t n te n Co 1 c i p o T -...

Mathematics SL guide17

Syllabus

Syllabus contentS

yllab

us co

nten

t

Math

ematics S

L g

uid

e 1

Topic 1—

Algebra

9 hours T

he aim

of th

is topic is to

intro

du

ce stud

ents to

som

e basic alg

ebraic co

ncep

ts and

app

lication

s.

Content

Further guidance Links

1.1

Arith

metic seq

uen

ces and

series; sum

of fin

ite arith

metic series; geo

metric seq

uen

ces and series;

sum

of fin

ite and in

finite geo

metric series.

Sig

ma n

otatio

n.

Tech

nolo

gy m

ay b

e used

to g

enerate an

d

disp

lay seq

uen

ces in sev

eral way

s.

Lin

k to

2.6

, exp

onen

tial functio

ns.

Int: Th

e chess leg

end

(Sissa ib

n D

ahir).

Int: Ary

abh

atta is som

etimes co

nsid

ered th

e “fath

er of alg

ebra”. C

om

pare w

ith

al-Kh

awarizm

i.

TOK

: Ho

w d

id G

auss ad

d u

p in

tegers fro

m

1 to

100

? Discu

ss the id

ea of m

athem

atical in

tuitio

n as th

e basis fo

r form

al pro

of.

TOK

: Deb

ate over th

e valid

ity o

f the n

otio

n o

f “in

finity

”: finitists su

ch as L

. Kro

neck

er co

nsid

er that “a m

athem

atical object d

oes n

ot

exist u

nless it can

be co

nstru

cted fro

m n

atural

nu

mb

ers in a fin

ite nu

mb

er of step

s”.

TOK

: What is Z

eno’s d

icho

tom

y p

arado

x?

Ho

w far can

math

ematical facts b

e from

in

tuitio

n?

A

pp

lications.

Ex

amp

les inclu

de co

mp

ou

nd

interest an

d

po

pu

lation

gro

wth

.

Mathematics SL guide 18

Syllabus contentSyllabus content

Mathem

atics SL guide 2

Content


1.2 Elem

entary treatment of exponents and

logarithms.

Examples:

3416

8

; 16

3log

84

;

log325log2

;

43

12(2

)2

��

.

Appl: C

hemistry 18.1 (C

alculation of pH ).

TOK

: Are logarithm

s an invention or discovery? (This topic is an opportunity for teachers to generate reflection on “the nature of m

athematics”.)

Laws of exponents; law

s of logarithms.

Change of base.

Examples:

4

ln7log

ln4

7

,

2555

loglog

log125

3125

252

§

· ¨

¸©

¹ .

Link to 2.6, logarithmic functions.

1.3 The binom

ial theorem:

expansion of ()

, nab

n�

�`

.

Counting principles m

ay be used in the developm

ent of the theorem.

Aim

8: Pascal’s triangle. Attributing the origin

of a mathem

atical discovery to the wrong

mathem

atician.

Int: The so-called “Pascal’s triangle” was

known in C

hina much earlier than Pascal.

Calculation of binom

ial coefficients using

Pascal’s triangle andnr §·

¨¸

©¹ .

nr §·

¨¸

©¹ should be found using both the form

ula

and technology.

Example: finding

6r §·

¨¸

©¹ from

inputting

6n

ry

CX

and then reading coefficients from

the table.

Link to 5.8, binomial distribution.

Not required:

formal treatm

ent of permutations and form

ula for

nrP

.


Syllabus content

Syllabus content

Mathem

atics SL guide 3

Topic 2—

Functions and equations 24 hours

The aims of this topic are to explore the notion of a function as a unifying them

e in mathem

atics, and to apply functional methods to a variety of

mathem

atical situations. It is expected that extensive use will be m

ade of technology in both the development and the application of this topic, rather than

elaborate analytical techniques. On exam

ination papers, questions may be set requiring the graphing of functions that do not explicitly appear on the

syllabus, and students may need to choose the appropriate view

ing window

. For those functions explicitly mentioned, questions m

ay also be set on com

position of these functions with the linear function y

axb

�

.

Content


2.1 C

oncept of function :

()

fx

fx

6.

Dom

ain, range; image (value).

Example: for

2x

x�

6, dom

ain is 2

xd

, range is

0yt

.

A graph is helpful in visualizing the range.

Int: The development of functions, Rene

Descartes (France), G

ottfried Wilhelm

Leibniz (G

ermany) and Leonhard Euler (Sw

itzerland).

Com

posite functions. �

� ()

((

))fgx

fgx

D

. TO

K: Is zero the sam

e as “nothing”?

TOK

: Is mathem

atics a formal language?

Identity function. Inverse function1

f�

. 1

1(

)()

()(

)ff

xf

fx

x�

�

D

D.

On exam

ination papers, students will only be

asked to find the inverse of a one-to-one function. N

ot required: dom

ain restriction.

2.2 The graph of a function; its equation

()

yfx

.

A

ppl: Chem

istry 11.3.1 (sketching and interpreting graphs); geographic skills.

TOK

: How

accurate is a visual representation of a m

athematical concept? (Lim

its of graphs in delivering inform

ation about functions and phenom

ena in general, relevance of modes of

representation.)

Function graphing skills.

Investigation of key features of graphs, such as m

aximum

and minim

um values, intercepts,

horizontal and vertical asymptotes, sym

metry,

and consideration of domain and range.

Note the difference in the com

mand term

s “draw

” and “sketch”.

Use of technology to graph a variety of

functions, including ones not specifically m

entioned.

An analytic approach is also expected for

simple functions, including all those listed

under topic 2.

The graph of 1(

)y

fx

�

as the reflection in the line y

x

of the graph of (

)y

fx

.

Link to 6.3, local maxim

um and m

inimum

points.



Mathem

atics SL guide 4

Content


2.3 Transform

ations of graphs. Technology should be used to investigate these transform

ations. A

ppl: Economics 1.1 (shifting of supply and

demand curves).

Translations: (

)y

fx

b

�;

()

yfxa

�

.

Reflections (in both axes):

()

yfx

�

; (

)y

fx

�

.

Vertical stretch w

ith scale factor p: (

)y

pfx

.

Stretch in the x-direction with scale factor 1q

:

��

yfqx

.

Translation by the vector 32

§·

¨¸

�©¹ denotes

horizontal shift of 3 units to the right, and vertical shift of 2 dow

n.

Com

posite transformations.

Example:

2yx

used to obtain

23

2y

x

� by

a stretch of scale factor 3 in the y-direction

followed by a translation of

02 §·

¨¸

©¹ .

2.4 The quadratic function

2x

axbx

c�

�6

: its graph, y-intercept (0,

)c . Axis of sym

metry.

The form

()(

)x

axpxq

��

6,

x-intercepts (,0)p

and (,0)q

.

The form

2(

)x

axh

k�

�6

, vertex (,

)hk

.

Candidates are expected to be able to change

from one form

to another.

Links to 2.3, transformations; 2.7, quadratic

equations.

Appl: C

hemistry 17.2 (equilibrium

law).

Appl: Physics 2.1 (kinem

atics).

Appl: Physics 4.2 (sim

ple harmonic m

otion).

Appl: Physics 9.1 (H

L only) (projectile m

otion).



Mathem

atics SL guide 5

Content


2.5 The reciprocal function

1x

x6

, 0

xz

: its

graph and self-inverse nature.

The rational function ax

bx

cxd

��6

and its

graph.

Examples:

42

()

, 3

23

hx

xx

z

�;

75

, 2

52

xy

xx �

z

�.

Vertical and horizontal asym

ptotes. D

iagrams should include all asym

ptotes and intercepts.

2.6 Exponential functions and their graphs:

xx

a6

, 0

a!

, ex

x6

.

Int: The B

abylonian method of m

ultiplication: 2

22

()

2ab

ab

ab�

��

. Sulba Sutras in ancient

India and the Bakhshali M

anuscript contained an algebraic form

ula for solving quadratic equations.

Logarithmic functions and their graphs:

loga

xx

6,

0x!

, ln

xx

6,

0x!

.

Relationships betw

een these functions: ln

ex

xa

a

; logx

a ax

;

logax

ax

,

0x!

.

Links to 1.1, geometric sequences; 1.2, law

s of exponents and logarithm

s; 2.1, inverse functions; 2.2, graphs of inverses; and 6.1, lim

its.


Syllabus contentS

ylla

bu

s c

onte

nt

Math

em

atic

s S

L g

uid

e

6

Content


2.7

Solv

ing e

quatio

ns, b

oth

gra

phic

ally

and

analy

tically

.

Use o

f technolo

gy to

solv

e a

varie

ty o

f

equatio

ns, in

clu

din

g th

ose w

here

there

is n

o

appro

pria

te a

naly

tic a

ppro

ach.

Solu

tions m

ay b

e re

ferre

d to

as ro

ots

of

equatio

ns o

r zero

s o

f functio

ns.

Lin

ks to

2.2

, functio

n g

raphin

g s

kills

; and 2

.3–

2.6

, equatio

ns in

volv

ing s

pecific

functio

ns.

Examples

: 4

56

0e

sin

,x

xx

x�

�

.

Solv

ing

20

axbx

c�

�

, 0

az

.

The q

uadra

tic fo

rmula

.

The d

iscrim

inant

24

bac

'

� a

nd th

e n

atu

re

of th

e ro

ots, th

at is

, two d

istin

ct re

al ro

ots

, two

equal re

al ro

ots

, no re

al ro

ots

.

Example

: Fin

d k

giv

en th

at th

e e

quatio

n

23

20

kxxk

��

h

as tw

o e

qual re

al ro

ots

.

Solv

ing e

xponentia

l equatio

ns.

Examples

: 1

210

x�

, 1

19

3

xx�

§·

¨¸

©¹

.

Lin

k to

1.2

, exponents

and lo

garith

ms.

2.8

Applic

atio

ns o

f gra

phin

g s

kills

and s

olv

ing

equatio

ns th

at re

late

to re

al-life

situ

atio

ns.

Lin

k to

1.1

, geom

etric

serie

s.

Appl: C

om

pound in

tere

st, gro

wth

and d

ecay;

pro

jectile

motio

n; b

rakin

g d

istance; e

lectric

al

circ

uits

.

Appl: P

hysic

s 7

.2.7

–7.2

.9, 1

3.2

.5, 1

3.2

.6,

13.2

.8 (ra

dio

activ

e d

ecay a

nd h

alf-life

)



Mathem

atics SL guide 7

Topic 3—

Circular functions and

trigonometry

16 hours The aim

s of this topic are to explore the circular functions and to solve problems using trigonom

etry. On exam

ination papers, radian measure should be

assumed unless otherw

ise indicated.

Content


3.1 The circle: radian m

easure of angles; length of an arc; area of a sector.

Radian m

easure may be expressed as exact

multiples of ʌ

, or decimals.

Int: Seki Takakazu calculating ʌ to ten

decimal places.

Int: Hipparchus, M

enelaus and Ptolemy.

Int: Why are there 360 degrees in a com

plete turn? Links to B

abylonian mathem

atics.

TOK

: Which is a better m

easure of angle: radian or degree? W

hat are the “best” criteria by w

hich to decide?

TOK

: Euclid’s axioms as the building blocks

of Euclidean geometry. Link to non-Euclidean

geometry.

3.2 D

efinition of cosT and sinT

in terms of the

unit circle.

Aim

8: Who really invented “Pythagoras’

theorem”?

Int: The first work to refer explicitly to the

sine as a function of an angle is the A

ryabhatiya of Aryabhata (ca. 510).

TOK

: Trigonometry w

as developed by successive civilizations and cultures. H

ow is

mathem

atical knowledge considered from

a sociocultural perspective?

Definition of tanT

as sincos TT

. The equation of a straight line through the origin is

tany

xT

.

Exact values of trigonometric ratios of

ʌʌʌʌ

0,,

,,

64

32

and their multiples.

Examples:

ʌ�

�ʌ�

�sin

,cos

,tan210

32

43

2

�

q .



Mathem

atics SL guide 8

Content


3.3 The Pythagorean identity

22

cossin

1T

T�

.

Double angle identities for sine and cosine.

Simple geom

etrical diagrams and/or

technology may be used to illustrate the double

angle formulae (and other trigonom

etric identities).

Relationship betw

een trigonometric ratios.

Examples:

Given sin T

, finding possible values of tanT

without finding T

.

Given

3cos

4x

, and x is acute, find sin2x

without finding x.

3.4 The circular functions sin

x, cosx and tan

x:

their domains and ranges; am

plitude, their periodic nature; and their graphs.

Appl: Physics 4.2 (sim

ple harmonic m

otion).

Com

posite functions of the form

��

()

sin(

)fx

abxc

d

��

. Exam

ples:

()

tan4

fx

xS

§·

�

¨¸

©¹ ,

��

()

2cos3(

4)1

fx

x

��

.

Transformations.

Example:

siny

x

used to obtain 3sin

2y

x

by a stretch of scale factor 3 in the y-direction

and a stretch of scale factor 12 in the

x-direction.

Link to 2.3, transformation of graphs.

Applications.

Examples include height of tide, m

otion of a Ferris w

heel.



Mathem

atics SL guide 9

Content


3.5 Solving trigonom

etric equations in a finite interval, both graphically and analytically.

Examples: 2sin

1x

, 02ʌ

xd

d,

2sin23cos

xx

,

oo

0180

xd

d,

��

2tan3(

4)1

x�

,

ʌ�ʌ

x�

dd

.

Equations leading to quadratic equations in sin

,cos

ortan

xx

x .

Not required:

the general solution of trigonometric equations.

Examples:

22sin

5cos1

0x

x�

�

for 04

xd

�S

,

2sincos2

xx

,

ʌʌ

x�

dd

.

3.6 Solution of triangles.

Pythagoras’ theorem is a special case of the

cosine rule. A

im 8: A

ttributing the origin of a m

athematical discovery to the w

rong m

athematician.

Int: Cosine rule: A

l-Kashi and Pythagoras.

The cosine rule.

The sine rule, including the ambiguous case.

Area of a triangle, 1

sin2

abC

.

Link with 4.2, scalar product, noting that:

22

22

�

�

��

�cab

ca

bab

.

Applications.

Examples include navigation, problem

s in two

and three dimensions, including angles of

elevation and depression.

TOK

: Non-Euclidean geom

etry: angle sum on

a globe greater than 180°.



Mathem

atics SL guide 10

Topic 4—Vectors

16 hours The aim

of this topic is to provide an elementary introduction to vectors, including both algebraic and geom

etric approaches. The use of dynamic geom

etry softw

are is extremely helpful to visualize situations in three dim

ensions.

Content


4.1 V

ectors as displacements in the plane and in

three dimensions.

Link to three-dimensional geom

etry, x, y and z-axes.

Appl: Physics 1.3.2 (vector sum

s and differences) Physics 2.2.2, 2.2.3 (vector resultants).

TOK: H

ow do w

e relate a theory to the author? W

ho developed vector analysis: JW

Gibbs or O

Heaviside?

Com

ponents of a vector; column

representation; 12

12

3

3

vvv

vv

v §·

¨¸

��

¨¸

¨¸

©¹

vi

jk

.

Com

ponents are with respect to the unit

vectors i, j and k (standard basis).

Algebraic and geom

etric approaches to the follow

ing: A

pplications to simple geom

etric figures are essential.

� the sum

and difference of two vectors; the

zero vector, the vector �v ; The difference of v

and w is

()

�

��

vw

vw

. Vector sum

s and differences can be represented by the diagonals of a parallelogram

.

� m

ultiplication by a scalar, kv ; parallel vectors;

Multiplication by a scalar can be illustrated by

enlargement.

� m

agnitude of a vector, v;

� unit vectors; base vectors; i, j and k;

� position vectors O

A o

a

;

� A

BO

BO

Ao

oo

�

�ba

. D

istance between points A

and B is the

magnitude of A

B o

.



Mathem

atics SL guide 11

Content


4.2 The scalar product of tw

o vectors. The scalar product is also know

n as the “dot product”.

Link to 3.6, cosine rule.

Perpendicular vectors; parallel vectors. For non-zero vectors,

0�

vw

is equivalent to the vectors being perpendicular.

For parallel vectors, k

w

v,

�

vw

vw

.

The angle between tw

o vectors.

4.3 V

ector equation of a line in two and three

dimensions:

t

�ra

b.

Relevance of a

(position) and b (direction).

Interpretation of t as time and b

as velocity, w

ith b representing speed.

Aim

8: Vector theory is used for tracking

displacement of objects, including for peaceful

and harmful purposes.

TOK

: Are algebra and geom

etry two separate

domains of know

ledge? (Vector algebra is a

good opportunity to discuss how geom

etrical properties are described and generalized by algebraic m

ethods.)

The angle between tw

o lines.

4.4 D

istinguishing between coincident and parallel

lines.

Finding the point of intersection of two lines.

Determ

ining whether tw

o lines intersect.



Mathem

atics SL guide 12

Topic 5—Statistics and probability

35 hours The aim

of this topic is to introduce basic concepts. It is expected that most of the calculations required w

ill be done using technology, but explanations of calculations by hand m

ay enhance understanding. The emphasis is on understanding and interpreting the results obtained, in context. Statistical tables w

ill no longer be allow

ed in examinations. W

hile many of the calculations required in exam

inations are estimates, it is likely that the com

mand term

s “write dow

n”, “find” and “calculate” w

ill be used.

Content


5.1 C

oncepts of population, sample, random

sam

ple, discrete and continuous data.

Presentation of data: frequency distributions (tables); frequency histogram

s with equal class

intervals;

Continuous and discrete data.

Appl: Psychology: descriptive statistics,

random sam

ple (various places in the guide).

Aim

8: Misleading statistics.

Int: The St Petersburg paradox, Chebychev,

Pavlovsky. box-and-w

hisker plots; outliers. O

utlier is defined as more than 1.5

IQR

u from

the nearest quartile.

Technology may be used to produce

histograms and box-and-w

hisker plots.

Grouped data: use of m

id-interval values for calculations; interval w

idth; upper and lower

interval boundaries; modal class.

Not required:

frequency density histograms.



Mathem

atics SL guide 13

Content


5.2 Statistical m

easures and their interpretations.

Central tendency: m

ean, median, m

ode.

Quartiles, percentiles.

On exam

ination papers, data will be treated as

the population.

Calculation of m

ean using formula and

technology. Students should use mid-interval

values to estimate the m

ean of grouped data.

Appl: Psychology: descriptive statistics

(various places in the guide).

Appl: Statistical calculations to show

patterns and changes; geographic skills; statistical graphs.

Appl: B

iology 1.1.2 (calculating mean and

standard deviation ); Biology 1.1.4 (com

paring m

eans and spreads between tw

o or more

samples).

Int: Discussion of the different form

ulae for variance.

TOK

: Do different m

easures of central tendency express different properties of the data? A

re these measures invented or

discovered? Could m

athematics m

ake alternative, equally true, form

ulae? What does

this tell us about mathem

atical truths?

TOK

: How

easy is it to lie with statistics?

Dispersion: range, interquartile range,

variance, standard deviation.

Effect of constant changes to the original data.

Calculation of standard deviation/variance

using only technology.

Link to 2.3, transformations.

Examples:

If 5 is subtracted from all the data item

s, then the m

ean is decreased by 5, but the standard deviation is unchanged.

If all the data items are doubled, the m

edian is doubled, but the variance is increased by a factor of 4.

Applications.

5.3 C

umulative frequency; cum

ulative frequency graphs; use to find m

edian, quartiles, percentiles.

Values of the m

edian and quartiles produced by technology m

ay be different from those

obtained from a cum

ulative frequency graph.



Mathem

atics SL guide 14

Content


5.4 Linear correlation of bivariate data.

Independent variable x, dependent variable y. A

ppl: Chem

istry 11.3.3 (curves of best fit).

Appl: G

eography (geographic skills).

Measures of correlation; geographic skills.

Appl: B

iology 1.1.6 (correlation does not im

ply causation).

TOK

: Can w

e predict the value of x from y,

using this equation?

TOK

: Can all data be m

odelled by a (known)

mathem

atical function? Consider the reliability

and validity of mathem

atical models in

describing real-life phenomena.

Pearson’s product–mom

ent correlation coefficient r.

Technology should be used to calculate r. H

owever, hand calculations of r m

ay enhance understanding.

Positive, zero, negative; strong, weak, no

correlation.

Scatter diagrams; lines of best fit.

The line of best fit passes through the mean

point.

Equation of the regression line of y on x.

Use of the equation for prediction purposes.

Mathem

atical and contextual interpretation.

Not required:

the coefficient of determination R

2.

Technology should be used find the equation.

Interpolation, extrapolation.

5.5 C

oncepts of trial, outcome, equally likely

outcomes, sam

ple space (U) and event.

The sample space can be represented

diagramm

atically in many w

ays. TO

K: To w

hat extent does mathem

atics offer m

odels of real life? Is there always a function

to model data behaviour?

The probability of an event A is (

)P(

)(

)n

AA

nU

.

The complem

entary events A and A c (not A).

Use of V

enn diagrams, tree diagram

s and tables of outcom

es.

Experiments using coins, dice, cards and so on,

can enhance understanding of the distinction betw

een (experimental) relative frequency and

(theoretical) probability.

Simulations m

ay be used to enhance this topic.

Links to 5.1, frequency; 5.3, cumulative

frequency.


Syllabus content

Syllabus content

Mathem

atics SL guide 15

Content


5.6 C

ombined events, P(

)A

B�

.

Mutually exclusive events: P(

)0

AB

�

.

Conditional probability; the definition

��

P()

P|

P()

AB

AB

B �

.

Independent events; the definition �

��

�P

|P(

)P

|AB

AABc

.

Probabilities with and w

ithout replacement.

The non-exclusivity of “or”.

Problems are often best solved w

ith the aid of a V

enn diagram or tree diagram

, without explicit

use of formulae.

Aim

8: The gambling issue: use of probability

in casinos. Could or should m

athematics help

increase incomes in gam

bling?

TOK

: Is mathem

atics useful to measure risks?

TOK

: Can gam

bling be considered as an application of m

athematics? (This is a good

opportunity to generate a debate on the nature, role and ethics of m

athematics regarding its

applications.)

5.7 C

oncept of discrete random variables and their

probability distributions. Sim

ple examples only, such as: 1

P()

(4)

18X

xx

� for

^`

1,2,3x�

;

56

7P(

),

,18

1818

Xx

.

Expected value (mean), E(

)X

for discrete data.

Applications.

E()

0X

indicates a fair gam

e where X

represents the gain of one of the players.

Examples include gam

es of chance.



Mathem

atics SL guide 16

Content


5.8 B

inomial distribution.

Mean and variance of the binom

ial distribution.

Not required:

formal proof of m

ean and variance.

Link to 1.3, binomial theorem

.

Conditions under w

hich random variables have

this distribution.

Technology is usually the best way of

calculating binomial probabilities.

5.9 N

ormal distributions and curves.

Standardization of normal variables (z-values,

z-scores).

Properties of the normal distribution.

Probabilities and values of the variable must be

found using technology.

Link to 2.3, transformations.

The standardized value (z ) gives the number

of standard deviations from the m

ean.

Appl: B

iology 1.1.3 (links to normal

distribution).

Appl: Psychology: descriptive statistics

(various places in the guide).



Mathem

atics SL guide 17

Topic 6—C

alculus 40 hours

The aim of this topic is to introduce students to the basic concepts and techniques of differential and integral calculus and their applications.

Content


6.1 Inform

al ideas of limit and convergence.

Example: 0.3, 0.33, 0.333, ... converges to 13

.

Technology should be used to explore ideas of lim

its, numerically and graphically.

Appl: Econom

ics 1.5 (marginal cost, m

arginal revenue, m

arginal profit).

Appl: C

hemistry 11.3.4 (interpreting the

gradient of a curve).

Aim

8: The debate over whether N

ewton or

Leibnitz discovered certain calculus concepts.

TOK

: What value does the know

ledge of lim

its have? Is infinitesimal behaviour

applicable to real life?

TOK

: Opportunities for discussing hypothesis

formation and testing, and then the form

al proof can be tackled by com

paring certain cases, through an investigative approach.

Limit notation.

Example:

23

lim1

x

xxof

��§

·¨

¸©

¹

Links to 1.1, infinite geometric series; 2.5–2.7,

rational and exponential functions, and asym

ptotes.

Definition of derivative from

first principles as

0

()

()

()

limhfxh

fx

fx

ho

��

§·

c

¨¸

©¹ .

Use of this definition for derivatives of sim

ple polynom

ial functions only.

Technology could be used to illustrate other derivatives.

Link to 1.3, binomial theorem

.

Use of both form

s of notation, dd yx and

��

fx

c,

for the first derivative.

Derivative interpreted as gradient function and

as rate of change. Identifying intervals on w

hich functions are increasing or decreasing.

Tangents and normals, and their equations.

Not required:

analytic methods of calculating lim

its.

Use of both analytic approaches and

technology.

Technology can be used to explore graphs and their derivatives.



Mathem

atics SL guide 18

Content


6.2 D

erivative of (

)nxn�_

, sinx

, cosx , tanx

, ex and ln

x .

Differentiation of a sum

and a real multiple of

these functions.

The chain rule for composite functions.

The product and quotient rules.

Link to 2.1, composition of functions.

Technology may be used to investigate the chain

rule.

The second derivative. U

se of both forms of notation,

2

2

ddyx

and (

)fx

cc.

Extension to higher derivatives. dd

n

n yx and

��(

)nf

x.



Mathem

atics SL guide 19

Content


6.3 Local m

aximum

and minim

um points.

Testing for maxim

um or m

inimum

.

Using change of sign of the first derivative and

using sign of the second derivative.

Use of the term

s “concave-up” for (

)0

fx

cc!

, and “concave-dow

n” for (

)0

fx

cc�

.

Appl: profit, area, volum

e.

Points of inflexion with zero and non-zero

gradients. A

t a point of inflexion , (

)0

fx

cc

and changes sign (concavity change).

()

0fx

cc

is not a sufficient condition for a point of inflexion: for exam

ple, 4

yx

at (0,0) .

Graphical behaviour of functions,

including the relationship between the

graphs of f, f c and f cc.

Optim

ization.

Both “global” (for large

x) and “local”

behaviour.

Technology can display the graph of a derivative w

ithout explicitly finding an expression for the derivative.

Use of the first or second derivative test to

justify maxim

um and/or m

inimum

values.

Applications.

Not required:

points of inflexion where

()

fx

cc is not defined:

for example,

13

yx

at (0,0) .

Examples include profit, area, volum

e.

Link to 2.2, graphing functions.



Mathem

atics SL guide

20

Content


6.4 Indefinite integration as anti-differentiation.

Indefinite integral of (

)nxn�_

, sinx

, cosx,

1x and e

x.

1d

lnx

xC

x

�³

, 0

x!

.

The com

posites of any of these with the linear

function axb

�.

Example:

1(

)cos(2

3)(

)sin(2

3)2

fx

xfx

xC

c

��

�

�.

Integration by inspection, or substitution of the

form

((

))'(

)dfgxgxx

³.

Examples:

��

42

22

1d

,sin

d,

dsin

cosxx

xx

xx

xxx

�³

³³

.

6.5 A

nti-differentiation with a boundary condition

to determine the constant term

. Exam

ple:

if 2

d3

d yx

xx

� and

10y

when

0x

, then

32

110

2yx

x

��

.

Int: Successful calculation of the volume of

the pyramidal frustum

by ancient Egyptians

(Egyptian M

oscow papyrus).

Use of infinitesim

als by Greek geom

eters.

Definite integrals, both analytically and using

technology. (

)d(

)(

)ba g

xxgb

ga

c

�³

.

The value of som

e definite integrals can only be found using technology.

Accurate calculation of the volum

e of a cylinder by C

hinese mathem

atician Liu H

ui

Areas under curves (betw

een the curve and the x-axis).

Areas betw

een curves.

Volum

es of revolution about the x-axis.

Students are expected to first write a correct

expression before calculating the area.

Technology m

ay be used to enhance understanding of area and volum

e.

Int: Ibn Al H

aytham: first m

athematician to

calculate the integral of a function, in order to find the volum

e of a paraboloid.

6.6 K

inematic problem

s involving displacement s,

velocity v and acceleration a. dd s

vt

;

2

2

dd

dd

vs

at

t

.

Appl: Physics 2.1 (kinem

atics).

Total distance travelled.

Total distance travelled

21

dttvt

³.

t n te n Co 1 c i p o T -...

Documents

Transcript of t n te n Co 1 c i p o T -...