Mathematics Methods Units 1 and 2

Transcription

1 Mathematics Methds Units 1 and 2 Mathematics Methds is an ATAR curse which fcuses n the use f calculus and statistical analysis. The study f calculus prvides a basis fr understanding rates f change in the physical wrld, and includes the use f functins, their derivatives and integrals, in mdelling physical prcesses. The study f statistics develps students abilities t describe and analyse phenmena that invlve uncertainty and variatin. WACE Year 11 Syllabus Aims The Mathematics Methds ATAR curse aims t develp students : Understanding f cncepts and techniques drawn frm algebra, the study f functins, calculus and prbability and statistics; Ability t slve applied prblems using cncepts and techniques drawn frm algebra, functins, calculus, prbability and statistics; Reasning in mathematical and statistical cntexts and interpretatin f mathematical and statistical infrmatin, including ascertaining the reasnableness f slutins t prblems; Capacity t cmmunicate in a cncise and systematic manner using apprpriate mathematical and statistical language; and Capacity t chees and use technlgy apprpriately and efficiently. Overview f units Mathematics Methds Unit One Unit Tw Unit Three Unit Fur 1.1 Functins and graphs 1.2 Trignmetric functins 2.1 Expnential functins 2.2 Arithmetic and gemetric sequences and series 3.1 Further differentiatin and applicatins 4.1 The lgarithmic functin 3.2 Integrals 4.2 Cntinuus randm variables and the nrmal distributin 1.3 Cunting and prbability 2.3 Intrductin t differential calculus 3.3 Discrete randm variables 4.3 Interval estimates fr prprtins

2 Assessment table Year 11 Type f assessment Respnse Students respnd using knwledge f mathematical facts, cncepts and terminlgy, applying prblem-slving skills and. Respnse tasks can include: tests, assignments, quizzes and bservatin checklists. Tests are administered under cntrlled and timed cnditins. Investigatin Students plan, research, cnduct and cmmunicate the findings f an investigatin. They can investigate prblems t identify the underlying mathematics, r select, adapt and apply mdels and prcedures t slve prblems. This assessment type prvides fr the assessment f general inquiry skills, curse-related knwledge and skills, and mdelling skills. Evidence can include: bservatin and interview, written wrk r multimedia presentatins. Examinatin Students apply mathematical understanding and skills t analyse, interpret and respnd t questins and situatins. Examinatins prvide fr the assessment f cnceptual understandings, knwledge f mathematical facts and terminlgy, prblem-slving skills, and the use f algrithms. Examinatin questins can range frm thse f a rutine nature, assessing lwer level cncepts, thrugh t pen-ended questins that require respnses at the highest level f cnceptual thinking. Students can be asked questins f an investigative nature fr which they may need t cmmunicate findings, generalise, r make and test cnjectures. Typically cnducted at the end f each semester and/r unit. In preparatin fr Unit 3 and Unit 4, the examinatin shuld reflect the examinatin design brief included in the ATAR Year 12 syllabus fr this curse. Where a cmbined assessment utline is implemented, the Semester 2 examinatin shuld assess cntent frm bth Unit 1 and Unit 2. Hwever, the cmbined weighting f Semester 1 and Semester 2 shuld reflect the respective weightings f the curse cntent as a whle. Weighting 40% 20% 40% Number f assessments Respnse Test 1 Test 2 Test 3 Test 4 Investigatin Investigatin 1 Investigatin 2 Investigatin 3 Exams Semester 1 Exam Semester 2 Exam Weighting 10% 10% 10% 10% 6% 7% 7% 18% 22%

3 Prgramme Wk Descriptin Syllabus Cntent Reference Assessment 1 Csine and Sine Rules Trig Functins (Exact Trig Values) Csine and sine rules review sine, csine and tangent as ratis f side lengths in right-angled triangles understand the unit circle definitin f and and peridicity using degrees examine the relatinship between the angle f inclinatin f a line and the gradient f that line establish and use the csine and sine rules, including cnsideratin f the ambiguus case and the frmula fr the area f a triangle recgnise the exact values f at integer multiples f and Ch 1 and 2 Circular measure and radian measure Circular measure and radian measure define and use radian measure and understand its relatinship with degree measure calculate lengths f arcs and areas f sectrs and segments in circles Ch 2 3 Functins , Functins understand the cncept f a functin as a mapping between sets and as a rule r a frmula that defines ne variable quantity in terms f anther use functin ntatin; determine dmain and range; recgnise independent and dependent variables understand the cncept f the graph f a functin recgnise the distinctin between functins and relatins and apply the vertical line test Ch 3 and Lines and Linear Relatinships Quadratic Relatinships Lines and linear relatinships determine the crdinates f the mid-pint between tw pints determine an end-pint given the ther endpint and the mid-pint examine examples f direct prprtin and linearly related variables recgnise features f the graph f, including its linear nature, its intercepts and its slpe r gradient determine the equatin f a straight line given sufficient infrmatin; including parallel and perpendicular lines slve linear equatins, including thse with algebraic fractins and variables n bth sides Quadratic relatinships examine examples f quadratically related variables recgnise features f the graphs f, and, including their parablic nature, turning pints, axes f symmetry and intercepts Ch 4 and Ch 5 and Ch 6 TEST 1 Ch 1-4

4 slve quadratic equatins, including the use f quadratic frmula and cmpleting the square determine the equatin f a quadratic given sufficient infrmatin determine turning pints and zers f quadratics and understand the rle f the discriminant recgnise features f the graph f the general quadratic INVEST 1 7 Inverse Prprtin Inverse prprtin examine examples f inverse prprtin recgnise features and determine equatins f the graphs f and, including their hyperblic shapes and their asympttes. Ch 7 and 8 Pwers and Plynmials Pwers and plynmials recgnise features f the graphs f fr, including shape, and behaviur as and identify the cefficients and the degree f a plynmial expand quadratic and cubic plynmials frm factrs recgnise features and determine equatins f the graphs f, and, including shape, intercepts and behaviur as and factrise cubic plynmials in cases where a linear factr is easily btained slve cubic equatins using technlgy, and algebraically in cases where a linear factr is easily btained Ch 7 and 9 Graphs f Relatins Graphs f relatins recgnise features and determine equatins f the graphs f and, including their circular shapes, their centres and their radii recgnise features f the graph f, including its parablic shape and its axis f symmetry examine translatins and the graphs f and examine dilatins and the graphs f and Ch 7 and 1 Trig Functins 1.2.7, (Unit circle, graphs, identities, mdelling, slving eqns) Trignmetric functins understand the unit circle definitin f and peridicity using radians recgnise the graphs f n extended dmains examine amplitude changes and the graphs f and examine perid changes and the graphs f and examine phase changes and the graphs f and Ch 8 TEST 2 Ch 4-7 examine the relatinships ( )

5 2 and ( ) prve and apply the angle sum and difference identities identify cntexts suitable fr mdelling by trignmetric functins and use them t slve practical prblems slve equatins invlving trignmetric functins using technlgy, and algebraically in simple cases 3 Language f Events and Sets Review the fundamentals f prbability Language f events and sets review the cncepts and language f utcmes, sample spaces, and events, as sets f utcmes use set language and ntatin fr events, including: (r fr the cmplement f an event and fr the intersectin and unin f events and respectively and fr the intersectin and unin f the three events respectively recgnise mutually exclusive events. use everyday ccurrences t illustrate set descriptins and representatins f events and set peratins Review f the fundamentals f prbability review prb bility s me sure f the likelih f ccurre ce f eve t review the prbability scale: fr each event with if is an impssibility and if is a certainty review the rules: and Ch 9 and use relative frequencies btained frm data as estimates f prbabilities Cnditinal prbability and independence understand the ntin f a cnditinal prbability and recgnise and use language that indicates cnditinality use the ntatin and the frmula Cnditinal prbability understand the ntin f independence f an Ch 9 and 4 and independence event A frm an event B, as defined by establish and use the frmula fr independent events and, and recgnise the symmetry f independence use relative frequencies btained frm data as estimates f cnditinal prbabilities and as indicatins f pssible independence f events 6-7 EXAMS EXAM 1 (18%) Cmbinatins understand the ntin f a cmbinatin as a set f bjects taken frm a set f distinct bjects 8 Cmbinatins use the ntatin ( ) and the frmula ( ) fr the number f cmbinatins f bjects taken frm a set f distinct bjects expand fr small psitive integers recgnise the numbers( ) as binmial cefficients (as cefficients in the expansin f Ch 10 and

6 ) use P sc l s tri gle its prperties 9 10 Indices and the Index Laws , Expnential functins Indices and the index laws review indices (including fractinal and negative indices) and the index laws use radicals and cnvert t and frm fractinal indices understand and use scientific ntatin and significant figures slve equatins invlving expnential functins using technlgy, and algebraically in simple cases Expnential functins establish and use the algebraic prperties f expnential functins recgnise the qualitative features f the graph f, including asympttes, and f its translatins ( and ) identify cntexts suitable fr mdelling by expnential functins and use them t slve practical prblems Ch1 and Ch 2 INVEST 2 11 Arithmetic sequences Arithmetic sequences recgnise and use the recursive definitin f an arithmetic sequence: develp and use the frmula fr the general term f an arithmetic sequence and recgnise its linear nature Ch Arithmetic sequences use arithmetic sequences in cntexts invlving discrete linear grwth r decay, such as simple interest establish and use the frmula fr the sum f the first terms f an arithmetic sequence Ch Gemetric sequences , Gemetric sequences 2.2.4, Rates f change Gemetric sequences recgnise and use the recursive definitin f a gemetric sequence: develp and use the frmula fr the general term f a gemetric sequence and recgnise its expnential nature use gemetric sequences in cntexts invlving gemetric grwth r decay, such as cmpund interest Gemetric sequences recgnise and use the recursive definitin f a gemetric sequence: develp and use the frmula fr the general term f a gemetric sequence and recgnise its expnential nature understand the limiting behaviur as f the terms in a gemetric sequence and its dependence n the value f the cmmn rati establish and use the frmula fr the sum f the first terms f a gemetric sequence use gemetric sequences in cntexts invlving gemetric grwth r decay, such as cmpund interest Ch 3 Ch 4 Rates f change Ch 5 TEST 3 Ch 1-4

7 interpret the difference qutient as the average rate f change f a functin use the Leibniz ntatin and fr changes r increments in the variables and use the ntatin fr the difference qutient where interpret the ratis and as the slpe r gradient f a chrd r secant f the graph f The cncept f the derivative examine the behaviur f the difference qutient as as an infrmal intrductin t the cncept f a limit define the derivative as 6 The cncept f the derivative use the Leibniz ntatin fr the derivative: where and the crrespndence interpret the derivative as the instantaneus rate f change interpret the derivative as the slpe r gradient f a tangent line f the graph f Ch 5 INVEST Cmputatins f derivatives Prperties f derivatives Cmputatin f derivatives estimate numerically the value f a derivative fr simple pwer functins examine examples f variable rates f change f nn-linear functins establish the frmula fr nnnegative integers expanding r by factrising Prperties f derivatives understand the cncept f the derivative as a functin identify and use linearity prperties f the derivative calculate derivatives f plynmials Ch 5 Ch 5 9 Applicatins f derivatives , (Tangents, terminlgy assciated with curves) Applicatins f derivatives determine instantaneus rates f change determine the slpe f a tangent and the equatin f the tangent sketch curves assciated with simple plynmials, determine statinary pints, and lcal and glbal maxima and minima, and examine behaviur as and Ch 5 10 Applicatins f derivatives , (Curve Sketching, Optimisatin) Applicatins f derivatives determine instantaneus rates f change determine the slpe f a tangent and the equatin f the tangent cnstruct and interpret psitin-time graphs with velcity as the slpe f the tangent recgnise velcity as the first derivative f Ch 6 TEST 4 Ch 5&6

8 displacement with respect t time sketch curves assciated with simple plynmials, determine statinary pints, and lcal and glbal maxima and minima, and examine behaviur as and slve ptimisatin prblems arising in a variety f cntexts invlving plynmials n finite interval dmains 1 2 Antiderivatives (Plynmial Functins) Applicatins f derivatives (Rectilinear Mtin) Anti-derivatives calculate anti-derivatives f plynmial functins Anti-derivatives cnstruct and interpret psitin-time graphs with velcity as the slpe f the tangent recgnise velcity as the first derivative f displacement with respect t time Ch 7 Ch 8 Anti-derivatives Antiderivatives 3 calculate anti-derivatives f plynmial Ch (Rect Mtin) functins 4 Revisin fr Exam UNIT 1 and UNIT EXAM 2 (22%) Please nte this prgram is a guide nly and may be subject t change. Fr further infrmatin please visit

9 Appendix 1 Glssary This glssary is prvided t enable a cmmn understanding f the key terms in this syllabus. Unit 1 Functins and graphs Asymptte A line is an asymptte t a curve if the distance between the line and the curve appraches zer as they tend t infinity. Fr example, the line with equatin is a vertical asymptte t the graph f and the line with equatin is a hrizntal asymptte t the graph f. Binmial distributin The expansin ( ) ( ) is knwn as the binmial therem. The numbers ( ) are called binmial cefficients. Cmpleting the square The quadratic expressin can be rewritten as ( ) ( ) Re-writing it in this way is called cmpleting the square. Discriminant The discriminant f the quadratic expressin is the quantity Functin A functin is a rule that assciates with each element in a set, a unique element in a set We write t indicate the mapping f t. The set is called the dmain f and the set is called the cdmain. The subset f cnsisting f all the elements is called the range f If we write we say that is the independent variable and is the dependent variable. Graph f a functin The graph f a functin is the set f all pints in Cartesian plane where is in the dmain f and. Quadratic frmula If with then called the quadratic frmula.. This frmula fr the rts is Vertical line test A relatin between tw real variables and is a functin and fr sme functin if and nly if each vertical line, i.e. each line parallel t the -axis, intersects the graph f the relatin in, at mst, ne pint. This test t determine whether a relatin is, in fact, a functin is knwn as the vertical line test. Trignmetric functins Angle sum and difference identites The angle sum and difference identites fr sine and csine are given by Area f a sectr The area f a sectr f a circle is given by, where is the sectr area, is the radius and is the angle subtended at the centre, measured in radians. Area f a segment The area f a segment f a circle is given by, where is the segment area, is the radius and is the angle subtended at the centre, measured in radians. Circular measure Circular measure is the measurement f angle size in radians. Length f an arc The length f an arc in a circle is given by, where is the arc length, is the radius and is the angle subtended at the centre, measured in radians. This is simply a rearrangement f the frmula defining the radian measure f an angle. Length f a chrd The length f a chrd in a circle is given by, where is the chrd length, is the radius and is the angle subtended at the centre, measured in radians.

10 Perid f a functin The perid f a functin is the smallest psitive number with the prperty that fr all. The functins and bth have perid and has perid. Radian measure Sine rule and csine rule The radian measure f an angle in a sectr f a circle is defined by, where is the radius and is the arc length. Thus, an angle whse degree measure is has radian measure. The lengths f the sides f a triangle are related t the sine f its angles by the equatins This is knwn as the sine rule. The lengths f the sides f a triangle are related t the csine f ne f its angles by the equatin This is knwn as the csine rule. Sine, csine and tangent functins Since each angle measured anticlckwise frm the psitive -axis determines a pint n the unit circle, we will define the csine f t be the crdinate f the pint the sine f t be the crdinate f the pint the tangent f is the gradient f the line segment. Cunting and prbability Cnditinal prbability The prbability that an event ccurs can change if it becmes knwn that anther event ccurs. The new prbability is knwn as a cnditinal prbability and is written as If has ccurred, the sample space is reduced by discarding all

11 Independent events Mutually exclusive utcmes that are nt in the event The new sample space, called the reduced sample space, is The cnditinal prbability f event is given by. Tw events are independent if knwing that ne ccurs tells us nthing abut the ther. The cncept can be defined frmally using prbabilities in varius ways: events and are independent if, if r if Fr events and with nn-zer prbabilities, any ne f these equatins implies any ther. Tw events are mutually exclusive if there is n utcme in which bth events ccur. Pascal s triangle Pascal s triangle is a triangular arrangement f binmial cefficients. The rw cnsists f the binmial cefficients ( ) fr sum f the tw entries abve it, and sum f the entries in the Fr example, 10 = , each interir entry is the rw is Relative frequency If an event ccurs times when a chance experiment is repeated times, the relative frequency f is. Unit 2 Expnential functins Algebraic prperties f expnential functins The algebraic prperties f expnential functins are the index laws:,,,, fr any real numbers and, with. Index laws The index laws are the rules:,,,, and Arithmetic and gemetric sequences and series Arithmetic sequence Gemetric sequence, fr any real numbers and, with and. An arithmetic sequence is a sequence f numbers such that the difference f any tw successive members f the sequence is a cnstant. Fr instance, the sequence 2, 5, 8, 11, 14, 17, is an arithmetic sequence with cmmn difference 3. If the initial term f an arithmetic sequence is and the cmmn difference f successive members is, then the th term f the sequence, is given by: fr. A recursive definitin is, where is the cmmn difference and. A gemetric sequence is a sequence f numbers where each term after the first is fund by multiplying the previus ne by a fixed number called the cmmn rati. Fr example, the sequence 3, 6, 12, 24,... is a gemetric sequence with cmmn rati 2. Similarly the sequence 40, 20, 10, 5, 2.5, is a gemetric sequence with cmmn rati. If the initial term f a gemetric sequence is and the cmmn rati f successive members is, then the th term f the sequence, is given by: fr. A recursive definitin is

12 Partial sums f a gemetric sequence (gemetric series) Partial sum f an arithmetic sequence (arithmetic series) Partial sums f a sequence (series) Intrductin t differential calculus fr and where is the cnstant rati. The partial sum f the first terms f a gemetric sequence with first term and cmmn rati, is,. The partial sums frm a sequence with The partial sum f the first terms f an arithmetic sequence with first term and cmmn difference. is where is the th term f the sequence. The partial sums frm a sequence with The sequence f partial sums f a sequence and is defined by Anti-differentiatin An anti-derivative, primitive r indefinite integral f a functin is a functin whse derivative is, i.e. The prcess f slving fr anti-derivatives is called anti-differentiatin. Anti-derivatives are nt unique. If is an anti-derivative f then s t is the functin where is any number. We write t dente the set f all anti-derivatives f The number is called the cnstant f integratin. Fr example, since we can write Gradient (Slpe) The gradient f the straight line passing thrugh pints and is the rati. Slpe is a synnym fr gradient. Linearity prperty f the derivative Lcal and glbal maximum and minimum Secant Simple plynmial The linearity prperty f the derivative is summarised by the equatins: fr any cnstant and. A statinary pint n the graph f a differentiable functin is a pint where. We say that is a lcal maximum f the functin if fr all values f near. We say that is a glbal maximum f the functin if fr all values f in the dmain f. We say that is a lcal minimum f the functin if fr all values f near. We say that is a glbal minimum f the functin if fr all values f in the dmain f. A secant f the graph f a functin is the straight line passing thrugh tw pints n the graph. The line segment between the tw pints is called a chrd. A simple plynmial is ne which is easily factrised and whse statinary pints maybe easily determined using traditinal calculus techniques. Tangent line The tangent line (r simply the tangent) t a curve at a given pint can be described intuitively as the straight line that "just tuches" the curve at that pint. At where the curve meets the tangent, the curve has "the same directin" as the tangent line. In this sense, it is the best straight-line apprximatin t the curve at the pint.

13 Appendix 2 Cntent as per WACE Syllabus Methds Unit One: Tpic 1.1: Functins and graphs (22 hurs) Lines and linear relatinships determine the crdinates f the mid-pint between tw pints determine an end-pint given the ther end-pint and the mid-pint examine examples f direct prprtin and linearly related variables recgnise features f the graph f, including its linear nature, its intercepts and its slpe r gradient determine the equatin f a straight line given sufficient infrmatin; including parallel and perpendicular lines slve linear equatins, including thse with algebraic fractins and variables n bth sides Quadratic relatinships examine examples f quadratically related variables recgnise features f the graphs f, and, including their parablic nature, turning pints, axes f symmetry and intercepts slve quadratic equatins, including the use f quadratic frmula and cmpleting the square determine the equatin f a quadratic given sufficient infrmatin determine turning pints and zers f quadratics and understand the rle f the discriminant recgnise features f the graph f the general quadratic Inverse prprtin examine examples f inverse prprtin recgnise features and determine equatins f the graphs f and, including their hyperblic shapes and their asympttes. Pwers and plynmials recgnise features f the graphs f fr, including shape, and behaviur as and identify the cefficients and the degree f a plynmial expand quadratic and cubic plynmials frm factrs recgnise features and determine equatins f the graphs f, and, including shape, intercepts and behaviur as and factrise cubic plynmials in cases where a linear factr is easily btained slve cubic equatins using technlgy, and algebraically in cases where a linear factr is easily btained Graphs f relatins recgnise features and determine equatins f the graphs f and, including their circular shapes, their centres and their radii recgnise features f the graph f, including its parablic shape and its axis f symmetry Functins understand the cncept f a functin as a mapping between sets and as a rule r a frmula that defines ne variable quantity in terms f anther use functin ntatin; determine dmain and range; recgnise independent and dependent variables understand the cncept f the graph f a functin examine translatins and the graphs f and examine dilatins and the graphs f and recgnise the distinctin between functins and relatins and apply the vertical line test

14 Methds Unit One: Tpic 1.2: Trignmetric functins (15 hurs) Csine and sine rules review sine, csine and tangent as ratis f side lengths in right-angled triangles understand the unit circle definitin f and and peridicity using degrees examine the relatinship between the angle f inclinatin f a line and the gradient f that line establish and use the csine and sine rules, including cnsideratin f the ambiguus case and the frmula fr the area f a triangle Circular measure and radian measure define and use radian measure and understand its relatinship with degree measure calculate lengths f arcs and areas f sectrs and segments in circles Trignmetric functins understand the unit circle definitin f and peridicity using radians recgnise the exact values f at integer multiples f and recgnise the graphs f n extended dmains examine amplitude changes and the graphs f and examine perid changes and the graphs f and examine phase changes and the graphs f and examine the relatinships ( ) and ( ) prve and apply the angle sum and difference identities identify cntexts suitable fr mdelling by trignmetric functins and use them t slve practical prblems slve equatins invlving trignmetric functins using technlgy, and algebraically in simple cases Methds Unit One: Tpic 1.3: Cunting and prbability (18 hurs) Cmbinatins understand the ntin f a cmbinatin as a set f bjects taken frm a set f distinct bjects use the ntatin ( ) and the frmula ( ) fr the number f cmbinatins f bjects taken frm a set f distinct bjects expand fr small psitive integers recgnise the numbers( ) as binmial cefficients (as cefficients in the expansin f ) use P sc l s tri gle its prperties Language f events and sets review the cncepts and language f utcmes, sample spaces, and events, as sets f utcmes use set language and ntatin fr events, including: a. (r fr the cmplement f an event b. and fr the intersectin and unin f events and respectively c. and fr the intersectin and unin f the three events respectively d. recgnise mutually exclusive events use everyday ccurrences t illustrate set descriptins and representatins f events and set peratins Review f the fundamentals f prbability review prb bility s me sure f the likelih f ccurre ce f eve t review the prbability scale: fr each event with if is an impssibility and if is a certainty review the rules: and use relative frequencies btained frm data as estimates f prbabilities

15 Cnditinal prbability and independence understand the ntin f a cnditinal prbability and recgnise and use language that indicates cnditinality use the ntatin and the frmula understand the ntin f independence f an event A frm an event B, as defined by establish and use the frmula fr independent events and, and recgnise the symmetry f independence use relative frequencies btained frm data as estimates f cnditinal prbabilities and as indicatins f pssible independence f events Methds Unit Tw: Tpic 2.1: Expnential functins (10 hurs) Indices and the index laws review indices (including fractinal and negative indices) and the index laws use radicals and cnvert t and frm fractinal indices understand and use scientific ntatin and significant figures Expnential functins establish and use the algebraic prperties f expnential functins recgnise the qualitative features f the graph f, including asympttes, and f its translatins ( and ) identify cntexts suitable fr mdelling by expnential functins and use them t slve practical prblems slve equatins invlving expnential functins using technlgy, and algebraically in simple cases Methds Unit Tw: Tpic 2.2: Arithmetic and gemetric sequences and series (15 hurs) Arithmetic sequences recgnise and use the recursive definitin f an arithmetic sequence: develp and use the frmula fr the general term f an arithmetic sequence and recgnise its linear nature use arithmetic sequences in cntexts invlving discrete linear grwth r decay, such as simple interest establish and use the frmula fr the sum f the first terms f an arithmetic sequence Gemetric sequences recgnise and use the recursive definitin f a gemetric sequence: develp and use the frmula fr the general term f a gemetric sequence and recgnise its expnential nature understand the limiting behaviur as f the terms in a gemetric sequence and its dependence n the value f the cmmn rati establish and use the frmula fr the sum f the first terms f a gemetric sequence use gemetric sequences in cntexts invlving gemetric grwth r decay, such as cmpund interest Methds Unit Tw: Tpic 2.3: Intrductin t differential calculus (30 hurs) Rates f change interpret the difference qutient as the average rate f change f a functin use the Leibniz ntatin and fr changes r increments in the variables and use the ntatin fr the difference qutient where interpret the ratis and as the slpe r gradient f a chrd r secant f the graph f

16 The cncept f the derivative examine the behaviur f the difference qutient as as an infrmal intrductin t the cncept f a limit define the derivative as use the Leibniz ntatin fr the derivative: and the crrespndence where interpret the derivative as the instantaneus rate f change interpret the derivative as the slpe r gradient f a tangent line f the graph f Cmputatin f derivatives estimate numerically the value f a derivative fr simple pwer functins examine examples f variable rates f change f nn-linear functins establish the frmula fr nn-negative integers expanding r by factrising Prperties f derivatives understand the cncept f the derivative as a functin identify and use linearity prperties f the derivative calculate derivatives f plynmials Applicatins f derivatives determine instantaneus rates f change determine the slpe f a tangent and the equatin f the tangent cnstruct and interpret psitin-time graphs with velcity as the slpe f the tangent recgnise velcity as the first derivative f displacement with respect t time sketch curves assciated with simple plynmials, determine statinary pints, and lcal and glbal maxima and minima, and examine behaviur as and slve ptimisatin prblems arising in a variety f cntexts invlving plynmials n finite interval dmains Anti-derivatives calculate anti-derivatives f plynmial functins