e between the &ines " N O' P , and " N PK' (O.

tanθ=333m1−m21+m1m2333tanθ=3333−(−2)1+(3)(−2)333

tanθ=|−1|=1

θ=tan−1(1)=45∘

E#ample ) 9ind the acute ang&e between the &ines Q' P " ( R N S and PO'P,," (,S N S

earrange the equation

y=6x+8

y=−311x−1011

m,NQ: mKNPOU,,

tanθ=333m1−m21+m1m2333

tanθ=33336−(−311)1+(6)(−311)3333

tanθ=333−697333θ=tan−1697=84.2∘

(h) find the point of intersection of two lines, and of a line and a plane;

9ind the =oint of Intersection %etween Two Dines

At the point of intersecting &ines: the points are equa&.

C'amp&e) 9ind the point of intersection between &ines " N O' P V and " NPK'(O.

y=3x−7−−−−−(1)

y=−2x+3−−−−−(1)8ubstitute ,! into K!

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3x−7=−2x+3

5x=10

x=2

'NK:y=3(2)−7

y=−1• Wence: the intersecting point is K: P,!

• =oogle

• P 8ee more at) http)UUcoo&mathso&utions.b&ogspot.comUKS,OUSOU#ndPpointPofPintersectionPbetweenPtwo.htm&Xsthash.h"vY


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Find the 6ectorDcross product o these norma4 6ectors

n1→6n2→=3333i23 j−54k3−33333n1→6n2→=333@543−3333i−333233−3333 j+33323−54333k

=3i+15 j+23kector product is N!, $, '!F

Find the position 6ector rom the ori!in

8ind some point which lies on both the planes because then it must lie on their line of

intersection. #ny point which lies on both planes will do. Bould be plane- xy, yz, xz.0f x4,

−5y+3z=12

4y−3z=6

y=−18,z=−269oint with position vector (, -$+, -'D) lies on the line of intersection.

The e5uation of the line of intersection is

r=(0,−18,−26)+t(3,15,23)

To chec" that point that we get does really lie on both planes and so on their line ofintersection. 0f t4$

r=(3,−3,−3)Substitute into the planes e5uations

2(3)−5(−3)+3(−3)=12

3(3)+4(−3)−3(−3)=6

I 57>

2x+3z=12

3x−3z=6


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x=3.6,z=1.69oint with position vector (, !.D, $.D) lies on the line of intersection.

The e5uation of the line of intersection is

r=(3.6,0,1.6)+t(3,15,23)

To chec" that point that we get does really lie on both planes and so on their line ofintersection. 0f t4$

r=(6.6,15,24.6)Substitute into the planes e5uations

2(6.6)−5(15)+3(24.6)=12

3(6.6)+4(15)−3(24.6)=6


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Chapter 9 Limits and Continuit5

9.1 Limits

(a) determine the existence and values of the left-hand limit, right-hand limit and limit ofa function;

DeftPhand Dimit and ightPhand Dimit

A &imit is the va&ue that a function or sequence [approaches[ as the input orinde' approaches some va&ue.

The &imit of f(x) as ' approaches a from the right.

limx→a+f(x)

The &imit of f(x) as ' approaches a from the left .

limx→a−f(x)

C'amp&e) 9ind

limx→2−(x3−1)

limx→2−(x3−1)=limx→2−(23−1)=7As ' approaches K from the &eft: 'Opproaches R and 'OP, approaches V.

9ind

limx→2+(x3−1)

limx→2+(x3−1)=limx→2+(23−1)=7

•

(b) use the properties of limits;=roperties of &imits

The &imit of a constant is the constant itse&f.

limx→ak=k

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The &imit of a function mu&tip&ied b" a constant is equa& to the va&ue of thefunction mu&tip&ied b" the constant.

limx→ak.f(x)=k.limx→af(x)

The &imit of a sum or dierence! of the functions is the sum or dierence! ofthe &imits of the individua& functions.

limx→a[f(x)+g(x)]=limx→af(x)+limx→ag(x)

limx→a[f(x)−g(x)]=limx→af(x)−limx→ag(x)

The &imit of a product is the product of the &imits.

limx→af(x).g(x)=limx→af(x).limx→ag(x)

The &imit of a quotient is the quotient of the &imits

limx→af(x)g(x)=limx→af(x)limx→ag(x)

The &imit of a power is the power of the &imit.

limx→axn=an

9.* Continuit5

(c) determine the continuity of a function at a point and on an interval;

Continuit5 o a Function at A Point and Qn An Inter6a

Continuit5 at a Point

# function f (x) is continuous at a if the following three conditions are valid%

i) The function is de fined at a% That is, a is in the domain of definition of f (x)

ii) if

limx→af(x)exists.

iii) if

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limx→af(x)=f(a)

0f any of the three conditions in the definition of continuity fails when x 4 c, the functionis discontinuous at that point.

Continuit5 on An Inter6a4

# function which is continuous at every point of an open interval I is called continuouson I .

(d) use the intermediate value theorem.

/et f (x) be a continuous function on the interval Ra :

0f d is between &a+ and &:+, then a corresponding c between a and :, exists, so that

f(c)=d

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Chapter "ierentiation

.1 "eri6ati6es

(a) identify the derivative of a function as a limit;

\erivative of a 9unction as a Dimit

9or a function f'!: its derivative is de#ned as

f′(x)=lim]x→0f(x+]x)−f(x)]x

C'amp&e ,) 7ompute the derivative of f'! b" using &imit de#nition

f(x)=13x−25


f′(x)=lim]x→013(x+]x)−25−{13x−25}]x=lim]x→013x+13]x−25−13x+25]x

=lim]x→013]x]x

=13

C'amp&e ,) 7ompute the derivative of f'! b" using &imit de#nition

f(x)=5−x+2−−−−√


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=lim]x→0{5−(x+]x)+2−−−−−−−−−−−√ }−{5−x+2−−−−√}]x

=lim]x→0x+2−−−−√−x+]x+2−−−−−−−−−√]x

=lim]x→0x+2−−−−√−x+]x+2−−−−−−−−−√]x.x+2−−−

−√+x+]x+2−−−−−−−−−√x+2−−−−√+x+]x+2−−−−−−

−−−√

=lim]x→0(x+3)−(x+]x+3)]x{x+3−−−−√+x+]x+3−−−−−−

−−−√}

=lim]x→0−]x]x{x+3−−−−√+x+]x+3−−−−−−−−−√}

=lim]x→0−1x+3−−−−√+x+]x+3−−−−−−−−−√

=−1x+3−−−−√+x+3−−−−√

=−12x+3−−−−√

• (b) find the derivatives of xn (n $ Q ), e x , ln x( sin x, cos x, tan x, sin -1 x, cos -1 x, tan-1 x, with constant multiples, sums, differences, products, 5uotients and composites;

Common "eri6ati6es

Ta:4e o "eri6ati6es

ddx(x)=1

ddx(xn)=nxn−1

ddx(ex)=ex

ddx(lnx)=1x,x>0

ddx(sinx)=cosx

ddx(cosx)=−sinx

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ddx(tanx)=sec2x

ddx(sin−1x)=11−x2−−−−−√

ddx(cos−1x)=−11−x2−−−−−√

ddx(secx)=secxtanx

ddx(cscx)=−cscxcotx

ddx(cotx)=−csc2x

(c) perform implicit differentiation;

C'p&icit and Imp&icit \ierentiation

There are two wa"s to de#ne functions: implicitly and e#plicitly. ?ost of the equations we have dea&t with have been e'p&icit equations: such as " NO'PK. This is ca&&ed e#plicit because given an ': "ou can direct&" get f'!.

The technique of implicit di$erentiation a&&ows "ou to #nd the derivative of " with respect to ' without having to so&ve the given equation for ". Ziven

x2+y2=10

=erforming a chain ru&e to get d"Ud': so&ve d"Ud' in terms of ' and ".9ind d"Ud' for

x2+y2=10

2x+2ydydx=0

2ydydx=−2x

dydx=−xy

(d) find the first derivatives of functions defined parametrically;

First "eri6ati6e o Parametric Functions

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Parametric deri6ati6e is a derivative in calculus that is ta"en when both the x and yvariables (independent and dependent, respectively) depend on an independent thirdvariable t, usually thought of as CtimeC.

The irst deri6ati6e o the parametric e;uations is

dydx=dydtdxdt

dydx=dydt.dtdx

x=f(t),y=g(t)

E3amp4e# 8ind the first derivative, given

x=t+cost

y=sint

dydx=dydtdxdt=cost1−sint

E3amp4e# 8ind the first derivative, given

x=t4−4t2

y=t3

dydx=dydtdxdt=2t23t3−8t

.* App4ications o dierentiation

(e) determine where a function is increasing, decreasing, concave upward and concavedownward

here is a unction increasin! or decreasin!2

• Increasin! unction# if f :(x)F, function is increasing.

• "ecreasin! unction# if f :(x)N, function is decreasing

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here is unction conca6e up or conca6e do


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C'tremum =oints

?a'ima and minima are points where a function reaches a highest or

&owest va&ue: respective&"g!

h!i!

Second %erivative

If fLLL'! is positive: then it is a minimum point.If fLLL'! is ne&ative: then it is a ma#imum point.

If fLLL'!N ^ero: then it cou&d be a ma'imum: minimum or point of in_e'ion.=oint of In_e'ionAn in_ection point is a point on a curve at which the sign of thecurvature changes.

d2ydx2=0

'hird %erivative

If fLLL'! ` S There is an in(e#ion point


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(g) s"etch the graphs of functions, including asymptotes parallel to the coordinateaxes;

(h) find the e5uations of tangents and normals to curves, including parametric curves;

Tan!ents and orma4s to a Cur6e

#t point (x$,y$) on the curve y4f(x) the e;uation o tan!ent is

y−y1=m1(x−x1)where

m1=f′(x)=dydxthe gradient to the function of f(x).

Tangent is perpendicular to norma4, thus

m1m2=−1

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E3amp4e# 8ind the e5uations of the tangent line and the normal line for the curve at t4$.

x=t2

y=2t+1

dxdt=2t

dydt=2

dydx=dydt.dtdx

=2t2t4$

dydx=2Since t 4 $, x 4 $, y 4 ! A5uation of tangent

y−y1=m1(x−x1)

y−3=1(x−1)

y=x+2A5uation of normal

m1m2=−1

m2=−1y−3=−1(x−1)

y=−x+4

(i) solve problems concerning rates of change, including related rates;

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ates of 7hange and e&ated ates

e&ated rates prob&ems invo&ve #nding a rate at which a quantit" changesb" re&ating that quantit" to other quantities whose rates of change areMnown. The rate of change is usua&&" with respect to time.

E#ample) Air is being pumped into a spherica& ba&&oon such that itsradius increases at a rate of .RS cmUmin. 9ind the rate of change of itsvo&ume when the radius is cm.

The vo&ume

V=43πr3\ierentiating above equation with respect to t

dVdt=43π.3r2.drdt

dVdt=4πr2.drdt The rate of change of the radius drUdt N S.RS cmUmin because the radius isincreasing with respect to time.

dVdt=4π(5)2(0.80)

dVdt=80π cm3/min

Wence: the vo&ume is increasing at a rate of RS cmOUmin when the radiushas a &ength of cm.

(H) solve optimisation problems.

The differentiation and its applications can be used to solve practical problems. Thisinclude minimiing costs, maximiing areas, minimiing distances and so on.

$. "ia!ram - 6raw a diagram.'. oa4 - Oaximie or minimie which un"nown3!. "ata - 0ntroduce variable names.


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• Pse the Qrst or second derivative test to determine whether the critical points are localmaxima, local minima, or neither.

• Bhec" end points of f, if applicable.

Chapter Inte!ration

.1 Indeinite inte!ra4s

(a) identify integration as the reverse of differentiation;

(b) integrate xn (n $ Q ), e x , sin x, cos x, sec2 x, with constant multiples, sums anddifferences;

Integra& 7ommon 9unction

Constant

∫adx=ax+C*ariable

∫xdx=x22+CPo+er

∫xndx=xn+1n+1+Ceciprocal

∫1xdx=ln|x|+CE#ponential

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∫exdx=ex+C∫axdx=axlna+C∫ln(x)dx=x(ln(x)−1)+C'ri&onometry

∫cos(x)dx=sin(x)+C∫sin(x)dx=−cos(x)+C∫sec2(x)dx=tan(x)+C

(c) integrate rational functions by means of decomposition into partial fractions;

Inte!ration :5 Partia4 Fractions "ecomposition

Ho< to inte!rate the rationa4 unction( ;uotient o t


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∫1x2+5x+6dx8actor (x) into linear and&or 5uadratic (irreducible) factors Z find the partial fractiondecomposition

1(x+2)(x+3)=Ax+2+Bx+31=A(x+3)+B(x+2)

x=−2,A=1

x=−3,B=−1

1(x+2)(x+3)=1x+2−1x+30ntegrate the result

∫1x2+5x+6dx=∫1(x+2)(x+3)dx=∫1x+2−1x+3dx=ln|x+2|−ln|x+3|+C

E3amp4e *# Avaluate the indefinite integral

1(x−1)(x+2)(x+1)=Ax−1+Bx+2+Cx+1

6ecomposition of rational functions into partial fractions

1=A(x+2)(x+1)+B(x−1)(x+1)+C(x−1)(x+2)

x=1,6A=1,A=16

x=−1−2C=1,C=−12

x=−23B=1,B=13

1(x−1)(x+2)(x+1)=16

(1x−1

)+13

(1x+2

)−12

(1x+1

)0ntegrate the result

∫1(x−1)(x+2)(x+1)dx


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=∫16(1x−1)+13(1x+2)−12(1x+1)=16ln|x−1|+13ln|x+2|−12ln|x+1|+C

(d) use trigonometric identities to facilitate the integration of trigonometric functions;

(e) use algebraic and trigonometric substitutions to find integrals;

Inte!ration :5 Tri!onometric Su:stitution V Identities

Su:stitute one o the o44o


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.* "einite inte!ra4s

(g) identify a definite integral as the area under a curve;

(h) use the properties of definite integrals;

Properties o Inte!ra4s

Additi6e Properties

Split a definite integral up into two integrals with the same integrand but different limits

∫baf(x)dx+∫cbf(x)dx=∫caf(x)dx0f the upper and lower bound are the same, the area is .

∫aaf(x)dx=00f an interval is bac"wards, the area is the opposite sign.

∫ba

f(x)dx=−∫ab

f(x)dx

Inte!ra4 o Sum

The integral of a sum can be split up into two integrands

∫ba[f(x)+g(x)]dx=∫baf(x)dx+∫bag(x)dx

Sca4in! :5 a constant

Bonstants can be distributed out of the integrand and multiplied afterwards.

∫bacf(x)dx=c∫baf(x)dx

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Tota4 Area ithin an Inter6a4

∫baf(x)dx=F(b)−F(a)∫ba|f(x)|dx=F(b)+F(a)

Inte!ra4 ine;ua4ities

0f

f(x)≥0and a


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0f y 4 f(x) is continuous and f(x) N on [a, b\, then the area under the curve from a to bis%

Area=−∫baf(x)dx

0f y 4 f(x) is continuous and f(x) N and f(x) F

Area=∫ba

f(x)d(x)+333∫cb

f(x)d(x)333+∫dc

f(x)d(x)

0f x 4 g (y ) is continuous and non-negative on [c, d\, then the area under the curve of gfrom c to d is%

Area=∫dcg(y)dy


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(") calculate volumes of solids of revolution about one of the coordinate axes.

Chapter 1>

"ierentia4 E;uations

• (a) find the general solution of a first order differential e5uation with separable variables;

• (b) find the general solution of a first order linear differential e5uation by means of anintegrating factor;


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• (c) transform, by a given substitution, a first order differential e5uation into one withseparable

• variables or one which is linear;

• (d) use a boundary condition to find a particular solution;

• (e) solve problems, related to science and technology, that can be modelled by differentiale5uations.

Chapter 11

Mac4aurin Series


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(a) find the Oaclaurin series for a function and the interval of convergence;

?ac&aurin 8eries

'aylor series is a representation of a function as an in#nite sum of termsthat are ca&cu&ated from the va&ues of the functionLs derivatives at a sing&epoint.'aylor Series

f(x)=∑ x=0∞fn(a)n!(x−a)n=f(a)+f′(a)(x−a)+f′(a)2!(x−a)2+f′′(a)3!(x−a)3+...+f(n)(a)n!

(x−a)n+...

-acLaurin series is the Ta"&or series of the function about #./0

f(x)=∑ x=0∞fn(0)n!xnf(x)=f(0)+f′(0)x+f′′(0)2!x2+f′′′(0)3!x3+...+f(n)(0)n!xn+...

E#ample ) 9ind the ?ac&aurin 8eries e'pansion of e'

f(x)=e5x

f′(x)=5e5x

f′′(x)=52e5x

f′′′(x)=53e5x

f(4)(x)=54e5x

f(0)=1

f′(0)=5

f′′(0)=52

f′′′(0)=53

f(4)=54

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f(x)=f(0)+f′(0)x+f′′(0)2!x2+f′′′(0)3!x3+...+f(n)(0)n!xn+...

e5x=1+5x1!+52x22!+53x33!+...+5nxnn!

=∑ n=0∞5nxnn!

Inter6a4 o Con6er!ence

The set of points where the series converges is called the inter6a4 o con6er!enceFindin! inter6a4 o con6er!ence$) perform ratio test to test for the convergence of a series.') Bhec" endpoint

$atio test

L=limn→∞333an+1an333

if / N $ then the series converges.if / F $ then the series does not converge;if / 4 $ or the limit fails to exist, then the test is inconclusive

E3amp4e# 6etermine the interval of convergence for the series

∑ n=1∞

(x−2)

nn.5n

#pply ratio test

limn→∞333an+1an333=333(x−2)n+1(n+1).5n+1.n.5n(x−2)n333

=limn→∞333x−25.nn+1333

=15|x−2|limn→∞333nn+1333

=15|x−2|#s n approcaches infinity, n&n>$ aprpoaches $

limn→∞333nn+1333]1 The series converges for

15|x−2|


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Bhec" end point x4E,

∑ n=1∞(5)nn.5n=∑ n=1∞1nThis is the harmonic series, and it diverges.

Bhec" end point x4-!,∑ n=1∞(−5)nn.5n=∑ n=1∞(−1)n1nThe series converges by the #lternating Series Test.

The inter6a4 o con6er!ence is

−3≤x


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x4+x2*ewrite the function as

x4+x2=x(14+x2)=x4EF11+x24GH

=x4EF11−(−x24)GHJThis is the sum of the infinite geometric series with the first term x&7 and ratio x'&7

=x4∑ n=0∞(−x24)=x4∑ n=0∞(−1)nx2n4n=∑ n=0∞(−1)nx2n+14n+1

(c) perform differentiation and integration of a power series;

(d) use series expansions to find the limit of a function.

Chapter 1* umerica4 Method

1*.1 umerica4 so4ution o e;uations

(a) locate a root of an e5uation approximately by means of graphical considerations and by searching for a sign change;

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(b) use an iterative formula of the form xn>$ 4f(xn) to find a root of an e5uation to a prescribed degree of accuracy;

(c) identify an iteration which converges or diverges;

(d) use the 1ewton-*aphson method;

1*.* umerica4 inte!ration

(e) use the trapeium rule;

(f) use s"etch graphs to determine whether the trapeium rule gives an over-estimate oran under-estimate in simple cases.

Chapter 1'

"ata "escription

(a) identify discrete, continuous, ungrouped and grouped data;

"iscrete( Continuous( Un!rouped and rouped "ata

"iscrete "ata

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6iscrete 6ata is counted and can only ta"e certain values (whole numbers).Ag 1umber of students in a class, 1umber of children in a playground, etc.

Continuous "ata

Bontinuous 6ata is data that can ta"e any value (within a range)Ag. eight of children, weights of car, etc.

rouped "ata

6ata that has been organied into groups (into a fre5uency distribution).

Ag.Class

0−5

6−10

11−15

16−20

Frequency

10

20

17

4

Un!rouped "ata

6ata that has not been organied into groups$ $ ' 7 + 7 DE E+ $ $$ '! D Y + $

(b) construct and interpret stem-and-leaf diagrams, box-and-whis"er plots, histogramsand cumulative fre5uency curves;

Stem0and0Lea "ia!rams

# stem0and04ea dia!rams presents 5uantitative data in a graphical format, similar to ahistogram, to assist in visualiing the shape of a distribution, giving the reader a 5uic"overview of distribution.

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-, - - - ' , 9' 9' 9, 9 1 , 1>9

7 M D + Y M

D M ! D + +E M ! ! D+ M $ +$ M E"ey 7 M means 7

Bo30and0hiser P4ots

Bo30and0hiser P4ots displays of the spread of a set of data through five-number

summaries% the minimum, lower 5uartile ($), median ('), upper 5uartile(!), andmaximum.

• The first and third 5uartiles are at the ends of the box,

• The median is indicated with a vertical line in the interior of the box

• Ands of the whis"ers indicated the maximum and minimum.

Histo!rams and Cumu4ati6e Fre;uenc5 Histo!rams

Histo!rams and Cumu4ati6e Fre;uenc5 Histo!rams

# histo!ram is constructed from a fre5uency tableThe cumu4ati6e re;uenc5 is the running total of the fre5uencies.


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(c) state the mode and range of ungrouped data;

Mode o un!rouped data

#n observation occurring most fre5uently in the data is called mode of the data

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E3amp4e# 8ind the median of the following observations

$, $7, $D, ', '7, '+, '+, !, !', 7

0n the given data, the observation '+ occurs maximum. So the mode is *.

$an!e o un!rouped data

$an!e 7 Hi!hest Oa4ue ) Lo


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E3amp4e 1# 8ind the median, lower 5uartile and upper 5uartile of the following numbers.

1*( ,( **( '>( 9( '( 1( -*( 1,( -'( ',

*earrange the data in ascending order%

$ 44 $&7 ($$>$) 4 !th observation, $4$'' 44 $&' ($$>$) 4 Dth observation, '4''! 44 !&7 ($$>$) 4 Yth observation, !4!+

Inter;uarti4e $an!e 4 ! -$ 4 !+ - $' 4 'D$an!e 4 /argest value - smallest value 4 7! - 4 !+

E3amp4e *# 8ind the median, lower 5uartile and upper 5uartile of the following numbers.

1*( ,( **( '>( 9( '( 1( -*( 1,( -'( ',( ,>

*earrange the data in ascending order%

Q1=(12+152)=13.5Q2=(22+302)=26Q3=

(38+422

)=40

Inter;uarti4e $an!e 4 ! -$ 4 7 - $!. 4 'D.$an!e 4 /argest value - smallest value 4 - 4 7

Median and Inter;uarti4e $an!e o rouped "ata


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Median=Lm+(n2−Ffm)Cn 4 the total fre5uency/m 4 the lower boundary of the class median8 4 the cumulative fre5uency before classmedianf 4 the fre5uency of the class medianf m4 the lower boundary of the class medianB4 the class width

Q1=LQ1+(n4−FfQ1)CQ3=LQ3+EF3n4−FfQ3GHC

E3amp4e % 8ind the median and inter5uartile range of below data.

Q2=20.5+EF502−2312GH10=22.167

Q1=10.5+EF504−1010GH10=13

Q3=30.5+EF3(50)4−358GH10=33.625

(e) calculate the mean and standard deviation of ungrouped and grouped data, from rawdata and from given totals such as

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∑ i=1n(xi−a)and ∑ i=1n(xi−a)2

Mean and Standard "e6iation o Un!rouped and rouped "ata

Un!rouped "ata

Oean

x̄=∑xn

Standard 6eviation

s=∑(x−x̄)2n−−−−−−−−−√ s=∑x2n−(∑xn)2−−−−−−−−−−−−−− ⎷ ̂ ^

rouped "ata

Oean

x̄=∑fx∑f

Standard 6eviation

s=∑f(x−x̄)2n−−−−−−−−−−√ s=∑fx2∑f−(∑fx∑f)2−−−−−−−−−−−−−−−− ⎷ ̂ ^

(f) select and use the appropriate measures of central tendency and measures ofdispersion;

Measures o Centra4 Tendenc5

Oeasure of central tendency is an average of a set of measurements.

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• Mode 0 the number that occurs most fre5uently.

• Median 0 the value of the middle item in a set of observations.

• Mean 0 average value of the distribution.

Measures o "ispersion

Oeasures of 6ispersion is group of analytical tools that describes the spread or variability ofa data set.

• $an!e - 6ifference between the largest and smallest sample values.

• OarianceD Standard "e6iation - Oeasures the dispersion around an average.

• Coeicient o 6ariation - Axpressed in a relative value.

• (g) calculate the 9earson coefficient of s"ewness;

Pearson coeicient o se


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Positi6e se


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Addition Princip4e

0f we want to find the probability that event # happens or event ? happens, we shouldadd the probability that # happens to the probability that ? happens.

Addition $u4e#

P&A or B+ 7 P&A+ 8 P&B+

E3amp4e# # single D-sided die is rolled.


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7omp&ementar": C'haustive and ?utua&&" C'c&usive Cvents

7omp&ementar" Cvent

The comp&ement of an" event A: AL is the event that A does not occur.

C'haustive Cvent

A set of events is co&&ective&" e'haustive if at &east one of the events mustoccur. 9or e'amp&e: when ro&&ing a si'Psided die: the outcomes ,: K: O: : :and Q are co&&ective&" e'haustive: because the" inc&ude the entire range of possib&e outcomes. Thus: a&& samp&e spaces are co&&ective&" e'haustive.

P(AB)=1

-utually E#clusive Event

Two events are Lmutua&&" e'c&usiveL if the" cannot occur at the same time.C'amp&e: tossing a coin once: which can resu&t in either heads or tai&s: but notboth.

P(A∩B)=0

P(AB)=P(A)+P(B)


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(e) use the formula

Independent Cvents 7onditiona& =robabi&ities

Formu4a

P(AB)=P(A)+P(B)−P(A∩B)

Independent E6ents

Two events are independent means that the occurrence o one does not aect thepro:a:i4it5 o the other. Two events, # and ?, are independent if and only if

P(A∩B)=P(A).P(B)

Conditiona4 Pro:a:i4it5

Conditiona4 Pro:a:i4it5 is the probability that an event will occur, when another event is"nown to occur or to have occurred.

=iven two events # and ?, , the conditional probability of # given ? is defined as the5uotient of the Hoint probability of # and ?, and the probability of ?

P(A|B)=P(A∩B)P(B)

P(A∩B)=P(A|B).P(B)

P&A B+ 7 P&A+ 8 P&B+ 0 P&A ∩ B+W

• (f) calculate conditional probabilities, and identify independent events;

• (g) use the formulae

P&A ∩ B+ 7 P&A+ 3 P&BA+ 7 P&B+ 3 P&A B+W

(h) use the rule of total probability.

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The 9undamenta& Daws of 8et A&gebra

Commutative la+s

AB=BA

A∩B=B∩A

2ssociative La+s

(AB)C=A(BC)

(A∩B)∩C=A∩(B∩C)

%istributive La+s

A(B∩C)=(AB)∩(AC)

A∩(BC)=(A∩B)(A∩C)

3mpotent La+s

A∩A=A

AA=A

%omination La+s

AU=U

A∩=

2bsorption La+s

A(A∩B)=A

A∩(AB)=A

3nverse La+s

AA′=U

A∩A′=

• La+ of Complement

(A′)′=A

U′=


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′=U

• %e-or&an4s La+

(AB)′=A′∩B′

(A∩B)′=A′B′

• elative Complement of 5 in 2

A−B=A∩B′

Chapter 1, Pro:a:i4it5 "istri:utions


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1,.1 "iscrete random 6aria:4es

Var(X)=E(X2)−[E(X)]2

Var(X)=σ2x=∑ [x2i6P(xi)]−µ2x

• (a) identify discrete random variables;

• (b) construct a probability distribution table for a discrete random variable;

• (c) use the probability function and cumulative distribution function of a discrete randomvariable;

• (d) calculate the mean and variance of a discrete random variable;

"iscrete random 6aria:4es

# discrete 6aria:4e is a variable which can only ta"e a counta:4e num:er o 6a4ues.

# discrete random variable _ is uni5uely determined by

•

0ts set of possible values _

• 0ts probability density function (pdf)% # real-valued function f (`) deQned for each x$_ as the probability that _ has the value x.

• Pro:a:i4it5 "ensit5 Function

f(x)=Pr(X=x)

∑ x=inf(xi)=1• Cumu4ati6e "istri:ution Function 8(x) is defined to be

F(x)=P(X≤x)

http://coolmathsolutions.blogspot.com/2013/03/discrete-distribution-discrete-random.htmlhttp://coolmathsolutions.blogspot.com/2013/03/discrete-distribution-discrete-random.htmlhttp://coolmathsolutions.blogspot.com/2013/03/discrete-distribution-discrete-random.htmlhttp://coolmathsolutions.blogspot.com/2013/03/discrete-distribution-discrete-random.htmlhttp://coolmathsolutions.blogspot.com/2013/03/discrete-distribution-discrete-random.htmlhttp://coolmathsolutions.blogspot.com/2013/03/discrete-distribution-discrete-random.htmlhttp://coolmathsolutions.blogspot.com/2013/03/discrete-distribution-discrete-random.htmlhttp://coolmathsolutions.blogspot.com/2013/03/discrete-distribution-discrete-random.htmlhttp://coolmathsolutions.blogspot.com/2013/03/discrete-distribution-discrete-random.htmlhttp://coolmathsolutions.blogspot.com/2013/03/discrete-distribution-discrete-random.htmlhttp://coolmathsolutions.blogspot.com/2013/03/discrete-distribution-discrete-random.htmlhttp://coolmathsolutions.blogspot.com/2013/03/discrete-distribution-discrete-random.htmlhttp://coolmathsolutions.blogspot.com/2013/03/discrete-distribution-discrete-random.html


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"iscrete Pro:a:i4it5 "istri:ution

The probability that random variable _ will ta"e the value xi is denoted by p(xi) whereP(xi)=P(X=xi)

Mean and Oariance of a Discrete Random Variable

Oean

E(X)=µx=∑ [xi6P(xi)]ariance

1,.* Continuous random 6aria:4es

• (e) identify continuous random variables;

• (f) relate the probability density function and cumulative distribution function of acontinuous random variable;

• (g) use the probability density function and cumulative distribution function of acontinuous random variable;

• (h) calculate the mean and variance of a continuous random variable;

Continuous $andom Oaria:4es

# random variable _ is continuous if its set of possible values is an entire interval ofnumbers

Pro:a:i4it5 "ensit5 Function

http://coolmathsolutions.blogspot.com/2013/03/continuous-distribution-continuous.htmlhttp://coolmathsolutions.blogspot.com/2013/03/continuous-distribution-continuous.htmlhttp://coolmathsolutions.blogspot.com/2013/03/continuous-distribution-continuous.htmlhttp://coolmathsolutions.blogspot.com/2013/03/continuous-distribution-continuous.htmlhttp://coolmathsolutions.blogspot.com/2013/03/continuous-distribution-continuous.htmlhttp://coolmathsolutions.blogspot.com/2013/03/continuous-distribution-continuous.htmlhttp://coolmathsolutions.blogspot.com/2013/03/continuous-distribution-continuous.htmlhttp://coolmathsolutions.blogspot.com/2013/03/continuous-distribution-continuous.htmlhttp://coolmathsolutions.blogspot.com/2013/03/continuous-distribution-continuous.htmlhttp://coolmathsolutions.blogspot.com/2013/03/continuous-distribution-continuous.htmlhttp://coolmathsolutions.blogspot.com/2013/03/continuous-distribution-continuous.htmlhttp://coolmathsolutions.blogspot.com/2013/03/continuous-distribution-continuous.htmlhttp://coolmathsolutions.blogspot.com/2013/03/continuous-distribution-continuous.htmlhttp://coolmathsolutions.blogspot.com/2013/03/continuous-distribution-continuous.html


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/et _ be a continuous random variables. Then a pro:a:i4it5 distri:ution or pro:a:i4it5densit5 unction &pd+ o is a function f (x) such that for any two numbers a and b,

P(a≤X≤b)=

∫baf(x)dx

The graph of f is the density curve.

#rea of the region between the graph of f and the x U axis is e5ual to $.

The Cumu4ati6e "istri:ution Function

The cumulative distribution function, 8(x) for a continuous random variables _ is definedfor every number x by

F(x)=P(X≤x)=∫x−∞f(y)dy

P(a≤X≤b)=F(b)−F(a)8or each x, 8(x) is the area under the density curve to the left of x.

E3pected Oa4ue

The expected or mean value of a continuous random variables _ with pdf f(x) is

E(X)=µx=∫∞−∞x.f(x)dx

Oariance

The variance of continuous random variables _ with pdf f(x) is

σ2x=Var(X)=∫∞−∞(x−µ)2.f(x)dxE[(x−µ)2]


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Var(X)=E(X2)−[E(x)]2

1,.' Binomia4 distri:ution

• (i) use the probability function of a binomial distribution, and find its mean and variance;

• (H) use the binomial distribution as a model for solving problems related to science andtechnology;

Binomia4 "istri:ution

The :inomia4 distri:ution is used when there are exactly two mutually exclusive outcomes

of a trial. These outcomes are appropriately labeled CsuccessC and CfailureC.

0f the random variable _ follows the binomial distribution with parameters n and p, _ (?(n,9), the probability of getting exactly k successes in n trials is given by the probability massfunction

f(k;n, p)=Pr(X=k)=(nk) pk(1− p)n−k0f _ ?(n, p), _ is a binomially distributed random variable, then the expected value of_ is

Mean =npand the variance is

Variance =np(1− p)

n number of successes p is the probability of success in ?inomial 6istribution, assumes that p is fixed for alltrials.

1,.- Poisson distri:ution

• (") use the probability function of a 9oisson distribution, and identify its mean andvariance;

• (l) use the 9oisson distribution as a model for solving problems related to science andtechnology;

http://coolmathsolutions.blogspot.com/2013/03/binomial-distribution.htmlhttp://coolmathsolutions.blogspot.com/2013/03/binomial-distribution.htmlhttp://coolmathsolutions.blogspot.com/2013/03/binomial-distribution.htmlhttp://coolmathsolutions.blogspot.com/2013/03/poisson-distribution.htmlhttp://coolmathsolutions.blogspot.com/2013/03/poisson-distribution.htmlhttp://coolmathsolutions.blogspot.com/2013/03/poisson-distribution.htmlhttp://coolmathsolutions.blogspot.com/2013/03/poisson-distribution.htmlhttp://coolmathsolutions.blogspot.com/2013/03/poisson-distribution.htmlhttp://coolmathsolutions.blogspot.com/2013/03/binomial-distribution.htmlhttp://coolmathsolutions.blogspot.com/2013/03/binomial-distribution.htmlhttp://coolmathsolutions.blogspot.com/2013/03/binomial-distribution.htmlhttp://coolmathsolutions.blogspot.com/2013/03/poisson-distribution.htmlhttp://coolmathsolutions.blogspot.com/2013/03/poisson-distribution.htmlhttp://coolmathsolutions.blogspot.com/2013/03/poisson-distribution.htmlhttp://coolmathsolutions.blogspot.com/2013/03/poisson-distribution.html


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=oisson \istribution

Poisson distribution is a discrete probabi&it" distribution that e'pressesthe probabi&it" of a given number of events occurring in a #'ed interva& of time andUor space if these events occur with a Mnown average rate and

independent&" of the time since the &ast event.A discrete stochastic variab&e is said to have a =oisson distribution withparameter S: if for M N S: ,: K: ... the probabi&it" mass function of isgiven b"

f(k;λ)=Pr(X=k)=λke−λk!

is the number of occurrences of an eventj the probabi&it" of which is givenb" the function is a positive rea& number.

?ean and


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The norma4 distri:ution is a continuous probability distribution, defined by the formula

f(x)=1σ2π−−√e−(x−µ)22σ2The normal distribution is also often denoted

Xk N(µ,σ2)

Standard orma4 "istri:ution

0f 4 and 4 $, the distribution is called the standard norma4 distri:ution.

8ormula for -score%

z=x−µσ

• is the C-scoreC (Standard Score)

• x is the value to be standardied

• is the mean

• is the standard deviation


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orma4 Appro3imations

Binomia4 Appro3imation

The normal distribution can be used as an approximation to the binomial distribution,under certain circumstances, namely%

0f _ ?(n, p) and if n is 4ar!e and&or p is c4ose to X, then _ is approximately 1(np,np5)


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Poisson Appro3imation

The normal distribution can also be used to approximate the 9oisson distribution for4ar!e 6a4ues o (the mean of the 9oisson distribution).

0f _ 9o() then for large values of l, _ 1(, ) approximately.


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Chapter 1

Samp4in! and Estimation

1.1 Samp4in!

• (a) distinguish between a population and a sample, and between a parameter and astatistic;

• (b) identify a random sample;

• (c) identify the sampling distribution of a statistic;

• (d) determine the mean and standard deviation of the sample mean;

• (e) use the result that _ has a normal distribution if _ has a normal distribution;

• (f) use the central limit theorem;

• (g) determine the mean and standard deviation of the sample proportion;

• (h) use the approximate normality of the sample

MATHEMATICS T (1).docx

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