Download Grade 12 advanced | Mathematics for science

Mathematics standards
Summary of students’ performance by the end of Grade 12
Grade 12
Advanced
(mathematics
for science)
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.
305 | Qatar mathematics standards | Grade 12 advanced | Mathematics for science
© Supreme Education Council 2004
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:
Number, algebra
Geometry, measures
Probability and
and calculus
and trigonometry
statistics
Grade 10
55%
30%
15%
Grade 11
55%
30%
15%
Grade 12
(quantitative)
40%
–
60%
75%
25%
–
Advanced
Grade 12
(for science)
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.
306 | Qatar mathematics standards | Grade 12 advanced | Mathematics for science
© Supreme Education Council 2004
Mathematics standards
Reasoning and problem solving
Grade 12
Advanced
(mathematics
for science)
Key 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.
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
Students should:
italics are intended to clarify
the standards, not to
1
represent the full range of
Use mathematical reasoning to solve problems
possible 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 realworld applications, and to compare and contrast two or more given models
of a particular situation.
Reasoning and problem
solving
Reasoning, generalisation
and problem solving should
be an integral part of the
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.
teaching and learning of
mathematics in all lessons.
State whether the following statements are true or false.
x2 = 16 ⇒ x = 4
x2 ≤ 16 ⇒ –4 ≤ x ≤ 4
4
≥3⇒ x ≤
x
4
3
307 | Qatar mathematics standards | Grade 12 advanced | Mathematics for science
© Supreme Education Council 2004
The Al Huda sisters made these statements.
Inas:
Safa:
Roza:
Farida:
‘If the rug is in the car, then it is not in the garage.’
‘If the rug is not in the car, then it is in the garage.’
‘If the rug is in the garage, it is in the car.’
‘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.
B.
C.
D.
Inas
Safa
Farida
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.
308 | Qatar mathematics standards | Grade 12 advanced | Mathematics for science
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.
© Supreme Education Council 2004
Students should:
2
2.1
Manipulate algebraic expressions
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.
Factorisation
Include the sum and
difference of two cubes.
Factorise 27x3 + 8y3. Use your factorisation to write an identity for 27x3 – 8y3.
Simplify
x3 + 1
, given that x ≠ ±1.
x2 − 1
Solve the equation (x + 3)(3x –1) – (x + 3)(5x + 2) = 0.
2.2
Combine and simplify rational algebraic fractions.
Simplify
2.3
5
3x − 2
.
−
x + 1 x2 − 1
Decompose a rational algebraic fraction into partial fractions (with
denominators not more complicated than repeated linear terms).
Show that
1
1
1
≡ −
.
r ( r + 1) r r + 1
Find the values of A, B and C in the identity
2.4
1
A
B
C
.
≡
+
+
(3r + 1)( r + 1) 2 3r + 1 r + 1 (r + 1) 2
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
3.1
Use index notation and logarithms to solve numerical
problems
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
8n × 25 n
.
42 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.
In addition to formulae for
the sum and difference of
Give the value of log10 1000.
Evaluate log2 64.
Express log 27 – 2 log 3 as a single logarithm.
Prove that log a b =
Logarithms
log c b
1
and use this result to show that log a b =
.
log b a
log c a
two logarithms, include:
loga a x = x
aloga x = x
loga a = 1
loga 1 = 0 for any a > 0
loga x n = n loga x
Given logb 2 = 1/3, logb 32 is equal to
A. 2
B. 5
C.
–3
/5
D. 5/3
E.
3
log 2 32
TIMSS Grade 12
309 | Qatar mathematics standards | Grade 12 advanced | Mathematics for science
© Supreme Education Council 2004
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
4.1
Using exponentials
Further examples of
modelling with exponential
functions are given in the
margin note at NAC 8.18.
Work with sequences, series, recurrence relations and
arrangements
Find permutations and combinations.
Permutations
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
from a combination.
Students should know that
n
Cr is the number of
combinations of r different
objects from n different
objects and that the number
of permutations of r
TIMSS Grade 12
9
The order of selection
distinguishes a permutation
different objects from n
n
10
different objects is r! Cr,
n
which is denoted by Pr.
Evaluate C6 and P5.
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
310 | Qatar mathematics standards | Grade 12 advanced | Mathematics for science
© Supreme Education Council 2004
4.2
Expand the binomial series (1 + x)n for any rational value of n.
Binomial theorem
The theorem is used to
6
Without using a calculator, find the value of (1.01) 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.
5.1
Work with functions and their graphs
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
has applications in
probability. The binomial
series is commonly used to
generate approximations.
1
≈ 1 − 2 x + 3x 2 .
Show that, for small values of x,
(1 + x) 2
5
expand positive integer
powers of binomial terms. It
Functions
Include use of the notation
Use the notation
f : x 6 f( x ) , as well as
y = f(x) or f(x) = …
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.
1
B. log e 12
C. log 2 e
D. (log e 2) 4
E. 2e 4
A. 14 log e 2
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 y = f −1 ( x) .
Show that the function f ( x) =
x
, where x ≠ 1, is its own inverse.
x −1
Find the inverse of the function f : t 6 t + 2 , defined for t ≥ 0.
5.6
Know that f −1 (f ( x)) = x and that the graph of y = f −1 ( x) is the reflection of
the graph of y = f ( x) about the straight line y = x.
Show that eln 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.
Discontinuous functions
Some exceptional functions
are discontinuous
everywhere.
The function f(n) = n is defined on ]+. Sketch the graph of this function.
311 | Qatar mathematics standards | Grade 12 advanced | Mathematics for science
© Supreme Education Council 2004
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 y = 5sin (2θ + 13 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.
dy
dy d y dx
dx
Find
. Verify that
and
.
=
÷
dt
dt
dx dt dt
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
7.1
7.2
Understand and use complex numbers
Understand that a complex number z = x + iy, where i2 = –1, consists of a
real part x and an imaginary part y.
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.
312 | Qatar mathematics standards | Grade 12 advanced | Mathematics for science
Complex numbers
Complex numbers could be
an extension topic for the
most able students.
© Supreme Education Council 2004
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
8
8.1
Use complex numbers to investigate functions of a complex variable.
Calculate the derivative of a function
Know that the derivative of f ′(x) is called the second derivative of the
Reading the second
2
function y = f(x) and that this can also be written in the forms f′′(x) or
d y
;
dx 2
know that higher derivatives may be taken in the same way.
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.
8.4
Higher derivatives
In general an nth order
derivative is denoted by
dn y
(n)
or f (x).
dx n
Cusps
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.
More able students might
Discuss the derivative function associated with the function f : x 6 | x | .
why a derivative cannot
discuss examples of curves
with cusp points, and show
exist at the cusp point.
Is there anywhere on this function where the derivative does not exist?
Justify your answer.
8.3
f′′(x) is read as ‘f-double2
2
dash of x’ and d y/dx is
read as ‘dee-two-y by deex-squared’.
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
derivative
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.
Maxima and minima
These are sometimes
referred to as turning
points.
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.
313 | Qatar mathematics standards | Grade 12 advanced | Mathematics for science
© Supreme Education Council 2004
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.
x 2 x3 x 4
+ + + ... represents the number ex.
2! 3! 4!
Show that if you differentiate the series term by term and add all these terms together,
you get back to what you started with.
It can be shown that the infinite series 1 + x +
8.8
Know that the derivative of the natural logarithm function ln x is 1/x.
z 2 z3 z 4
+ − + ... (where
2 3 4
−1 < z ≤ 1). Use this expansion and the properties of logarithms to calculate the
derivative of f(x) = ln x from first principles.
It can be shown that, for small values of z, ln (1 + z ) = z −
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
8.12
dy
when y = 10x ex.
dx
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.
314 | Qatar mathematics standards | Grade 12 advanced | Mathematics for science
© Supreme Education Council 2004
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
dy
= −20e −2t .
dx
An alternative and equally
acceptable way of writing
dy dy dz
this is
,
=
×
dx dz dx
where the composite
Find the derivative with respect to x of y = (3x + 1)2 (2 – x).
Prove that
Composite functions
function is formed by first
mapping x to z and then
d(ln x) 1
= .
dx
x
mapping z to y. This rule is
often called the chain rule
dx
1
=
.
Show that
dy dy
dx
because it extends for
composite functions formed
in more than two stages.
Differentiate ln (3x2 + 1).
Take natural logarithms to find the derivative of y = ax.
Given f : θ 6 sin 5θ find
df(θ )
π
when θ = .
6
dθ
π
Differentiate cos 2  2θ +  with respect to θ.
2

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
dy
for the implicit function x2 + y2 = 25 for y ≥ 0.
dx
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 y =
1
, showing clearly its turning points and asymptotes.
x( x − 2)
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
Exponential functions
Students should explore
exponential growth or
decay, through a range of
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.
problems such as
population growth, interest
on loans, radio-carbon
dating, cooling, half-life of
radioactive elements, and
the absorption of a medical
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.
315 | Qatar mathematics standards | Grade 12 advanced | Mathematics for science
drug into the body. See
also NAC 11.2.
© Supreme Education Council 2004
9.3
Know the approximations sin θ ≈ θ and cos θ ≈ 1 − 12 θ 2 for small values of
θ in radians.
Find lim
θ →0
sin θ
θ
.
9.4
Understand and use the Taylor series expansion
f ′′(0) x 2 f ′′′(0) x 3
f ( n ) (0) x n
f ( x) ≈ f (0) + f ′(0) x +
+
+ ... +
+ ...
2!
3!
n!
to approximate functions and numerical values.
9.5
Perform simple iterations to find roots of equations, including xn+1 = f(xn)
f ( xn )
, where f′(xn) ≠ 0.
and the Newton–Raphson iteration xn +1 = xn −
f ′( xn )
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 ′( x) dx = f ( 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.
Integration
The word integration comes
from integrating, i.e. adding,
the contributions of many
small parts. The symbol
∫ ...dx 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.
The definite integral
10.3
Use the definition of the definite integral:
∫
b
a
f ′( x) dx = f (b) − f (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
b
c
b
integration; know that ∫a = ∫a + ∫c .
Evaluate
10.4
∫
π
3
π
cos x dx .
6
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.
316 | Qatar mathematics standards | Grade 12 advanced | Mathematics for science
© Supreme Education Council 2004
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
∫
b
a
f ( x) dx .
The line l in the figure is the graph of y = f(x).
∫
3
−2
f ( x) dx is equal to
A. 3
B. 4
C. 4.5
D. 5
E. 5.5
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.
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
b
∫a f ( x)dx 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
A.
B.
C.
D.
E.
∫
b
a
f ( x) dx is
S1 + S2
S1 – S2
S2 – S1
| S1 – S2 |
1
S + S2 )
2( 1
TIMSS Grade 12
10.8
10.9
10.10
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.
Physical integrals
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.
Include integrating force
with respect to distance and
integrating momentum with
respect to velocity.
Use the integration by parts formula
∫ uv′ dx = uv − ∫ vu ′ dx
and understand that it reverses the derivative of the product of two
functions.
Find
∫ xe
Find
∫ x cos x dx .
x
dx .
317 | Qatar mathematics standards | Grade 12 advanced | Mathematics for science
© Supreme Education Council 2004
10.11
Understand that ∫ g ′(f ( x)) f ′( x) dx = g(f ( x)) + c reverses the derivative of a
composite function; recognise ‘simple’ functions for which this formula can
be instantly applied.
Explain why
∫x
2x
dx = ln ( x 2 + 1) + c .
+1
2
10.12
Use the terminology that if z is a function of x then the derivative of z with
dz
and the differential of z is the symbolic expression
respect to x is
dx
dz
dz = dx ; understand that z and its differential can be used to replace the
dx
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
∫
1
0
x e x dx using the substitution w = x2.
2
Use the substitution z = 2 – 3x to evaluate ∫ ( x + 4) 2 − 3 x dx .
10.14
Use partial fractions to integrate.
Find
10.15
Analyse simple instances of convergent definite integrals in which the
upper limit tends to infinity.
Find
11
11.1
1
∫ ( x + 1)( x + 2) dx .
∫
∞
0
e−3 x dx . [Hint: replace the upper limit by b and let b tend to infinity.]
Solve simple differential equations
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
dy
= ky , where k is a constant.
dx
Show that the solution of the differential equation
dy
x
= − is the family of circles
y
dx
x2 + y2 = c, where c is a positive constant.
11.2
11.3
Solve a range of physical problems involving simple differential equations
for exponential growth and decay.
Know that the differential equation
Exponential models
See the note at NAC 8.18.
d2 y
= −ky , where k > 0, represents
dx 2
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
d2 y
= − ky .
dx 2
318 | Qatar mathematics standards | Grade 12 advanced | Mathematics for science
© Supreme Education Council 2004
Geometry and measures
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
Geometry and measures
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.
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 sin 125 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
X +Y
X −Y
sin X + sin Y ≡ 2sin
cos
; and corresponding identities for
2
2
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.
319 | Qatar mathematics standards | Grade 12 advanced | Mathematics for science
© Supreme Education Council 2004
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;
JJJG
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
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).
15.1
These are called
parameters.
Use a range of measures and compound measures to solve
problems
14.1
15
Variable scalars
Use ICT to explore geometric relationships
Compound measures
Reinforce links with
physics, using compound
measures such as
pressure, power, velocity
and acceleration.
Use ICT to explore geometric relationships.
320 | Qatar mathematics standards | Grade 12 advanced | Mathematics for science
© Supreme Education Council 2004