Download Integration by Substitution

YEAR 13 (Level 3)
MATHEMATICS WITH CALCULUS
2016
Mathematics with Calculus 2016 Year Planner
Week 1
Week 1
Week 1
Week 1
Mon 1-Fri 5 Feb
Mon 2 May- Fri 6 May
Mon 25- Fri 29 July
Mon 10 Oct - Fri 15 Oct
TRIGONOMETRY 3.3
INTERNAL
DIFFERENTIATION
INTEGRATION 3.7
EXTERNAL
Week 2
Week 2
Week 2
Week 2
Mon 8- Thur 12 Feb
Mon 9- Fri 13 May
Mon 1 Aug - Fri 5 July
Mon 17- Fri 21 Oct
TRIGONOMETRY
DIFFERENTIATION
INTEGRATION
Week 3
Week 3
Week 3
Week 3
Mon 15- Fri 19 Feb
Mon 16 - Fri 20 May
Mon 8- Fri 12 Aug
Mon 24- Fri 28 Oct
TRIGONOMETRY
DIFFERENTIATION
INTEGRATION
Week 4
Week 4
Week 4
Week 4
Mon 22- Fri 26 Feb
Mon 23- Fri 27 May
Mon 15- Fri 19 Aug
Mon 31- Fri 4 Nov
TRIGONOMETRY
DIFFERENTIATION
INTEGRATION
Week 5
Week 5
Week 5
Week 5
Mon 29- Fri 4 Mar
Mon 30- Fri 3 June
Mon 22- Fri 26 Aug
Mon 7- Fri 11 Nov
TRIGONOMETRY
COMPLEX NUMBERS 3.5
EXTERNAL
INTEGRATION
Week 6
Week 6
Week 6
Week 6
Mon 7- Fri 11 March
Mon 6 - Fri 10 June
Mon 29- Fri 2 Sep
Mon 14- Fri 18 Nov
TRIGONOMETRY
ASSESSMENT 3.3
COMPLEX NUMBERS
INTEGRATION
Week 7
Week 7
Week 7
Week 7
Mon 14- Fri 18 March
Mon 13- Fri 17 June
Mon 5 Sep- Fri 9 Sept
Mon 21- Fri 25 Nov
DIFFERENTIATION 3.6
EXTERNAL
COMPLEX NUMBERS
INTEGRATION
Week 8
Week 8
Week 8
Week 8
Mon 21- Fri 25 March Good
Friday
Mon 20 - Fri 24 June
Mon 12 - Fri 16 Sept
Mon 28 Nov- Fri 2 Dec
COMPLEX NUMBERS
CONIC SECTIONS 3.1
INTERNAL
Week 9
Week 9
Week 9
Week 9
Mon 28- Fri 1 April
Easter Monday
Mon 27- Fri 1 July
Mon 19 - Fri 23 Sept
Mon 5- Fri 9 Dec
COMPLEX NUMBERS
CONIC SECTIONS
ASSESSMENT
DIFFERENTIATION
DIFFERENTIATION
Week 10
Week 10
Week 10
Mon 4 April- Friday 8 April
Mon 4- Fri 8 July
Mon12- Fri 16 Dec
DIFFERENTIATION
COMPLEX NUMBERS
Week 11
Mon 11 April – Friday 15 April
DIFFERENTIATION
2
MATHS WITH CALCULUS 2016
Course Outline
A graphics calculator is helpful.
SUBJECT:
Calculus
SUBJECT CODE:
3MCAL
LEVEL:
3
LEARNING AREA:
Mathematics
QUALIFICATION:
3.1
91573
3.3
91575
3.5
91577
3.6
91578
3.7
91579
Level 3 NCEA
Apply the geometry of conic sections
in solving problems.
Internal
3
3 credits
Apply trigonometric methods in
solving problems.
Internal
3
4 credits
Apply the algebra of complex
numbers in solving problems.
External
3
5 credits
Apply differentiation methods in
solving problems.
External
3
6 credits
External
3
6 credits
Apply integration methods in
solving problems.
TOTAL CREDITS:
24
ASSESSMENT:
Two internal standards and three external standards.
The external standards will be assessed in a 3-hour examination
at the end of the year.
The internal standards will be assessed shortly after they have been
completed.
FEES:
Nil
PREREQUISITES:
All 18 Level 2 credits.
Two of the internal assessments must be merit or better.
(If students do not meet the prerequisites but are close, then permission
may be granted by the HOD.)
3
ORDER OF WORK
Standard
1. Level 2 essential algebra.
2. Trigonometry (3.3)
Textbook
Chapters
Notes
These are essential skills necessary for all
the achievement standards. Students will
need to complete this revision in their own
time during the first few weeks of term 1.
Internally assessed about a week after the
conclusion of teaching, approximately
week six of term 1.
3. Differentiation (3.6)
4. Complex numbers (3.5)
5. Integration (3.7)
6. Conic Sections (3.1)
Internally assessed about a week after the
conclusion of teaching, approximately at
the end of term 3.
In addition to the content of the text book we also have to cover:
Irrational equations, logarithmic equations, cubic equations and the general solutions to trigonometric
equations.
Strategies for success:
1
2
3
4
Organise your work logically and neatly. Show all working.
Revise notes weekly.
The best way to master mathematics is to practise as many examples as possible.
Do not let your problems grow. You are encouraged to ask questions. Take class tests and
assignments seriously. They are important chances for you to determine what you really
understand and they let you know where you may need extra help.
They can also build your confidence with mathematics.
Give your best effort at all times.
4
SUGGESTED LEARNING ORDER FOR CALCULUS 2016
Essential Algebra
Exponentials and Logarithms
FROM DELTA MATHEMATICS TEXTBOOK
ACHIEVEMENT STANDARD
Trigonometric Graphs
Trigonometric Identities
Trigonometric Equations
Trigonometric Modelling
3.3 Trigonometry [Internal]
3.3 Trigonometry [Internal]
3.3 Trigonometry [Internal]
3.3 Trigonometry [Internal]
Limits and First Principles
Derivatives of Polynomials and Composite
Functions
Derivatives of Exponentials and Logs
Product and Quotient Rules
Derivatives of Trigonometric Functions
Calculus and Curve Properties
Applications of Differentiation
Tangents and Normals
Rates of Change
Parametric Functions and Derivatives
Implicit Derivatives
3.6 Differentiation
3.6 Differentiation
Surds
Complex Numbers
Complex Numbers and Polynomials
Complex Roots
Complex Loci
3.5 Algebra of Complex Numbers
3.5 Algebra of Complex Numbers
3.5 Algebra of Complex Numbers
3.5 Algebra of Complex Numbers
3.5 Algebra of Complex Numbers
Integration of Polynomials and Exponentials
Integration of Trigonometric Functions
Integration by Substitution
Definite Integration
Areas under Curves
Numerical Integration
Differential Equations (Verification)
Differential Equations
Differential Equations (Applications)
3.7 Integration
3.7 Integration
3.7 Integration
3.7 Integration
3.7 Integration
3.7 Integration
3.7 Integration
3.7 Integration
3.7 Integration
Conic Sections (Standard Forms)
Conic Sections (Parameters, Tangents)
3.1 Conics [Internal]
3.1 Conics [Internal]
3.6 Differentiation
3.6 Differentiation
3.6 Differentiation
3.6 Differentiation
3.6 Differentiation
3.6 Differentiation
3.6 Differentiation
3.6 Differentiation
3.6 Differentiation
We also have to teach: Irrational Equations, Logarithmic Equations, General Solutions of
Trigonometric Equations and possibly other Integration Techniques.
5
Essential Algebra Skills
Do this assignment on your own paper.
Set your work out to the highest standards of neatness, clarity and mathematical correctness that
you are capable of, showing whatever working you think is necessary at this level.
DO NOT HAND IN A SHABBY PIECE OF WORK. TAKE PRIDE IN WHAT YOU DO.
1
If p = 2.1a + 3.3b2 , then find the value of p when a = 3.4 and b = 2.6
L2 H  5.78
find the value of v when L = 3.62, H = 2.84  104
2
v
3
The value of H to two significant figures is 6.3 when T = 5.9 and U = 0.97. What is the correct
formula for H?
(a) H = 0.78T + 2.1U
(b) H = 0.56T + 1.3U
(c) H = 4.4TU
(d) H = 0.87T + 1.2U
(e) H = 0.77(T - U)
4
Simplify as much as possible.
(a) 7a3 + 3a2 -a2 + 2a3 + 11
(b) (2a2)2 + 3a4
(c) 2(x2 + 3x - 2) - (x + 1)(x - 2)
5
Expand the following.
(a) (x2 - x + 1)(x - 1)
6
3
2.73  10
(b)
Factorise the following.
(a) a2 + 3a
(b)
(a + 2)(a3 + 2a2 - a + 3)
a2 -13a + 22
(c)
3x2 - 15x -42
7
Write the following as single fractions as simply as possible.
A B
A B
A B
(a)
(b)
(c)



3 4
B C
3 3
A 1 A 1
x2  2x  3
(d)
(e)

2
3
x2  9
8
Make T the subject of the formula ( A T  C)2  D , simplifying your answer as much as
possible.
10
Evaluate
(a) log5.6
(b) 2log56
(c) sin(2.3)
11
If 2000(3-0.125t ) = 700 then find the value of t.
12
Solve the equation (x+1)(x-7) – (7-x) =0
13
Solve the equation √x +4 = x
(d) 1.22.4 – 1.2-2.4
6
Basic Year 12 Differentiation
Differentiate
(a)
f ( x )  3x 8
f (x ) 
(f)
(b)
f ( x )  43 x 8
f (x ) 
(c)
f ( x )  (3x  2) 2
(d)
f ( x) 
f (x ) 
(e)
4
2x
5
f (x ) 
f ( x) 
3
5 x
f (x ) 
Find the gradient of the tangent to the curve y  3x 2  2 x  4 at the point where x = 2
Gradient =
(g)
Find the stationary points for the function f ( x ) 
First point = (
,
)
4x 3
 2 x 2  8x  7
3
Second point = (
,
)
7
Achievement Standard
Subject Reference
Mathematics and Statistics 3.1
Title
Apply the geometry of conic sections in solving problems
Level
3
Subfield
Mathematics
Domain
Algebra
Credits
3
Status
Assessment
Internal
Status date
Planned review date
31 December 2016
Date version published
This achievement standard involves applying the geometry of conic sections in solving problems.
Achievement Criteria
Achievement
Achievement with Merit
Achievement with Excellence
 Apply the geometry of conic
sections in solving problems.
 Apply the geometry of conic
sections, using relational thinking, in
solving problems.
 Apply the geometry of conic sections,
using extended abstract thinking, in
solving problems.
Explanatory Notes
1
This achievement standard is derived from Level 8 of The New Zealand Curriculum, Learning Media, Ministry of Education,
2007; and is related to the achievement objective:
 Apply the geometry of conic sections
in the Mathematics strand of the Mathematics and Statistics Learning Area. It is also related to the material in the Teaching
and Learning Guide for Mathematics and Statistics, Ministry of Education, 2012, at http://seniorsecondary.tki.org.nz.
2
Apply the geometry of conic sections in solving problems involves:
 selecting and using methods
 demonstrating knowledge of concepts and terms
 communicating using appropriate representations.
Relational thinking involves one or more of:
 selecting and carrying out a logical sequence of steps
 connecting different concepts or representations
 demonstrating understanding of concepts
 forming and using a model;
and also relating findings to a context, or communicating thinking using appropriate mathematical statements.
Extended abstract thinking involves one or more of:
 devising a strategy to investigate or solve a problem
 identifying relevant concepts in context
 developing a chain of logical reasoning, or proof
 forming a generalisation;
and also using correct mathematical statements, or communicating mathematical insight.
3
Problems are situations that provide opportunities to apply knowledge or understanding of mathematical concepts and methods.
Situations will be set in real-life or mathematical contexts.
4
Methods include a selection from those related to:
 graphs and equations of the circle, ellipse, parabola, and hyperbola
 Cartesian and parametric forms
 properties of conic sections
 tangents and normals.
5
Conditions of Assessment related to this achievement standard can be found at www.tki.org.nz/e/community/ncea/conditionsassessment.php.
8
Replacement Information
This achievement standard replaced unit standard 20661 and AS90639.
Quality Assurance
1
Providers and Industry Training Organisations must have been granted consent to assess by NZQA before they can register
credits from assessment against achievement standards.
2
Organisations with consent to assess and Industry Training Organisations assessing against achievement standards must engage
with the moderation system that applies to those achievement standards.
Consent and Moderation Requirements (CMR) reference
0233
9
Achievement Standard
Subject Reference
Mathematics and Statistics 3.3
Title
Apply trigonometric methods in solving problems
Level
3
Subfield
Mathematics
Domain
Trigonometry
Credits
4
Status
Assessment
Internal
Status date
Planned review date
31 December 2016
Date version published
This achievement standard involves applying trigonometric methods in solving problems.
Achievement Criteria
Achievement
Achievement with Merit
Achievement with Excellence
 Apply trigonometric methods
in solving problems.
 Apply trigonometric methods, using
relational thinking, in solving
problems.
 Apply trigonometric methods, using
extended abstract thinking, in solving
problems.
Explanatory Notes
4
This achievement standard is derived from Level 8 of The New Zealand Curriculum, Learning Media, Ministry of Education,
2007; and is related to the achievement objectives:
 Manipulate trigonometric expressions
 Form and use trigonometric equations
in the Mathematics strand of the Mathematics and Statistics Learning Area. It is also related to the material in the Teaching
and Learning Guide for Mathematics and Statistics, Ministry of Education, 2012, at http://seniorsecondary.tki.org.nz.
5
Apply trigonometric methods in solving problems involves:
 selecting and using methods
 demonstrating knowledge of concepts and terms
 communicating using appropriate representations.
Relational thinking involves one or more of:
 selecting and carrying out a logical sequence of steps
 connecting different concepts or representations
 demonstrating understanding of concepts
 forming and using a model;
and also relating findings to a context, or communicating thinking using appropriate mathematical statements.
Extended abstract thinking involves one or more of:
 devising a strategy to investigate or solve a problem
 identifying relevant concepts in context
 developing a chain of logical reasoning, or proof
 forming a generalisation;
and also using correct mathematical statements, or communicating mathematical insight.
6
Problems are situations that provide opportunities to apply knowledge or understanding of mathematical concepts and methods.
Situations will be set in real-life or mathematical contexts.
7
Methods include a selection from those related to:
 trigonometric identities
 reciprocal trigonometric functions
 properties of trigonometric functions
 solving trigonometric equations
 general solutions.
10
8
Conditions of Assessment related to this achievement standard can be found at www.tki.org.nz/e/community/ncea/conditionsassessment.php.
Replacement Information
This achievement standard replaced unit standard 5268 and AS90637.
Quality Assurance
3
Providers and Industry Training Organisations must have been granted consent to assess by NZQA before they can register
credits from assessment against achievement standards.
4
Organisations with consent to assess and Industry Training Organisations assessing against achievement standards must engage
with the moderation system that applies to those achievement standards.
Consent and Moderation Requirements (CMR) reference
0233
11
Achievement Standard
Subject Reference
Mathematics and Statistics 3.4
Title
Use critical path analysis in solving problems
Level
3
Subfield
Mathematics
Domain
Geometry
Credits
2
Status
Assessment
Internal
Status date
Planned review date
31 December 2016
Date version published
This achievement standard involves using critical path analysis in solving problems.
Achievement Criteria
Achievement
Achievement with Merit
Achievement with Excellence
 Use critical path analysis in
solving problems.
 Use critical path analysis, with
relational thinking, in solving
problems.
 Use critical path analysis, with
extended abstract thinking, in solving
problems.
Explanatory Notes
9
This achievement standard is derived from Level 8 of The New Zealand Curriculum, Learning Media, Ministry of Education,
2007; and is related to the achievement objective:
 Develop network diagrams to find optimal solutions, including critical paths
in the Mathematics strand of the Mathematics and Statistics Learning Area. It is also related to the material in the Teaching
and Learning Guide for Mathematics and Statistics, Ministry of Education, 2012, at http://seniorsecondary.tki.org.nz.
10
Use critical path analysis in solving problems involves:
 selecting and using methods
 demonstrating knowledge of concepts and terms
 communicating using appropriate representations.
Relational thinking involves one or more of:
 selecting and carrying out a logical sequence of steps
 connecting different concepts or representations
 demonstrating understanding of concepts
 forming and using a model;
and also relating findings to a context, or communicating thinking using appropriate mathematical statements.
Extended abstract thinking involves one or more of:
 devising a strategy to investigate or solve a problem
 identifying relevant concepts in context
 developing a chain of logical reasoning, or proof
 forming a generalisation;
and also using correct mathematical statements, or communicating mathematical insight.
11
Problems are situations that provide opportunities to apply knowledge or understanding of mathematical concepts and methods.
Situations will be set in real-life or mathematical contexts.
12
Methods include a selection from those related to:
 precedence tables
 network diagrams
 critical events
 scheduling
 float times.
12
13
Conditions of Assessment related to this achievement standard can be found at www.tki.org.nz/e/community/ncea/conditionsassessment.php.
Quality Assurance
5
Providers and Industry Training Organisations must have been granted consent to assess by NZQA before they can register
credits from assessment against achievement standards.
6
Organisations with consent to assess and Industry Training Organisations assessing against achievement standards must engage
with the moderation system that applies to those achievement standards.
Consent and Moderation Requirements (CMR) reference
0233
13
Achievement Standard
Subject Reference
Mathematics and Statistics 3.5
Title
Apply the algebra of complex numbers in solving problems
Level
3
Subfield
Mathematics
Domain
Algebra
Credits
5
Status
Assessment
External
Status date
Planned review date
31 December 2016
Date version published
This achievement standard involves applying the algebra of complex numbers in solving problems.
Achievement Criteria
Achievement
Achievement with Merit
Achievement with Excellence
 Apply the algebra of complex
numbers in solving problems.
 Apply the algebra of complex
numbers, using relational thinking, in
solving problems.
 Apply the algebra of complex
numbers, using extended abstract
thinking, in solving problems.
Explanatory Notes
14
This achievement standard is derived from Level 8 of The New Zealand Curriculum, Learning Media, Ministry of Education,
2007; and is related to the achievement objectives:
 Manipulate complex numbers and present them graphically
 Form and use polynomial, and other non-linear equations
in the Mathematics strand of the Mathematics and Statistics Learning Area. It is also related to the material in the Teaching
and Learning Guide for Mathematics and Statistics, Ministry of Education, 2012, at http://seniorsecondary.tki.org.nz/.
15
Apply the algebra of complex numbers in solving problems involves:
 selecting and using methods
 demonstrating knowledge of concepts and terms
 communicating using appropriate representations.
Relational thinking involves one or more of:
 selecting and carrying out a logical sequence of steps
 connecting different concepts or representations
 demonstrating understanding of concepts
 forming and using a model;
and also relating findings to a context, or communicating thinking using appropriate mathematical statements.
Extended abstract thinking involves one or more of:
 devising a strategy to investigate or solve a problem
 identifying relevant concepts in context
 developing a chain of logical reasoning, or proof
 forming a generalisation;
and also using correct mathematical statements, or communicating mathematical insight.
16
Problems are situations that provide opportunities to apply knowledge or understanding of mathematical concepts and methods.
Situations will be set in real-life or mathematical contexts.
17
Methods are selected from those related to:
 quadratic and cubic equations with complex roots
 Argand diagrams
 polar and rectangular forms
 manipulation of surds
 manipulation of complex numbers
14



18
loci
De Moivre’s theorem
equations of the form zn = r cis , or zn = a + b i where a, b are real and n is a positive integer.
Assessment Specifications for this achievement standard can be accessed through the Mathematics and Statistics Resources
page found at http://www.nzqa.govt.nz/qualifications-standards/qualifications/ncea/subjects/.
Replacement Information
This achievement standard replaced unit standard 5267 and AS90638.
Quality Assurance
7
Providers and Industry Training Organisations must have been granted consent to assess by NZQA before they can register
credits from assessment against achievement standards.
8
Organisations with consent to assess and Industry Training Organisations assessing against achievement standards must engage
with the moderation system that applies to those achievement standards.
Consent and Moderation Requirements (CMR) reference
0233
15
Achievement Standard
Subject Reference
Mathematics and Statistics 3.6
Title
Apply differentiation methods in solving problems
Level
3
Subfield
Mathematics
Domain
Calculus
Credits
6
Status
Assessment
External
Status date
Planned review date
31 December 2016
Date version published
This achievement standard involves applying differentiation methods in solving problems.
Achievement Criteria
Achievement
Achievement with Merit
Achievement with Excellence
 Apply differentiation
methods in solving
problems.
 Apply differentiation methods, using
relational thinking, in solving
problems.
 Apply differentiation methods, using
extended abstract thinking, in solving
problems.
Explanatory Notes
19
This achievement standard is derived from Level 8 of The New Zealand Curriculum, Learning Media, Ministry of Education,
2007; and is related to the achievement objectives:
 Identify discontinuities and limits of functions
 Choose and apply a variety of differentiation techniques to functions and relations using analytical methods
in the Mathematics strand of the Mathematics and Statistics Learning Area. It is also related to the material in the Teaching
and Learning Guide for Mathematics and Statistics, Ministry of Education, 2012, at http://seniorsecondary.tki.org.nz.
20
Apply differentiation methods in solving problems involves:
 selecting and using methods
 demonstrating knowledge of concepts and terms
 communicating using appropriate representations.
Relational thinking involves one or more of:
 selecting and carrying out a logical sequence of steps
 connecting different concepts or representations
 demonstrating understanding of concepts
 forming and using a model;
and also relating findings to a context, or communicating thinking using appropriate mathematical statements.
Extended abstract thinking involves one or more of:
 devising a strategy to investigate or solve a problem
 identifying relevant concepts in context
 developing a chain of logical reasoning, or proof
 forming a generalisation;
and also using correct mathematical statements, or communicating mathematical insight.
3
Problems are situations that provide opportunities to apply knowledge or understanding of mathematical concepts and methods.
Situations will be set in real-life or mathematical contexts.
4
Methods are selected from those related to:
 derivatives of power, exponential, and logarithmic (base e only) functions
 derivatives of trigonometric (including reciprocal) functions
 optimisation
 equations of normals
 maxima and minima and points of inflection
16




5
related rates of change
derivatives of parametric functions
chain, product, and quotient rules
properties of graphs (limits, differentiability, continuity, concavity).
Assessment Specifications for this achievement standard can be accessed through the Mathematics and Statistics Resources
page found at http://www.nzqa.govt.nz/qualifications-standards/qualifications/ncea/subjects/.
Replacement Information
This achievement standard replaced unit standard 5265 and AS90635.
Quality Assurance
9
Providers and Industry Training Organisations must have been granted consent to assess by NZQA before they can register
credits from assessment against achievement standards.
10
Organisations with consent to assess and Industry Training Organisations assessing against achievement standards must engage
with the moderation system that applies to those achievement standards.
Consent and Moderation Requirements (CMR) reference
0233
17
Achievement Standard
Subject Reference
Mathematics and Statistics 3.7
Title
Apply integration methods in solving problems
Level
3
Subfield
Mathematics
Domain
Calculus
Credits
6
Status
Assessment
External
Status date
Planned review date
31 December 2016
Date version published
This achievement standard involves applying integration methods in solving problems.
Achievement Criteria
Achievement
Achievement with Merit
Achievement with Excellence
 Apply integration methods in
solving problems.
 Apply integration methods, using
relational thinking, in solving
problems.
 Apply integration methods, using
extended abstract thinking, in solving
problems.
Explanatory Notes
21
This achievement standard is derived from Level 8 of The New Zealand Curriculum, Learning Media, Ministry of Education,
2007; and is related to the achievement objectives:
 Choose and apply a variety of integration and anti-differentiation techniques to functions and relations using both
analytical and numerical methods
 Form differential equations and interpret the solutions
in the Mathematics strand of the Mathematics and Statistics Learning Area. It is also related to the material in the Teaching
and Learning Guide for Mathematics and Statistics, Ministry of Education, 2012, at http://seniorsecondary.tki.org.nz.
22
Apply integration methods in solving problems involves:
 selecting and using methods
 demonstrating knowledge of concepts and terms
 communicating using appropriate representations.
Relational thinking involves one or more of:
 selecting and carrying out a logical sequence of steps
 connecting different concepts or representations
 demonstrating understanding of concepts
 forming and using a model;
and also relating findings to a context, or communicating thinking using appropriate mathematical statements .
Extended abstract thinking involves one or more of:
 devising a strategy to investigate or solve a problem
 identifying relevant concepts in context
 developing a chain of logical reasoning, or proof
 forming a generalisation;
and also using correct mathematical statements, or communicating mathematical insight.
3
Problems are situations that provide opportunities to apply knowledge or understanding of mathematical concepts and methods.
Situations will be set in real-life or mathematical contexts.
4
Methods are selected from those related to:
 integrating power, polynomial, exponential (base e only), trigonometric, and rational functions
 reverse chain rule, trigonometric formulae
 rates of change problems
 areas under or between graphs of functions, by integration
18


5
finding areas using numerical methods, e.g. the rectangle or trapezium rule
differential equations of the forms y' = f(x) or y" = f(x) for the above functions or situations where the variables are
separable (e.g. y' = ky) in applications such as growth and decay, inflation, Newton's Law of Cooling and similar situations.
Assessment Specifications for this achievement standard can be accessed through the Mathematics and Statistics Resources
page found at http://www.nzqa.govt.nz/qualifications-standards/qualifications/ncea/subjects/.
Replacement Information
This achievement standard replaced unit standard 20660, unit standard 20905, and AS90636.
Quality Assurance
11
Providers and Industry Training Organisations must have been granted consent to assess by NZQA before they can register
credits from assessment against achievement standards.
12
Organisations with consent to assess and Industry Training Organisations assessing against achievement standards must engage
with the moderation system that applies to those achievement standards.
Consent and Moderation Requirements (CMR) reference
0233
19