# Learning Objectives of Module 1 (Calculus and Statistics) Notes:

In document Mathematics Curriculum and Assessment Guide (Secondary 4 - 6) (Page 56-74)

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 Area 1. Binomial

expansion

1.1 recognise the expansion of (a+b)n, where n is a positive integer

3 The use of the summation notation (∑) should be introduced.

The following are not required:

• expansion of trinomials

• the greatest coefficient, the greatest term and the properties of binomial coefficients

applications to numerical approximation

49

Learning Unit Learning Objective Time Remarks

2. Exponential and logarithmic functions

2.1 recognise the definition of the number e and the exponential series

2 3

1 ...

2! 3!

x x x

e = + +x + +

7

2.2 recognise exponential functions and logarithmic functions The following functions are required:

y =ex

y=lnx

2.3 use exponential functions and logarithmic functions to solve problems

Students are expected to know how to solve problems including those related to compound interest, population growth and radioactive decay.

50

Learning Unit Learning Objective Time Remarks

2.4 transform y=kxn and y=kax to linear relations, where a, n and k are real numbers, a>0 and a≠1

When experimental values of x and y are given, students can plot the graph of the corresponding linear relation from which they can determine the values of the unknown constants by considering its slope and intercept.

Subtotal in hours 10 Calculus Area

Differentiation and Its Applications 3. Derivative of a

function

3.1 recognise the intuitive concept of the limit of a function 5 The concepts of continuous function and discontinuous function are not required.

Theorems on the limits of sum, difference, product, quotient, scalar multiplication of functions and the limits of composite functions should be stated without proof.

51

Learning Unit Learning Objective Time Remarks

3.2 find the limits of algebraic functions, exponential functions and logarithmic functions

The following types of algebraic functions are required:

polynomial functions

rational functions

power functions x α

functions derived from the above ones through addition, subtraction,

multiplication, division and composition, for example,

2 1

x +

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.

Notations including y', f ' x( ) and dx

dy should be introduced.

3.4 recognise the slope of the tangent of the curve y= f(x) at a point x=x0

Notations including f '(x0) and

x0

dx x

dy

=

should be introduced.

52

Learning Unit Learning Objective Time Remarks

4. Differentiation of a function

4.1 understand the addition rule, product rule, quotient rule and chain rule of differentiation

7 The following rules are required:

dx

dv dx v du dx u

d ( + )= +

dx

vdu dx udv dx uv

d ( )= +

• ( ) 2

v dx udv dx vdu

v u dx

d

=

dx

du du dy dx

dy = ⋅

53

Learning Unit Learning Objective Time Remarks

4.2 find the derivatives of algebraic functions, exponential functions and logarithmic functions

The following formulae are required:

• (C)'=0

• (xn)'=nxn1

• (ex)'=ex

x 1x )' ln

( =

ax x a

ln )' 1

log

( =

• (ax)'=ax lna

Implicit differentiation is not required.

Logarithmic differentiation is not required.

54

Learning Unit Learning Objective Time Remarks

5. Second derivative

5.1 recognise the concept of the second derivative of a function 2 Notations including y", f" x( ) and

2 2

dx y

d should be introduced.

Third and higher order derivatives are not required.

5.2 find the second derivative of an explicit function 6. Applications of

differentiation

6.1 use differentiation to solve problems involving tangent, rate of change, maximum and minimum

9 Local and global extrema are required.

Subtotal in hours 23 Integration and Its Applications

7. Indefinite integrals and their

applications

7.1 recognise the concept of indefinite integration 10 Indefinite integration as the reverse process of differentiation should be introduced.

55

Learning Unit Learning Objective Time Remarks

7.2 understand the basic properties of indefinite integrals and basic integration formulae

The notation

### ∫

f x dx( ) should be introduced.

The following properties are required:

k f x dx( ) =k

f x dx( )

### ∫

[ ( )f x ±g x dx( )] =

f x dx( ) ±

### ∫

g x dx( )

The following formulae are required and the meaning of the constant of

integration C should be explained:

k dx=kx C+

1

1

n

n x

x dx C

### ∫

= n++ + , where n1

1

lndx x C

x = +

### ∫

e dxx =ex+ C

56

Learning Unit Learning Objective Time Remarks

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

integrals and their

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.

The notation b ( )

a f x dx

### ∫

should be

introduced.

The knowledge of dummy variables, i.e.

( ) ( )

b b

a f x dx a f t dt

=

### ∫

is required.

57

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 refers to b ( ) ( ) ( )

a f x dx F b F a

### ∫

= ,

where F(x) f(x) dx

d = .

The following properties are required:

b ( ) a ( )

a f x dxb 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 ( )

a k f x dx k a f x dx

=

b[ ( ) ( )]

a f x g x dx

±

( ) ( )

ab f x dx±

### ∫

ab g x dx

58

Learning Unit Learning Objective Time Remarks

8.3 find the definite integrals of algebraic functions and exponential functions

8.4 use integration by substitution to find definite integrals

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.

Subtotal in hours 26

59

Learning Unit Learning Objective Time Remarks

Statistics Area Further Probability 10. Conditional

probability and independence

10.1 understand the concepts of conditional probability and independent events

3

10.2 use the laws P(A ∩ B) = P(A) P(B | A) and P(D | C) = P(D) for independent events C and D to solve problems

11. Bayes’ theorem 11.1 use Bayes’ theorem to solve simple problems 4 Subtotal in hours 7 Binomial, Geometric and Poisson Distributions and Their Applications

12. Discrete random variables

12.1 recognise the concept of a discrete random variable 1

13. Probability distribution, expectation and variance

13.1 recognise the concept of discrete probability distribution and its representation in the form of tables, graphs and

mathematical formulae

5

13.2 recognise the concepts of expectation E( X) and variance )

(

Var X and use them to solve simple problems

60

Learning Unit Learning Objective Time Remarks

13.3 use the formulae E aX

+b

=aE X

+ and b

2

### ( )

Var aX +b =a Var X to solve simple problems 14. Binomial

distribution

14.1 recognise the concept and properties of the binomial distribution

5 Bernoulli distribution should be introduced.

The mean and variance of the binomial distribution should be introduced (proofs are not required).

14.2 calculate probabilities involving the binomial distribution Use of the binomial distribution table is not required.

15. Geometric distribution

15.1 recognise the concept and properties of the geometric distribution

4 The mean and variance of geometric distribution should be introduced (proofs are not required).

15.2 calculate probabilities involving the geometric distribution 16. Poisson

distribution

16.1 recognise the concept and properties of the Poisson distribution

4 The mean and variance of Poisson distribution should be introduced (proofs are not required).

61

Learning Unit Learning Objective Time Remarks

16.2 calculate probabilities involving the Poisson distribution Use of the Poisson distribution table is not required.

17. Applications of binomial, geometric and Poisson distributions

17.1 use binomial, geometric and Poisson distributions to solve problems

5

Subtotal in hours 24 Normal Distribution and Its Applications

18. Basic definition and properties

18.1 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.

The formulae written in Learning Objective 13.3 are also applicable to continuous random variables.

62

Learning Unit Learning Objective Time Remarks

18.2 recognise the concept and properties of the normal distribution

Properties of the normal distribution include:

the curve is bell-shaped and symmetrical about the mean

the mean, mode and median are equal

the dispersion can be determined by the value of σ

the area under the curve is 1 19. Standardisation

of a normal variable and use of the standard normal table

19.1 standardise a normal variable and use the standard normal table to find probabilities involving the normal distribution

2

63

Learning Unit Learning Objective Time Remarks

20. Applications of the normal distribution

20.1 find the values of P(X >x1), P(X < x2), P(x1 < X < x2) and related probabilities, given the values of x1, x2, µ and σ , where X ~ N(µ, σ2)

7

20.2 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(µ, σ2)

20.3 use the normal distribution to solve problems

Subtotal in hours 12 Point and Interval Estimation

21. Sampling distribution and point estimates

21.1 recognise the concepts of sample statistics and population parameters

7

21.2 recognise the sampling distribution of the sample mean from a random sample of size n

If the population mean is µ and

population variance is σ2, then the mean of the sample mean is µ and the variance of the sample mean is

2

n σ .

64

Learning Unit Learning Objective Time Remarks

21.3 recognise the concept of point estimates including the sample mean, sample variance and sample proportion

The concept of “estimator” should be introduced.

If the population mean is µ and the population size is N, then the population

variance is

N x

N i

### ∑

i

=

= 1

2 2

)

( µ

σ .

If the sample mean is x and the sample size is n, then the sample variance is

1 ) (

1

2 2

=

### ∑

=

n x x s

n i

i

.

Recognising the concept of “unbiased estimator” is required.

21.4 recognise Central Limit Theorem 22. Confidence

interval for a population mean

22.1 recognise the concept of confidence interval 6

22.2 find the confidence interval for a population mean a 100(1 − α)% confidence interval for the mean µ of a normal population with known variance σ2 is given by

65

Learning Unit Learning Objective Time Remarks

)

, (

2

2 x z n

z n

x σ σ

α

α +

when the sample size n is sufficiently large, a 100(1 − α)% confidence interval for the mean µ of a population with unknown variance is given by

)

, (

2

2 n

z s x n z s

xα + α ,

where s is the sample standard deviation

23. Confidence interval for a population proportion

23.1 find an approximate confidence interval for a population proportion

3 For a random sample of size n, where n is sufficiently large, drawn from a Bernoulli distribution, a 100(1 − α)% confidence interval for the population proportion p is given by

)) ˆ 1 ( ˆ ˆ

), ˆ 1 ( ˆ ˆ

(

2

2 n

p z p

n p p z p

pα − + α − ,

where ˆp is an unbiased estimator of the population proportion.

Subtotal in hours 16

66

Learning Unit Learning Objective Time Remarks

Further Learning Unit 24. 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

In document Mathematics Curriculum and Assessment Guide (Secondary 4 - 6) (Page 56-74)