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HSC Head Start lecture EXTENSION MATHEMATICS Integration, Acceleration and Induction
Presented by: JACOB SILOVE
Today’s Lecture TIMETABLE • • • •
Who am I? General HSC tips Course-specific tips Content
Who am I? HIGH SCHOOL Year 11 (2013): • Pacific Project • Mathematics- 96 Year 12 (2014): • International Science Forum • ATAR- 99.80 – – – – – –
Physics- 93 Chemistry- 93 Modern History- 93 Extension 2 Maths- 94 Advanced English- 95 Extension 1 Maths- 98
GENERAL HSC TIPS
DON’T PANIC! MESSING UP ONE ASSESSMENT ISN’T THE END OF THE WORLD
GENERAL HSC TIPS
DON’T COMPETE WITH EACH OTHER. IT ISN’T REALLY WORTH TRYING TO UNDERSTAND WHY THE HSC WORKS: LEAVE THAT TO THE MAGICIANS AT THE BOARD OF STUDIES.
GENERAL HSC TIPS
DON’T BE AFRAID TO USE COLOURS
GENERAL HSC TIPS
DON’T JUST WRITE NOTES: WRITE WORKSHEETS
GENERAL HSC TIPS
STUDY IN GROUPS
GENERAL HSC TIPS
TAILOR YOUR STUDY FOR YOUR SUBJECT
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CONTENT • • • • • • •
Methods of Integration Projectile Motion Induction Binomial Theorem Velocity and Acceleration as a function of x* Simple Harmonic Motion** Further Probability***
* If we have time ** Highly doubt that we will have time *** Wow, I’ve seriously misjudged the timing for this presentation
INTEGRATION • • • •
Method of finding the area under a curve Think of it as reverse-differentiation ∫𝑎↑𝑏▒𝑓(𝑥)𝑑𝑥 is the standard form for integration Eg. 𝑑/𝑑𝑥 𝑥↑2 =2𝑥, therefore ∫↑▒2𝑥𝑑𝑥=𝑥↑2 +𝐶 (Will explain +C)
INTEGRATION •
INTEGRATION Why plus C? • Differentiate 𝑦=3𝑥↑2 +2𝑥+5 • Differentiate 𝑦=3𝑥↑2 +2𝑥+4 • Differentiate 𝑦=3𝑥↑2 +2𝑥+100000 • Differentiate 𝑦=3𝑥↑2 +2𝑥−36.27
INTEGRATION Why plus C? Integral of 𝑦↑′ =2𝑥 could be : -‐ y=𝑥↑2 -‐ y=𝑥↑2 +10000 -‐ y=𝑥↑2 −10003938882181902890182098120291029 - Etc. - ALWAYS PLUS C FOR INDEFINITE INTEGRALS!!!!
INTEGRATION Indefinite vs definite integrals • Indefinite – ∫↑▒𝑓(𝑥)𝑑𝑥 – Finds the primitive graph – Eg ∫↑▒2𝑥𝑑𝑥=𝑥↑2 +𝐶
• Definite – ∫𝑎↑𝑏▒𝑓(𝑥)𝑑𝑥 – Finds an area beneath a graph – Eg ∫0↑1▒2𝑥𝑑𝑥=[𝑥↑2 ]↓0↑1 =(1)↑2 −(0)↑2 =1
INTEGRATION Some questions Find the indefinite integral of −𝒙↑𝟐 +𝟐𝒙+𝟏
INTEGRATION Some questions Find the definite integral of 𝒙↑𝟏/𝟐 +𝟒𝒙+𝟗 and x = 1
between x = 0
METHODS OF INTEGRATION Substitution • Always given Three easy steps • Find the derivative • Cancel terms and substitute • Convert back into the original variable
METHODS OF INTEGRATION Find ∫𝟎↑𝟏▒𝒕/√𝟏+𝒕↑ 𝒅𝒕 using the substitution 𝒕=𝒖↑𝟐 −𝟏
METHODS OF INTEGRATION Find ∫↑▒𝒙√𝟏+𝒙↑𝟐 𝒅𝒙, using the substitution 𝒖=𝒙↑𝟐 +𝟏
METHODS OF INTEGRATION • Find the derivative • Make substitutions and cancel • Make sure to change variable back into the original one
PROJECTILE MOTION
𝑥=𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑑𝑥/𝑑𝑡 =𝑥 =𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 ↑2 𝑥/𝑑𝑡↑2 =𝑑𝑥 /𝑑𝑡 =𝑥 =𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑
PROJECTILE MOTION Particle projected vertically Initially, 𝑡=0, 𝑥=0, 𝑣=𝑉 Vertical
Horizontal
PROJECTILE MOTION Particle projected at an angle alpha Initially, 𝑡=0, 𝑥=0, 𝑣=𝑉 Vertical
V
α
𝑉 ↓ 𝑥
𝑉 ↓ Horizontal 𝑦
PROJECTILE MOTION
PROJECTILE MOTION
PROJECTILE MOTION
PROJECTILE MOTION • • • •
Understand initial conditions Be able to quickly integrate Understand what the question is actually asking There are only like 3 possible questions!
INDUCTION • A method used to prove relationships true • We do this by utilising the following steps for set S(n): – – – –
Prove that 𝑆(1) is true, Assume 𝑆(𝑘) is true, Prove 𝑆(𝑘+1) is true, State true for all n
INDUCTION • Prove true for n = 1
INDUCTION • Assume true for n = k
INDUCTION • Prove true for n = k+1
INDUCTION • State true for all n
INDUCTION • Prove true for n = 1
INDUCTION • Assume true for n = k
INDUCTION • Prove true for n = k+1
INDUCTION • State true for all n
INDUCTION • Remember the steps required to prove an induction question true • Make intelligent substitutions • Make sure to write the conclusions!
BINOMIAL THEORUM Find the 34th term:
(𝑥+1/3 )↑45
BINOMIAL THEORUM Find the 34th term:
(𝑥+1/3 )↑45 2.876002165∗10↑10 ∗(1/3 )↑33 ∗𝑥↑12 0.00000517353∗𝑥↑12
BINOMIAL THEORUM Let’s examine an easier example, to find a general rule.
(1+𝑥)↑𝑛 (1+𝑥)∗(1+𝑥)∗(1+𝑥)∗(1+𝑥)∗…∗(1+𝑥) 𝑛 𝑡𝑖𝑚𝑒𝑠
BINOMIAL THEORUM Let’s examine an easier example, to find a general rule.
(1+𝑥)↑𝑛 (1+𝑥)∗(1+𝑥)∗(1+𝑥)∗(1+𝑥)∗…∗(1+𝑥) 𝑛 𝑡𝑖𝑚𝑒𝑠 1+𝐴𝑥+𝐵𝑥↑2 +𝐶𝑥↑3 +…+𝑍𝑥↑𝑛 nC (𝑛¦𝑘 ) k
BINOMIAL THEORUM Let’s examine an easier example, to find a general rule.
(1+𝑥)↑𝑛 (1+𝑥)∗(1+𝑥)∗(1+𝑥)∗(1+𝑥)∗…∗(1+𝑥) 𝑛 𝑡𝑖𝑚𝑒𝑠 1+𝐴𝑥+𝐵𝑥↑2 +𝐶𝑥↑3 +…+𝑍𝑥↑𝑛 nC (𝑛¦𝑘 ) k
(𝑛¦0 )+(𝑛¦1 )𝑥+(𝑛¦2 )𝑥↑2 +(𝑛¦3 )𝑥↑3 +…+(𝑛¦𝑛 )𝑥↑𝑛
BINOMIAL THEORUM Let’s examine an easier example, to find a general rule.
( 1+𝑥)↑0 =𝟏 (1+𝑥)↑1 =𝟏+𝟏𝑥 (1+𝑥)↑2 =𝟏+𝟐𝑥+𝟏𝑥↑2 (1+𝑥)↑3 =𝟏+𝟑𝑥+𝟑𝑥↑2 +𝟏𝑥↑3 (1+𝑥)↑4 =𝟏+𝟒𝑥+𝟔𝑥↑2 +𝟒𝑥↑3 +𝟏𝑥↑4 ( 1+𝑥)↑5 =𝟏+𝟓𝑥+𝟏𝟎𝑥↑2 +𝟏𝟎𝑥↑3 +𝟓𝑥↑4 +𝟏𝑥↑5
BINOMIAL THEORUM Let’s examine an easier example, to find a general rule. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1
BINOMIAL THEORUM
(𝑛¦𝑘 )=𝑛!/(𝑛−𝑘)!𝑘!
BINOMIAL THEORUM Let’s examine an easier example, to find a general rule.
(𝑢+𝑥)↑𝑛
(𝑛¦0 )𝑢↑𝑛 +(𝑛¦1 )𝑢↑𝑛−1 𝑥+(𝑛¦2 )𝑢↑𝑛−2 𝑥↑2 +…+(𝑛¦𝑛−1 )𝑢𝑥↑𝑛 −1 +(𝑛¦𝑛 )𝑥↑𝑛
BINOMIAL THEORUM
BINOMIAL THEORUM
BINOMIAL THEORUM Greatest term questions (3+5𝑥)↑20
BINOMIAL THEORUM Greatest term questions (3+5𝑥)↑20 (𝑛¦𝑘 )𝑎↑𝑛−𝑘 𝑏↑𝑘 (20¦𝑘 )3↑20−𝑘 5↑𝑘
BINOMIAL THEORUM Greatest term questions (3+5𝑥)↑20 (𝑛¦𝑘 )𝑎↑𝑛−𝑘 𝑏↑𝑘 (20¦𝑘 )3↑20−𝑘 5↑𝑘 If Tk > Tk+1, then Tk will be the greatest term. 𝑇↓𝑘+1 /𝑇↓𝑘 =𝑛!/(𝑛−𝑘+1)!(𝑘+1)! ∗(𝑛−𝑘)!𝑘!/𝑛! ∗5↑𝑘+1 ∗ 3↑19−𝑘 /5↑𝑘 ∗3↑20−𝑘 20−𝑘/(𝑘+1) ∗5/3
BINOMIAL THEORUM 20−𝑘/(𝑘+1) ∗5/3 This exceeds one between 𝑘=1 𝑎𝑛𝑑 𝑘=12, but then is less than one. The greatest term is therefore 𝑘=13, which has coefficient (20¦13 )3↑20−13 5↑13 =(20¦13 )3↑7 5↑13
BINOMIAL THEORUM • Remember that the term expansion equation is on the formula sheet. • Try to memorise the rough ways you solve various types of questions. • This is a tough topic; practice practice practice!
VELOCITY AND ACCELERATION From the Mathematics course: 𝑥=𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑑𝑥/𝑑𝑡 =𝑥 =𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑑↑2 𝑥/𝑑𝑡↑2 =𝑑𝑥 /𝑑𝑡 =𝑥 =𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛
VELOCITY AND ACCELERATION From the Mathematics course: 𝑥=𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑑𝑥/𝑑𝑡 =𝑥 =𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑑↑2 𝑥/𝑑𝑡↑2 =𝑑𝑥 /𝑑𝑡 =𝑥 =𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑥 =𝑎=𝑑𝑣/𝑑𝑡 =𝑑𝑣/𝑑𝑥 ∗𝑑𝑥/𝑑𝑡
VELOCITY AND ACCELERATION From the Mathematics course: 𝑥=𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑑𝑥/𝑑𝑡 =𝑥 =𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑑↑2 𝑥/𝑑𝑡↑2 =𝑑𝑥 /𝑑𝑡 =𝑥 =𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑥 =𝑎=𝑑𝑣/𝑑𝑡 =𝑑𝑣/𝑑𝑥 ∗𝑑𝑥/𝑑𝑡 𝑥 =𝑎=𝑑𝑣/𝑑𝑡 =𝑑𝑣/𝑑𝑥 ∗𝑑𝑥/𝑑𝑡 =𝑑𝑣/𝑑𝑥 ∗𝑣
VELOCITY AND ACCELERATION From the Mathematics course: 𝑥=𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑑𝑥/𝑑𝑡 =𝑥 =𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑑↑2 𝑥/𝑑𝑡↑2 =𝑑𝑥 /𝑑𝑡 =𝑥 =𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑥 =𝑎=𝑑𝑣/𝑑𝑡 =𝑑𝑣/𝑑𝑥 ∗𝑑𝑥/𝑑𝑡 𝑥 =𝑎=𝑑𝑣/𝑑𝑡 =𝑑𝑣/𝑑𝑥 ∗𝑑𝑥/𝑑𝑡 =𝑑𝑣/𝑑𝑥 ∗𝑣 𝑥 =𝑎=𝑑𝑣/𝑑𝑡 =𝑑𝑣/𝑑𝑥 ∗𝑑𝑥/𝑑𝑡 =𝑑𝑣/𝑑𝑥 ∗𝑣=𝑑/𝑑𝑥 (1/2 𝑣↑2 )
VELOCITY AND ACCELERATION
VELOCITY AND ACCELERATION
VELOCITY AND ACCELERATION • Remember that the relevant formulas are on your formula sheet. • Never forget to + C, because that’s usually half of the answer! • Remember that you can play with derivatives as much as you want. Flip em, multiply em, divide em up using the chain rule.
QUESTIONS
QUESTIONS
QUESTIONS
QUESTIONS
Thanks! Jacob Silove
[email protected]
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