Mathematics standards Grade 12 - sec.gov.qa And Subject/Math-Grade 12 (advanced...306 | Qatar...

305 | Qatar mathematics standards | Grade 12 advanced | Mathematics for science © Supreme Education Council 2004

Mathematics standards

Summary of students’ performance by the end of Grade 12

Reasoning and problem solving

Students solve a wide range of problems in mathematical and other contexts. They use mathematics to model and predict outcomes of substantial real-world applications. They break problems into smaller tasks, and set up and perform relevant manipulations and calculations. They identify and use interconnections between mathematical topics. They develop and explain chains of logical reasoning, using correct mathematical notation and terms, including logic symbols. They generate mathematical proofs. They generalise when possible and remark on special cases. They approach problems systematically, knowing when and how to enumerate all outcomes. They conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions. They synthesise, present, interpret and criticise mathematical information. They work to expected degrees of accuracy, and understand error bounds. They recognise when to use ICT efficiently and use it efficiently.

Number, algebra and calculus

Students continue to develop skills of algebraic manipulation through further work on factorisation, exponents and logarithms, partial fractions, summation of series and combinatorics. They understand and use the remainder theorem and the factor theorem. They expand and use the binomial series (1 + x)n for any rational value of n. They work with a range of functions and their inverses, including polynomial functions up to order four; the reciprocal, exponential and logarithmic functions, and the modulus function. They plot and describe the features of the circular functions. They understand the more detailed behaviour of these functions through their awareness of the associated differential and integral calculus. They find higher order derivatives of functions, and work out approximations. They find indefinite and definite integrals and solve simple differential equations. They use these functions and the calculus to model a range of substantial real-world scenarios. They use realistic data and ICT to analyse problems.

Geometry and measures

Students are aware of links between geometry and algebra, which deepens their understanding of space and movement. They understand the roles that trigonometry and circular functions play in modelling and in mathematical transformation. They use trigonometric identities to solve trigonometric equations. They use vectors to extend the study of space and motion into three dimensions, and they are familiar with curves represented by parametric equations. They use dimensionally correct units for length, area and volume and for a range of measures, including velocity, acceleration and other compound measures. They find areas and volumes by integration and volumes of revolution. They use ICT to explore geometrical relationships.

Grade 12 Advanced

(mathematics for science)


Probability and statistics

Students apply and use the work on probability and statistics learned in earlier grades to solve problems.

Content and assessment weightings for Grade 12

The advanced mathematics standards for Grade 12 have two pathways: mathematics for science and quantitative mathematics, to support the social sciences and economics. Each pathway includes reasoning and problem solving, and number, algebra and calculus. The mathematics for science standards include substantial work on calculus but no new work on probability and statistics, whereas the quantitative methods standards include substantial work on probability and statistics, less calculus, and no new work on geometry and measures.

The reasoning and problem solving strand cuts across the other strands. Reasoning, generalisation and problem solving should be an integral part of the teaching and learning of mathematics in all lessons.

The weightings of the strands relative to each other are as follows:

Advanced Number, algebra

and calculus Geometry, measures

and trigonometry Probability and

statistics

Grade 10 55% 30% 15%

Grade 11 55% 30% 15%

Grade 12 (quantitative)

40% – 60%

Grade 12 (for science)

75% 25% –

The standards are numbered for easy reference. Those in shaded rectangles, e.g. 1.2, are the performance standards for all mathematics for science students. The national tests for advanced mathematics for science will be based on these standards. Grade 12 teachers should consolidate earlier standards as necessary.


Mathematics standards


By the end of Grade 12, students solve a wide range of problems in mathematical and other contexts. They use mathematics to model and predict outcomes of substantial real-world applications. They break problems into smaller tasks, and set up and perform relevant manipulations and calculations. They identify and use interconnections between mathematical topics. They develop and explain chains of logical reasoning, using correct mathematical notation and terms, including logic symbols. They generate mathematical proofs. They generalise when possible and remark on special cases. They approach problems systematically, knowing when and how to enumerate all outcomes. They conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions. They synthesise, present, interpret and criticise mathematical information. They work to expected degrees of accuracy, and understand error bounds. They recognise when to use ICT efficiently and use it efficiently.

Students should:

1 Use mathematical reasoning to solve problems

1.1 Solve routine and non-routine problems in a range of mathematical and other contexts, including open-ended and closed problems.

1.2 Use mathematics to model and predict the outcomes of substantial real-world applications, and to compare and contrast two or more given models of a particular situation.

1.3 Identify and use interconnections between mathematical topics.

1.4 Break down complex problems into smaller tasks.

1.5 Use a range of strategies to solve problems, including working the problem backwards and redirecting the logic forwards; set up and solve relevant equations and perform appropriate calculations and manipulations; change the viewpoint or mathematical representation; and introduce numerical, algebraic, graphical, geometrical or statistical reasoning as necessary.

1.6 Develop chains of logical reasoning, using correct terminology and mathematical notation, including symbols for logical implication.

State whether the following statements are true or false.

x2 = 16 ⇒ x = 4

x2 ≤ 16 ⇒ –4 ≤ x ≤ 4

43

4 3≥ ⇒ ≤xx

Grade 12 Advanced

(mathematics for science)

Key standards

Key performance standards are shown in shaded rectangles, e.g. 1.2.

Cross-references

Standards are referred to using the notation RP for reasoning and problem solving, NAC for number, algebra and calculus, GM for geometry and measures, e.g. standard GM 2.3

Examples of problems

Examples of problems in italics are intended to clarify the standards, not to represent the full range of possible problems.


Reasoning, generalisation and problem solving should be an integral part of the teaching and learning of mathematics in all lessons.


The Al Huda sisters made these statements.

Inas: ‘If the rug is in the car, then it is not in the garage.’ Safa: ‘If the rug is not in the car, then it is in the garage.’ Roza: ‘If the rug is in the garage, it is in the car.’ Farida: ‘If the rug is not in the car, then it is not in the garage.’

If Roza told the truth, who else must have told the truth?

A. Inas B. Safa C. Farida D. None need have told the truth.

TIMSS Grade 12

1.7 Explain their reasoning, both orally and in writing.

1.8 Understand and generate mathematical proofs, and discuss exceptional cases, knowing the importance of a counter-example.

1.9 Generalise whenever possible.

1.10 Approach complex problems systematically, recognising when it is important to enumerate all outcomes.

1.11 Conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions.

1.12 Synthesise, present, discuss, interpret and criticise mathematical information presented in various mathematical forms.

1.13 Work to expected degrees of accuracy, and know when an exact solution is appropriate.

1.14 Identify error bounds on measurements.

1.15 Recognise when to use ICT and when not to, and use it efficiently.

Number, algebra and calculus

By the end of Grade 12, students continue to develop skills of algebraic manipulation through further work on factorisation, exponents and logarithms, partial fractions, summation of series and combinatorics. They understand and use the remainder theorem and the factor theorem. They expand and use the binomial series (1 + x)n for any rational value of n. They work with a range of functions and their inverses, including polynomial functions up to order four; the reciprocal, exponential and logarithmic functions, and the modulus function. They plot and describe the features of the circular functions. They understand the more detailed behaviour of these functions through their awareness of the associated differential and integral calculus. They find higher order derivatives of functions, and work out approximations. They find indefinite and definite integrals and solve simple differential equations. They use these functions and the calculus to model a range of substantial real-world scenarios. They use realistic data and ICT to analyse problems.

Algebra and calculus

Students should know that algebra enables generalisation and the establishment of relationships between quantities and/or concepts. They should understand the nature and place of algebraic reasoning and proof, and how algebra may be related to geometric concepts, and vice versa. They should appreciate how calculus furthers the study of functions and of mathematical applications.


Students should:

2 Manipulate algebraic expressions

2.1 Multiply, factorise and simplify expressions and divide a polynomial by a linear or quadratic expression.

Write down in ascending powers of x the expansion of (2 – 3x)3.

Factorise 27x3 + 8y3. Use your factorisation to write an identity for 27x3 – 8y3.

Simplify 3

211

xx

+−

, given that x ≠ ±1.

Solve the equation (x + 3)(3x –1) – (x + 3)(5x + 2) = 0.

2.2 Combine and simplify rational algebraic fractions.

Simplify 25 3 2

1 1x

x x−−+ −

.

2.3 Decompose a rational algebraic fraction into partial fractions (with denominators not more complicated than repeated linear terms).

Show that 1 1 1( 1) 1r r r r≡ −+ + .

Find the values of A, B and C in the identity 2 21

3 1 1(3 1)( 1) ( 1)A B C

r rr r r≡ + ++ ++ + +

.

2.4 Understand and use the remainder theorem. 3x3 – 2x2 – ax – 28 has a remainder of 5 when divided by x – 3. What is the value of a?

2.5 Understand and use the factor theorem.

Show that 2x3 + x2 – 13x + 6 is divisible by x – 2 and find the other factors. Hence find the solution set in of the equation 2x3 + x2 – 13x + 6 = 0. What is the solution set in ?

3 Use index notation and logarithms to solve numerical problems

3.1 Understand exponents and nth roots, and apply the laws of indices to simplify expressions involving exponents; use the xy key and its inverse on a calculator.

Without using a calculator, simplify 5

28 2

4n n

n× .

Plot a graph of y = 1000 × 2–t for values of t from 0 to 1, increasing in steps of 0.2.

3.2 Know the definition of a logarithm in number base a (a > 0), and the rules of combination of logarithms, including change of base.

Give the value of log10 1000.

Evaluate log2 64.

Express log 27 – 2 log 3 as a single logarithm.

Prove that loglog logc

ac

bb a= and use this result to show that 1log logab

b a= .

Given logb 2 = 1/3, logb

32 is equal to

A. 2 B. 5 C. –3/5 D. 5/3 E. 2

3log 32

TIMSS Grade 12

Factorisation

Include the sum and difference of two cubes.

Logarithms

In addition to formulae for the sum and difference of two logarithms, include:

==== >

=

a

xa

x

a

an

a a

a xa x

aa

x n x

log

log

log 1log 1 0 for any 0log log


Explain why the number base of a logarithm must be positive, but why the logarithm itself may take any value.

3.3 Use the ln and log keys on a calculator and the corresponding inverse function keys.

Use a calculator to evaluate log5 4 correct to three decimal places.

In 1916, two scientists each named du Bois derived a formula to estimate the surface area S m2 of human beings in terms of their mass M kg and their height H cm. The formula is S = 0.000 718 4 × M0.425 × H0.725. A boy has a mass of 60 kg and a height of 160 cm. Calculate the boy’s surface area to three significant figures.

The intensity of sound, N, is measured in decibels (dB) and is defined by the formula N = 10 log (I / 10–16), where I is the power of sound measured in watts. Find N for normal speech with a power of 10–10 watts. Find the power of a jet aircraft with a sound intensity of 150 dB.

In chemistry, the pH value of a solution is defined as pH = – log [H], where [H] is the concentration of hydrogen ions in the solution. Solutions with pH < 7 are acidic and solutions with pH > 7 are alkaline. Solutions with pH = 7 are neutral. A solution has [H] = 5 × 10–7. Find its pH value and state whether it is acid, neutral or alkaline.

4 Work with sequences, series, recurrence relations and arrangements

4.1 Find permutations and combinations.

An examination consists of 13 questions. A student must answer only one of the first two questions and only nine of the remaining ones. How many choices of question does the student have? A. 13C10 = 286 B. 11C8 = 165 C. 2 × 11C9 = 110 D. 2 × 11P2 = 220 E. some other number

TIMSS Grade 12

Evaluate 9C6 and 10P5.

Prove that nCr = nCn–r and that nCr + nCr+1 = n+1Cr+1.

A committee of 6 people is to be chosen from 6 men and 4 women. In how many ways can the committee be chosen to include 3 men and 3 women?

Show that nC0 + nC1 + nC2 + …+ nCn = 2n.

In how many ways can 5 thick books, 4 medium sized books and 3 thin books be arranged on a bookshelf so that the books of the same size remain together? A. 5! 4! 3! 3! = 103 680 B. 5! 4! 3! = 17 280 C. (5! 4! 3!) × 3 = 51 840 D. 5 × 4 × 3 × 3 = 180 E. 212 × 3 = 12 288

TIMSS Grade 12

Using exponentials

Further examples of modelling with exponential functions are given in the margin note at NAC 8.18.

Permutations

The order of selection distinguishes a permutation from a combination.

Students should know that nCr is the number of combinations of r different objects from n different objects and that the number of permutations of r different objects from n different objects is r! nCr, which is denoted by nPr.


4.2 Expand the binomial series (1 + x)n for any rational value of n.

Without using a calculator, find the value of (1.01)6 to four decimal places.

By writing 3.75 as 3(1 + 0.25), apply the binomial series to evaluate √3.75 correct to three decimal places.

Show that, for small values of x, 22

1 1 2 3(1 )

x xx

≈ − ++

.

5 Work with functions and their graphs

5.1 Use a graphics calculator to plot exponential functions of the form y = ekx; describe these functions, distinguishing between cases when k is positive or negative, and the special case when k is zero.

Investigate the behaviour of the tangent lines to these curves and observe that these follow a similar pattern of increase or decrease as the function itself.

5.2 Plot and describe the features of the natural logarithm function y = ln x; understand that the natural logarithm function is inverse to the exponential function (see NAC 5.5 below).

A radioactive element decomposes according to the formula y = y0e–kt, where y is the mass of the element remaining after t days and y0 is the value of y for t = 0. Find the value of the constant k for an element whose half-life (i.e. the time taken to decompose half of the material) is 4 days. A. 1

e4 log 2 B. 1e 2log C. 2log e D. 1

4e(log 2) E. 42e

TIMSS Grade 12

5.3 Understand the modulus function y = | x | and sketch its graph; sketch the modulus of the functions in NAC 5.2–5.5.

Sketch the graph of the function f(x) = | x – 2 | + | x | for –3 ≤ x ≤ 3.

Sketch the curve with equation y = | sin x |.

5.4 Form composite functions and use the notation y = g(f(x)).

The function f(x) = 4x – 7 is defined on . A second function g(x) = (x + 1)2 is also

defined on . Find f(g(x)) and g(f(x)). Comment on why these functions are not the

same.

5.5 Form inverse functions (on a restricted domain, if necessary) and use the notation 1f ( )y x−= .

Show that the function f ( ) 1xx x= − , where x ≠ 1, is its own inverse.

Find the inverse of the function f : 2t t + , defined for t ≥ 0.

5.6 Know that 1f (f ( ))x x− = and that the graph of 1f ( )y x−= is the reflection of the graph of f ( )y x= about the straight line y = x.

Show that lne x x= .

Sketch the graph of y x= for non-negative values of x.

5.7 Understand that some functions are continuous everywhere and that some are piecewise continuous.

Sketch the graph of y = 3 + (x + 2)–1. Show clearly the asymptotes of the graph and the intercepts with each axis.

The function f(n) = n is defined on +. Sketch the graph of this function.

Binomial theorem

The theorem is used to expand positive integer powers of binomial terms. It has applications in probability. The binomial series is commonly used to generate approximations.

Functions

Include use of the notation Use the notation

x xf : f( ) , as well as

y = f(x) or f(x) = …

Discontinuous functions

Some exceptional functions are discontinuous everywhere.


Comment on the difference between the possible domains of the functions defined implicitly by the equations y – y1 = m(x – x1) and m = (y – y1) / (x – x1). How does this difference affect the graphs of the two functions?

Rewrite the function f(x) = (x – 1) / (x + 1) in the form A + B / (x + 1) and find the values of A and B. Hence sketch the curve y = f(x). Show clearly the values of the intercepts on each axis and give the equations of each of its asymptotes.

The function f is defined on . Sketch the function f(x) = [ x ], where [ x ] means the

greatest integer less than or equal to x.

5.8 Understand that functions which repeat at regular intervals are called periodic functions, and that the smallest of these intervals is the period of the function.

A function on is defined by f(x) = 3x for 0 ≤ x < 3 with f(x) = f(x + 3). Sketch the graph

of this function and state its period.

A circular function is given as 135sin (2 )y θ π= + . State the amplitude of this function

and its periodicity. Sketch the graph of this function.

5.9 Understand that relations (one-to-many mappings) that represent looped curves are often described in terms of a parameter; consider simple examples of this type.

A curve is described by the parametric equations x = t2 and y = 2t. By eliminating t between these equations find the Cartesian equation of the curve. Sketch the curve.

Find ddyt and d

dxt . Verify that d d d

d d dy y xx t t= ÷ .

6 Solve equations associated with functions

6.1 Given a quadratic equation of the form ax2 + bx + c = 0, know that if the discriminant ∆ = b2 − 4ac is negative, there are two complex roots, which are conjugate to each other.

6.2 Solve exponential and logarithmic equations of the form ekx = A, where A is a positive constant, and ln kx = B, where B is constant.

6.3 Solve trigonometric equations of the form: sin (ax + b) = A, where –1 ≤ A ≤ 1; cos (ax + b) = A, where –1 ≤ A ≤ 1; tan (ax + b) = A, where A is constant; and find all the solutions in a stated interval.

6.4 Find approximate solutions for the intersection of any two functions from the intersection points of their graphs, and interpret this as the solution set of pairs of simultaneous equations.

An approximate solution of the equation x = ex can be found by plotting on the same axes the curves y = x and y = ex and finding the x-coordinate of their point of intersection.

7 Understand and use complex numbers

7.1 Understand that a complex number z = x + iy, where i2 = –1, consists of a real part x and an imaginary part y.

7.2 Know the rules for the addition, subtraction and multiplication of two complex numbers z1= x1 + iy1 and z2 = x2 + iy2; know that the complex conjugate of z is z* = x – iy and that zz* = x2 + y2; use this to divide one complex number by another.

Complex numbers

Complex numbers could be an extension topic for the most able students.


7.3 Represent a complex number as a point in the complex plane or as a vector in this plane; understand and use Argand diagrams to add or subtract two complex numbers; understand that addition or subtraction on an Argand diagram is analogous to the addition or subtraction of two component vectors.

7.4 Know the polar form of a complex number, using the modulus r and the argument θ, and the results that x = r cos θ and y = r sin θ and r2 = x2 + y2 = zz*.

7.5 Understand and use De Moivre’s theorem that z = r eiθ and zn = reinθ.

7.6 Perform multiplication and division of complex numbers using polar form, and apply this to representing multiplication and division of complex numbers on an Argand diagram.

7.7 Use de Moivre’s theorem to calculate roots of complex numbers.

7.8 Know that every polynomial of order n with real coefficients can be factorised into n linear factors in which complex number factors always occur in pairs that are complex conjugate to each other.

7.9 Use complex numbers to generate trigonometric identities.

7.10 Use complex numbers to investigate functions of a complex variable.

8 Calculate the derivative of a function

8.1 Know that the derivative of f ′(x) is called the second derivative of the

function y = f(x) and that this can also be written in the forms f′′(x) or 2

2

dd

yx

;

know that higher derivatives may be taken in the same way.

For the function f(x) = x3 – 6x, show that f′′(x) = 6x. What is the value of f′′(x) at the points where f′(x) is zero?

8.2 Interpret the numerical value of the derivative at a point on the curve of the function; know that:

• when the derivative is positive the function is increasing at the point;

• when the derivative is negative the function is decreasing at the point;

• when the derivative is zero the function is stationary at the point.

Discuss how knowledge of when a function increases, when it decreases and when it is stationary gives important information about the function as a whole and helps to analyse what it looks like.

Discuss the derivative function associated with the function f : | |x x .

Is there anywhere on this function where the derivative does not exist? Justify your answer.

8.3 Understand that stationary points of any function may correspond to a local maximum or minimum of the function, or may be a point of inflexion; understand how the derivative changes as the point at which the derivative is calculated moves through the local maximum or minimum, or through an inflexion; understand that not all points of inflexion are stationary points.

8.4 Understand and use the second derivative to test whether a stationary point is a local maximum, or a local minimum, or a point of inflexion.

Reading the second derivative

f′′(x) is read as ‘f-double-dash of x’ and d2y/dx2 is read as ‘dee-two-y by dee-x-squared’.

Higher derivatives

In general an nth order derivative is denoted by

n

ny

xdd

or f(n)(x).

Cusps

More able students might discuss examples of curves with cusp points, and show why a derivative cannot exist at the cusp point.

Maxima and minima

These are sometimes referred to as turning points.


Which of the following graphs has these features: f′(0) > 0, f′(1) < 0, and f′′(x) is always negative?

TIMSS Grade 12

8.5 Know that of all exponential functions, the exponential function y = ex is defined as the one in which the slope at the y-intercept point (0, 1) has the value 1.

8.6 Use the definition in NAC 8.5 and properties of exponents to find the derivative of y = ex from first principles.

8.7 Know that the function f(x) = ex is the only non-zero function in mathematics for which the derivative f′(x) = ex gives back the original function.

It can be shown that the infinite series 2 3 4

1 ...2! 3! 4!x x xx+ + + + + represents the number ex.

Show that if you differentiate the series term by term and add all these terms together, you get back to what you started with.

8.8 Know that the derivative of the natural logarithm function ln x is 1/x.

It can be shown that, for small values of z, 2 3 4

ln (1 ) ...2 3 4z z zz z+ = − + − + (where

−1 < z ≤ 1). Use this expansion and the properties of logarithms to calculate the derivative of f(x) = ln x from first principles.

Discuss, with examples, the process of logarithmic differentiation.

8.9 Know, without proof, the derivatives of the circular functions sin θ, cos θ and tan θ, and that the domain set for these functions must be in radians.

Derivative of combinations of functions

8.10 Understand that given any function f(x) = f1(x) + f2(x) then the derivative of this sum of two functions is f′(x) = f1′(x) + f2′(x).

Find the derivative of the function given by y = 5x + ex.

8.11 Understand that given a function f(x) = u(x) v(x) then the derivative of this product of two functions is given by f′ = uv′ + vu′; use this result in calculating the derivative of the product of two functions; know the special case of this result that if y = a f(x), where a is constant, then y′ = a f′.

Find ddyx when y = 10x ex.

8.12 Understand that given a function f(x) = u(x) / v(x) then the derivative of this quotient of two functions is given by f′ = (uv′ – vu′) / v2; use this result in calculating the derivative of the quotient of two functions.


8.13 Understand that given a composite function h(x) = g(f(x)) then the derivative of this composite function is given by h′(x) = g′(f(x))f′(x); use this result in calculating the derivative of the composite of two functions.

Use the chain rule to show that if y = 10 e–2t then 2d 20edty

x−= − .

Find the derivative with respect to x of y = (3x + 1)2 (2 – x).

Prove that d(ln ) 1d

xx x= .

Show that d 1d d

d

xy y

x

= .

Differentiate ln (3x2 + 1).

Take natural logarithms to find the derivative of y = ax.

Given f : sin5θ θ find df( )d

θθ when 6

πθ = .

Differentiate 2cos 2 2πθ +

with respect to θ.

8.14 Recognise that the derivative of f(x) = A ekx, where A and k are constants, is f′(x) = kA ekx.

8.15 Find the derivative of a function defined implicitly.

Find ddyx for the implicit function x2 + y2 = 25 for y ≥ 0.

Applications using derivatives

8.16 Use the first and second derivatives of functions to analyse the behaviour of functions and to sketch curves.

Sketch the curve 1( 2)y x x= − , showing clearly its turning points and asymptotes.

Sketch the curve y = xe–x for x ≥ 0, showing clearly its turning points and any points of inflexion.

8.17 Use the derivative to explore a range of optimisation problems in which a function is maximised or minimised.

8.18 Analyse a range of problems using exponential functions.

8.19 Analyse a range of problems involving periodicity or oscillation using circular functions.

8.20 Use polynomial and other functions to model a range of phenomena, including some relating to mechanics and motion, knowing that the derivative of distance with respect to time is a speed (or velocity) and that the derivative of speed (or velocity) with respect to time is acceleration.

9 Perform numerical approximation

9.1 Understand the error bounds on measurements recorded to a given number of significant figures.

9.2 Understand and use the tangent line approximation of f(x) near x = a in the form f(x) ≈ f(a) + f′(a) (x – a) and in the special case near the origin when a = 0.

Composite functions

An alternative and equally acceptable way of writing

this is = ×y y zx z x

d d dd d d ,

where the composite function is formed by first mapping x to z and then mapping z to y. This rule is often called the chain rule because it extends for composite functions formed in more than two stages.

Exponential functions

Students should explore exponential growth or decay, through a range of problems such as population growth, interest on loans, radio-carbon dating, cooling, half-life of radioactive elements, and the absorption of a medical drug into the body. See also NAC 11.2.


9.3 Know the approximations sin θ ≈ θ and cos θ ≈ 2121 θ− for small values of

θ in radians.

Find 0

sinlimθ

θθ→

.

9.4 Understand and use the Taylor series expansion 2 3 ( )f (0) f (0) f (0)f ( ) f (0) f (0) ... ...

2! 3! !

n nx x xx xn

′′ ′′′′≈ + + + + + +

to approximate functions and numerical values.

9.5 Perform simple iterations to find roots of equations, including xn+1 = f(xn)

and the Newton–Raphson iteration 1f ( )f ( )

nn n

n

xx x

x+ = − ′ , where f′(xn) ≠ 0.

10 Reconstruct a function from its derivative

The indefinite integral

10.1 Understand integration as the inverse process to differentiation; use the notation for indefinite integrals, knowing that f ( )d f ( )x x x c′ = +∫ , where c

is any constant, and that there is a whole family of curves y = f(x) + c, each member of which has derivative function f′(x).

Discuss how the individual members of the family of curves represented by y = f(x) + c are related to each other.

The graph of the function g passes through the point (1, 2). The slope of the tangent to the graph at any point (x, y) is given by g′(x) = 6x – 12. What is g(x)? Show all your work.

TIMSS Grade 12

10.2 Know the integrals of the functions: xn, where n ≠ –1 1/x, with x ≠ 0 ekx sin kx , cos kx and sec2 kx, where k is constant; write the integrals of multiples of these functions and of linear combinations of these functions.

The definite integral

10.3 Use the definition of the definite integral: f ( )d f ( ) f ( )

b

ax x b a′ = −∫ , where f(x) is a function of x and a ≤ x ≤ b;

interpret this as ‘the integral of a rate of change of a function is the total change of that function’; understand the effect of interchanging the limits of integration; know that .

b c b

a a c= +∫ ∫ ∫

Evaluate 3

6

cos dx xπ

π∫ .

10.4 Use summation of areas of rectangles to calculate lower and upper bounds for the area between the x-axis and a curve y = f(x) with y > 0, bounded on either side by lines x = constant; understand that as the width δx of each of the rectangles tends to zero the sums Σ f(x) δx for the lower and upper bounds of the area under the curve tend to the same value, and that this value is called the area under the curve.

Integration

The word integration comes from integrating, i.e. adding, the contributions of many small parts. The symbol

∫ x...d denotes summation

with respect to x. In the

definite integral ∫b

a, a and b

are called the limits of the integral, or the limits of integration.


10.5 Understand that the area bounded by a positive function y = f(x), the x-axis and the lines x = a and x = b, with a ≤ x ≤ b, is the definite integral

f ( )db

ax x∫ .

The line l in the figure is the graph of y = f(x). 3

2f ( )dx x

−∫ is equal to

A. 3 B. 4 C. 4.5 D. 5 E. 5.5

TIMSS Grade 12

10.6 Use the trapezium rule to find an approximation to the area represented by

the definite integral of a particular function when it is not easy or possible to integrate the function.

10.7 Understand that if a curve y = f(x) lies entirely below the x-axis, so that its

y-value is always negative, then the definite integral f ( )db

ax x∫ over the

interval a ≤ x ≤ b has a negative value.

Calculate the area between the curve y = x2 + 5, the x-axis and the lines x = –2 and x = 3.

Find the area between the curves y = x2 – 4 and y = 4 – x2.

Find the area between the curves y = x3 and y = x.

This figure shows the graph of y = f(x). S1 is the area enclosed by the x-axis, x = a and y = f(x); S2 is the area enclosed by the x-axis, x = b and y = f(x); where a < b and 0 < S2 < S1.

The value of f ( )db

ax x∫ is

A. S1 + S2

B. S1 – S2 C. S2 – S1 D. | S1 – S2

| E. ( )1

1 22 S S+

TIMSS Grade 12

10.8 Interpret and use an integral of velocity with respect to time as distance travelled, and an integral of acceleration with respect to time as velocity.

10.9 Solve other physical problems in which the integral of the rate of change of a physical quantity has to be interpreted as a total change in that quantity.

10.10 Use the integration by parts formula uv d uv vu d′ ′= −∫ ∫x x

and understand that it reverses the derivative of the product of two functions.

Find e dxx x∫ .

Find cos dx x x∫ .

Areas under curves

Students should work with ‘area-so-far’ for the area under a curve y = f(x), using definite integrals as in NAC 10.5, or the trapezium rule as in NAC 10.6.

Physical integrals

Include integrating force with respect to distance and integrating momentum with respect to velocity.


10.11 Understand that g (f ( )) f ( )d g(f ( ))x x x x c′ ′ = +∫ reverses the derivative of a

composite function; recognise ‘simple’ functions for which this formula can be instantly applied.

Explain why 222 d ln ( 1)

1x x x c

x= + +

+∫ .

10.12 Use the terminology that if z is a function of x then the derivative of z with

respect to x is dd

zx

and the differential of z is the symbolic expression

dd dd

zz xx

= ; understand that z and its differential can be used to replace the

variable of integration in an integral.

10.13 Perform simple cases of integration by substitution to undo the ‘chain rule’; perform integration with a given substitution.

Evaluate 21

0e dxx x∫ using the substitution w = x2.

Use the substitution z = 2 – 3x to evaluate ( 4) 2 3 dx x x+ −∫ .

10.14 Use partial fractions to integrate.

Find 1 d( 1)( 2) xx x+ +∫ .

10.15 Analyse simple instances of convergent definite integrals in which the upper limit tends to infinity.

Find 30

e dx x∞

−∫ . [Hint: replace the upper limit by b and let b tend to infinity.]

11 Solve simple differential equations

11.1 Recognise when an equation is a differential equation, and how such an equation can be formed; solve a differential equation that can be solved by separation of variables.

Find the general solution of the equation ddy kyx = , where k is a constant.

Show that the solution of the differential equation ddy xx y= − is the family of circles

x2 + y2 = c, where c is a positive constant.

11.2 Solve a range of physical problems involving simple differential equations for exponential growth and decay.

11.3 Know that the differential equation 2

2

dd

y kyx

= − , where k > 0, represents

simple harmonic motion (SHM) and that the solution of this equation has the form y = A sin x + B cos x; investigate some common cases of SHM.

Show that the function y = A sin x + B cos x satisfies the differential equation 2

2dd

y kyx

= − .

Exponential models

See the note at NAC 8.18.



By the end of Grade 12, students are aware of links between geometry and algebra, which deepens their understanding of space and movement. They understand the roles that trigonometry and circular functions play in modelling and in mathematical transformation. They use trigonometric identities to solve trigonometric equations. They use vectors to extend the study of space and motion into three dimensions, and they are familiar with curves represented by parametric equations. They use dimensionally correct units for length, area and volume and for a range of measures, including velocity, acceleration and other compound measures. They find areas and volumes by integration and volumes of revolution. They use ICT to explore geometrical relationships.

Students should:

12 Extend their understanding of circular functions

Sum or difference of two angles

12.1 Know, but not prove, identities for: sin (A + B); sin (A – B); cos (A + B); cos (A – B); tan (A + B); tan (A – B).

Show that sin 2A = 2 sin A cos A.

Find an exact expression for 512sin π .

By writing 7 sin θ + 5 cos θ in the form R sin (θ + α) find R and α, and hence the greatest value of the expression.

Show that cos 3A = 4 cos3 A – 3 cos A.

12.2 Know corresponding identities for double or half angles.

Sum or difference of two sines or cosines

12.3 Use the relevant identities from GM 12.1 to find the ‘sum–product’ identity

sin sin 2sin cos2 2

X Y X YX Y + −+ ≡ ; and corresponding identities for

sin X – sin Y; cos X + cos Y; cos X – cos Y.

Solution of trigonometric equations

12.4 Use trigonometric identities to solve trigonometric equations over specified angle domains.

Solve the equation cos 2θ + 3 sin θ = 2 for 0 ≤ θ ≤ 2π.

Solve the equation sin 2x = cos x for 0 ≤ x ≤ 2π.


Students should appreciate the importance and range of geometrical applications in the real world. They should understand the nature and place of geometric reasoning and proof, and how geometry may be related to algebraic concepts, and vice versa. They should know how a dynamic geometry system, or DGS, can be used to investigate results.


13 Use vectors to study position, displacement and motion

13.1 Use vectors in up to three dimensions; identify the components of the vector in relation to three orthogonal directions; use unit vectors i, j and k in these directions; use column matrix form for vectors, including unit vectors; use the notation AB to denote the vector from point A to point B; use and understand the terms position vector and displacement vector.

13.2 Know the rules for the addition and subtraction of two vectors; represent addition and subtraction of two vectors diagrammatically; know that there exists a null vector 0 such that a – a = 0 for any vector; know that vector addition is commutative and associative.

13.3 Find the magnitude | a | of any vector a and the direction of a in relation to specified axes.

13.4 Know the distinction between a vector and a scalar; know that any vector can be multiplied by a positive scalar to rescale it, or by a negative scalar to rescale it and reverse its direction.

13.5 Know the notation a.b for the scalar product of two vectors a and b; form and calculate the scalar product, and interpret the scalar product in terms of the magnitudes of the two vectors and the angle between them; know that a.a is the square of the magnitude of a.

Find the angle between the two vectors a = 3i + j – 2k and b = 2i – 5j – k.

Show that | a + b |2 = a2 + b2 + 2a.b and use this result to prove the cosine rule.

13.6 Know that if a and b are two non-zero vectors and a.b = 0 then a and b are perpendicular to each other.

Prove that the diagonals of any rhombus are perpendicular to each other.

13.7 Find the mid-point of a line segment AB given the position vectors of A and B.

13.8 Find the vector equation of a straight line in the form r = a + λb, where r is the position vector of any point on the line, a is the position vector of a given point on the line, b is a vector in the direction of the line and λ is a variable scalar.

13.9 Use vectors to represent velocity and know that speed is the magnitude of velocity; use vectors to represent acceleration, force and momentum.

13.10 Solve dynamical problems by differentiating or integrating vectors that are functions of position, or time, or velocity.

14 Use a range of measures and compound measures to solve problems

14.1 Find areas and volumes by integration; find volumes of revolution.

14.2 Solve problems using a range of compound measures using appropriate units and dimensions: for example, density (mass per unit volume), pressure (force per unit area) and power (energy per unit time).

15 Use ICT to explore geometric relationships

15.1 Use ICT to explore geometric relationships.

Variable scalars

These are called parameters.

Compound measures

Reinforce links with physics, using compound measures such as pressure, power, velocity and acceleration.

Mathematics standards Grade 12 - sec.gov.qa And Subject/Math-Grade 12 (advanced...306 | Qatar...

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