1. Learning units are grouped under three areas (“Foundation Knowledge”, “Calculus” and “Statistics”) and a Further Learning Unit.
2. Related learning objectives are grouped under the same learning unit.
3. The notes in the “Remarks” column of the table may be considered as supplementary information about the learning objectives.
4. To aid teachers in judging how far to take a given topic, a suggested lesson time in hours is given against each learning unit. However, the lesson time assigned is for their reference only. Teachers may adjust the lesson time to meet their individual needs.
Learning Unit Learning Objective Time Remarks
Foundation Knowledge
1. Binomial expansion 1.1 recognise the expansion of (ab)n , where n is a positive integer
3 Students are required to recognise the summation notation ().
The following contents are not required:
expansion of trinomials
the greatest coefficient, the greatest term and the properties of binomial coefficients
applications to numerical approximation
4646
Learning Unit Learning Objective Time Remarks
2. Exponential and logarithmic functions
2.1 recognise the definition of e and the exponential series
2 3
1 2! 3!
x x x
e x
8
2.2 understand exponential functions and logarithmic functions
The following functions are required:
yex
ylnx
2.3 use exponential functions and logarithmic functions to solve problems
Students are required to solve problems including those related to compound interest, population growth and radioactive decay.
2.4 transform y kax and yk f x
( )
n to linear relations, where a , n and k are real numbers,0
a , a1, f x( )0 and f x( ) 1
When experimental values of x and y are given, students are required to plot the graph of the corresponding linear relation from which they can determine the values of the unknown constants by considering its slope and intercepts.
Subtotal in hours 11
4747
Learning Unit Learning Objective Time Remarks
Calculus
3. Derivative of a function
3.1 recognise the intuitive concept of the limit of a function
5 Student are required to recognise the theorems on the limits of sum, difference, product, quotient, scalar multiplication of functions and the limits of composite functions (the proofs are not required).
3.2 find the limits of algebraic functions, exponential functions and logarithmic functions
The following algebraic functions are required:
polynomial functions
rational functions
power functions x
functions derived from the above ones through addition, subtraction, multiplication, division and composition, such as x21
3.3 recognise the concept of the derivative of a function from first principles
Students are not required to find the derivatives of functions from first principles.
Students are required to recognise the notations: y,f x( ) and dy
dx .
4848
Learning Unit Learning Objective Time Remarks
3.4 recognise the slope of the tangent of the curve ( )
y f x at a point xx0
Students are required to recognise the notations: f x and '( )0
x x0
dy dx .
4. Differentiation of a function
4.1 understand the addition rule, product rule, quotient rule and chain rule of differentiation
8 The rules include:
d ( ) du dv u v
dx dx dx
d ( ) du dv
uv v u
dx dx dx
( ) 2
du dv
v u
d u dx dx
dx v v
dy dy du dx du dx
4949
Learning Unit Learning Objective Time Remarks
4.2 find the derivatives of algebraic functions, exponential functions and logarithmic functions
The formulae that students are required to use include:
( )C 0
(xn) nxn1
( )ex ex
1
ln ) ( x
x
(log ) 1
ax ln
x a
(ax) axlna
Implicit differentiation and logarithmic differentiation are not required.
5. Second derivative 5.1 recognise the concept of the second derivative of a function
2 Students are required to recognise the notations: y, f( )x and
2 2
d y dx .
Third and higher order derivatives are not required.
5050
Learning Unit Learning Objective Time Remarks
5.2 find the second derivative of an explicit function Students are required to recognise the second derivative test and concavity.
6. Applications of differentiation
6.1 use differentiation to solve problems involving tangent, rate of change, maximum and minimum
10 Local and global extrema are required.
7. Indefinite integration and its applications
7.1 recognise the concept of indefinite integration 10 Indefinite integration as the reverse process of differentiation should be introduced.
7.2 understand the basic properties of indefinite integrals and basic integration formulae
Students are required to recognise the notation:
f x dx( ) .The properties include:
kf x dx( ) k f x dx
( )
f x( )g x dx( )
( ) ( )
f x dx g x dx
The formulae include:
kdxkxC
1
1
n
n x
x dx C
n
5151
Learning Unit Learning Objective Time Remarks
1
ln x xdx C
e dxx exCStudents are required to understand the meaning of the constant of integration C.
7.3 use basic integration formulae to find the indefinite integrals of algebraic functions and exponential functions
7.4 use integration by substitution to find indefinite integrals
Integration by parts is not required.
7.5 use indefinite integration to solve problems 8. Definite integration
and its applications
8.1 recognise the concept of definite integration 12 The definition of the definite integral as the limit of a sum of the areas of rectangles under a curve should be introduced.
Students are required to recognise the notation: b ( )
a f x dx
.The concept of dummy variables is required, for example: b ( ) b ( )
a f x dx a f t dt
.5252
Learning Unit Learning Objective Time Remarks
8.2 recognise the Fundamental Theorem of Calculus and understand the properties of definite integrals
The Fundamental Theorem of Calculus that students are required to recognise is:
( ) ( ) ( )
b
a f x dxF b F a
, where( ) ( ) d F x f x
dx .
The properties include:
b ( ) a ( )
a f x dx b f x dx
a ( ) 0
a f x dx
b ( ) c ( ) b ( )
a f x dx a f x dx c f x dx
b ( ) b ( )
akf x dxk a f x dx
b
( ) ( )
a f x g x dx
( ) ( )
b b
a f x dx a g x dx
8.3 find the definite integrals of algebraic functions and exponential functions
8.4 use integration by substitution to find definite integrals
5353
Learning Unit Learning Objective Time Remarks
8.5 use definite integration to find the areas of plane figures
Students are not required to use definite integration to find the area between a curve and the y-axis and the area between two curves.
8.6 use definite integration to solve problems 9. Approximation of
definite integrals using the trapezoidal rule
9.1 understand the trapezoidal rule and use it to estimate the values of definite integrals
4 Error estimation is not required.
Students are required to determine whether an estimate is an over-estimate or under-estimate by using the second derivative and concavity.
Subtotal in hours 51 Statistics
10. Conditional
probability and Bayes’
theorem
10.1 understand the concept of conditional probability 6 10.2 use Bayes’ theorem to solve simple problems
11. Discrete random variables
11.1 recognise the concept of discrete random variables 1
5454
Learning Unit Learning Objective Time Remarks
12. Probability distribution, expectation and variance
12.1 recognise the concept of discrete probability distribution and represent the distribution in the form of tables, graphs and mathematical formulae
7
12.2 recognise the concepts of expectation E X
and variance Var( )X and use them to solve simple problems
The formulae that students are required to use include:
E X
xP X( x) Var( )X E(X )2
E g X
( )
g x P X( ) ( x) E aX
b
aE X
b Var( )X E X 2(E X
)2 Var(aXb)a2Var( )X Notation E X
can also be used.5555
Learning Unit Learning Objective Time Remarks
13. The binomial distribution
13.1 recognise the concept and properties of the binomial distribution
5 The Bernoulli distribution should be introduced.
The mean and variance of the binomial distribution are required (the proofs are not required).
13.2 calculate probabilities involving the binomial distribution
Use of the binomial distribution table is not required.
14. The Poisson distribution
14.1 recognise the concept and properties of the Poisson distribution
5 The mean and variance of Poisson distribution are required (the proofs are not required).
14.2 calculate probabilities involving the Poisson distribution
Use of the Poisson distribution table is not required.
15. Applications of the binomial and the Poisson distributions
15.1 use the binomial and the Poisson distributions to solve problems
5
16. Basic definition and properties of the normal distribution
16.1 recognise the concepts of continuous random variables and continuous probability distributions, with reference to the normal distribution
3 Derivations of the mean and variance of the normal distribution are not required.
Students are required to recognise that the formulae in Learning Objective 12.2 are also applicable to continuous random variables.
5656
Learning Unit Learning Objective Time Remarks
16.2 recognise the concept and properties of the normal distribution
The properties include:
the curve is bell-shaped and symmetrical about the mean
the mean, mode and median are all equal
the flatness can be determined by the value of
the area under the curve is 1 17. Standardisation of a
normal variable and use of the standard normal table
17.1 standardise a normal variable and use the standard normal table to find probabilities involving the normal distribution
2
18. Applications of the normal distribution
18.1 find the values of P X( x1) , P X( x2) ,
1 2
( )
P x X x and related probabilities, given the values of x1 , x2 , and , where
~ ( , 2) X N
7
18.2 find the values of x, given the values of P X( x),
( )
P X x , P a( X x) , P x( X b) or a related probability, where X ~N( , 2)
18.3 use the normal distribution to solve problems
5757
Learning Unit Learning Objective Time Remarks
19. Sampling distribution and point estimates
19.1 recognise the concepts of sample statistics and population parameters
9 Students are required to recognise:
If the population mean is and the population size is N, then the population variance is
2
2 1
( )
N i i
x N
. 19.2 recognise the sampling distribution of the sample
mean X from a random sample of size n
Students are required to recognise:
If the population mean is and the population variance is 2 , then
E X and
2
Var(X) n
.
IfX ~N( , 2), then
2
~ ( , ) X N
n
(the proof is not required).
19.3 use the Central Limit Theorem to treat X as being normally distributed when the sample size n is sufficiently large
5858
Learning Unit Learning Objective Time Remarks
19.4 recognise the concept of point estimates including the sample mean and sample variance
Students are required to recognise:
If the sample mean is x and the sample size is n, then the sample variance is
2
2 1
( )
1
n i i
x x
s n
.
Students are required to recognise the concept of unbiased estimator.
20. Confidence interval for a population mean
20.1 recognise the concept of confidence interval 6
20.2 find the confidence interval for a population mean Students are required to recognise:
A 100(1)% confidence interval for the mean of a normal population with known variance2, based on a random sample of size n, is given by
2 2
(x z ,x z )
n n
.
When the sample size n is sufficiently large, a 100(1)% confidence interval for the mean of a population with unknown variance is given by
2 2
( s , s )
x z x z
n n
, where s is the
5959
Learning Unit Learning Objective Time Remarks
sample standard deviation.
Subtotal in hours 56 Further Learning Unit
21. Inquiry and investigation
Through various learning activities, discover and construct knowledge, further improve the ability to inquire, communicate, reason and conceptualise mathematical concepts
7 This is not an independent and isolated learning unit. The time is allocated for students to engage in learning activities from different learning units.
Subtotal in hours 7