Download 1~ a < 10, k E 7/,.

As noted in the previous section on prior learning, it is expected that all students have extensive previous
mathematical experiences, but these will vary. It is expected that mathematics HL students will be familiar with the
following topics before they take the examinations, because questions assume knowledge of them. Teachers must
therefore ensure that any topics listed here that are unknown to their students at the start of the course are included at
an early stage. They should also take into account the existing mathematical knowledge of their students to design an
appropriate course of study for mathematics HL. This table lists the knowledge, together with the syllabus content,
that is essential to successful completion of the mathematics HL course.
Students must be familiar with SI (Systeme International)
units oflength, mass and time, and their derived units.
Topic
Content
Number
Routine use of addition, subtraction, multiplication and division, using integers, decimals
and fractions, including order of operations.
Rational exponents.
Simplification of expressions involving roots (surds or radicals), including rationalizing the
denominator.
Prime numbers and factors (divisors), including greatest common divisors and least
common multiples.
Simple applications of ratio, percentage and proportion, linked to similarity.
Definition and elementary treatment of absolute value (modulus),
lal.
Rounding, decimal approximations and significant figures, including appreciation of errors.
Expression ofnumbers in standard form (scientific notation), that is, a x 10k, 1 ~ a < 10,
k
E
7/,.
f---------j--------------------
Sets and numbers
..
------------------.
Concept and notation of sets, elements, universal (reference) set, empty (null) set,
complement, subset, equality of sets, disjoint sets. Operations on sets: union and
intersection. Commutative, associative and distributive properties. Venn diagrams.
Number systems: natural numbers; integers, Z; rationals, IQ, and irrationals; real numbers, lit
Intervals on the real number line using set notation and using inequalities. Expressing the
solution set of a linear inequality on the number line and in set notation.
Mappings of the elements of one set to another; sets of ordered pairs.
.Jb
Mathematics
HL guide
15
Prior learning topics
Topic
Algebra
Content
Manipulation
completing
oflinear
and quadratic expressions,
including
factorization,
expansion,
the square and use of the formula.
Rearrangement,
evaluation
and combination
subject areas, particularly
of simple formulae.
Examples
from other
the sciences, should be included.
Linear functions, their graphs, gradients and y-intercepts.
Addition
and subtraction
The properties
of simple algebraic fractions.
s.>. ~.
of order relations: <,
Solution oflinear
equations
and inequalities
in one variable, including cases with rational
coefficients.
Solution of quadratic equations and inequalities, using factorization and completing the square.
Solution of simultaneous
Trigonometry
Geometry
Angle measurement
linear equations
in degrees. Compass directions.
applications
for solving triangles.
Pythagoras'
theorem and its converse.
Simple geometric
Congruence
in two variables.
transformations:
and similarity,
translation,
Right-angle
reflection,
trigonometry.
Simple
rotation, enlargement.
including the concept of scale factor of an enlargement.
The circle, its centre and radius, area and circumference.
The terms arc, sector, chord,
tangent and segment.
Perimeter and area of plane figures. Properties of triangles and quadrilaterals,
parallelograms,
compound
Elementary
rectangles,
squares, kites and trapeziums
of prisms and pyramids, including tetrahedra.
geometry of the plane, including the concepts of dimension
and space. The equation of a line in the form y
including
including
(trapezoids);
shapes. Volumes of cuboids, pyramids, spheres, cylinders and cones.
Classification
Coordinate
geometry
rhombuses,
m,
=
m2
and
m, m2
= mx + c . Parallel
for point, line, plane
and perpendicular
lines,
= -\ .
The Cartesian plane: ordered pairs (x, y) , origin, axes. Mid-point
of a line segment and
distance between two points in the Cartesian plane.
Statistics and
probability
Descriptive
statistics: collection of raw data, display of data in pictorial and diagrammatic
forms, including frequency histograms,
Obtaining
16
frequency
simple statistics from discrete and continuous
mode, quartiles, range, interquartile
Calculating
cumulative
probabilities
graphs.
data, including mean, median,
range and percentiles.
of simple events.
Mathematics
HL guide
Jb
Topic I-Core: Algebra
1.1
The ntlt term of an
arithmetic sequence
The sum of n terms of
an arithmetic sequence
The ntlt term of a
geometric sequence
The sum of n terms of a
finite geometric sequence
The sum of an infinite
geometric sequence
1.2
Exponents and
logarithms
s=...!:!L,
a'
= b <=> x = log,
=x=
log, a'
_loge
Iogba---
1.3
Combinations
1rI<1
l-r
n)
(r
alog.
b
X
a
loge b
n!
r!(n-r)!
Binomial theorem
1.5
Complex numbers
z=a+ib=r(cosB+isinB)
1.7
De Moivre's theorem
[r(cosB + isinB))"
=rei() =rcisB
= r"
(cosnB + isin nB)
= rneinfJ= r"
cisnB
Topic 2 -Core: Functions and equations
2.5
-------
2.6
2
Axis of symmetry of the
graph of a quadratic
function
------------.------._--------.
f(x)
= ax'
+ bx + c ~ axis of symmetry x
=-~
2a
-------------------------------------------------------------------------------
Solution of a quadratic
equation
ax' + bx + c = 0 ~ x =
Discriminant
f..=b2 -4ac
-b±.Jb2
2a
-4ac
, a:;t:O
© Intemational Baccalaureate Organization 2004
Topic 3 -Core: Circular functions and trigonometry
3.1
3.2
Length of an arc
I = Or , where 0 is the angle measured in radians, r is the radius
Area of a sector
A = ..!..Orz , where 0 is the angle measured in radians, r is the
2
radius
sinO
tan O =--
Identities
Pythagorean
cosO
identities
Z
0 + sin? 0 =1
COS
1 + tan ' 0 = see' 0
1 + cor'
3.3
Compound angle
identities
0
= csczO
sin(A ± B) = sin AcosB± cosAsin B
cos(A± B) =cosAcosB=tsin
Asin B
tan(A ± B) = tan A ± tan B
l=ttanAtanB
Double angle identities
sin 20 = 2sinOcosO
cos20 = cos ' 0 =sin ' 0 =2cosz 0-1 = 1-Zsin ' 0
tan 20 =
3.6
2 tan 0
I-tanZO
a' + b' _cz
2ab
Cosine rule
cosC=----
a
b
c
sin A
sin B
sin C
Sine rule
--=--=--
Area of a triangle
A =..!..ab sin C , where a and b are adjacent sides, C is the included
2
angle
Topic 4 -Core: Matrices
4.3
Determinant of a 2 x 2
matrix
A=(:
!)
Inverse of a 2 x 2 matrix
A=(:
b) =>A-' =
1
(d
d
ad -be -e
Determinant of a 3 x 3
matrix
A=
© Intemational Baccalaureate Organization 2004
=> det A = ad - be
(ad bel
e f
g
=>detA=al:
-b) .od
a
{I-bl~
e bc
{I+el~
:1
h k
3
Topic 5-Core: Vectors
5.1
Magnitude of a vector
Distance between two
points (X"y"ZI)
and
(X2'Y2,Z2)
Coordinates of the
midpoint of a line
segment with endpoints
(X"y"ZI),
(X2'Y2,Z2)
5.2
Scalar product
v· w = Iviiwicos B , where B is the angle between vand w
Angle between two
vectors
5.3
5.5
Vector equation of a
line
r=a+Xb
Parametric form of
equations of a line
x = Xo + AI, Y = Yo + Am,
Cartesian equations of
a line
X-Xo
Vector product
(Determinant
representation)
= Y-
I
vx w =
Yo
m
Z
=
Zo
+ An
= Z-Zo
n
i
j
k
VI
V2
V3
WI
W2
W3
Ivx wi = IvllwlsinB, whereB is the angle between vand w
Area of a triangle
--
- - -- -
5.6
4
1
A =-Ivx
2
wi
- - - -- - - -- - - - - -- - - - --- - - - -- - - - - - - - -- - - - - --- - - - - - - - -- - - - - -- - - - - - - - - - -- - - - - -- - - - -- - - - - - - - - - - - - - - - - - - --_.Vector equation of a
plane
r=a+Xb+uc
Equation of a plane
(using the normal vector)
r . n = a .n
Cartesian equation of a
plane
ax + by + cz + d = 0
- -- - - --
© International Baccalaureate Organization 2004
Topic 6-Core: Statistics and probability
k
Ih
Let n=
6.3
;=1
Population parameters
k
IhX;
f..l=~
Mean f..l
n
k
k
Ih(x;-f..l)2
Ihx;2
(Y2 = ...:..:;=:..:...1
Variance (Y2
=
;-1
_
n
Standard deviation
f..l2
n
(Y
Sample statistics
k
Ihx;
Mean
x=~
x
Variance
n
s;
n
n
Standard deviation sn
k
2
Sn_1=r+:s;
n-l
Probability ofan event A
P(A) = n(A)
Complementary
P(A)
S;_I
6.5
-- - - -
6.6
n
Unbiased estimate of
population variance
events
i»:
k
Ih(x;
_X)2
2
I
=...:;;=::!-I----=
n-l
I
;=1
n-l
n-l
n(U)
+ P(A') = 1
-- - - --- - - --- - - - -- - - - - -- - - - --- - - - - - - -- - -- - - -- - - -- - - - - - - - - - -- - - - - - - - - --- - - -- - - - -
Combined events
P(A u B) = P(A)
+ P(B)
Mutually exclusive
events
P(A u B) = P(A)
+ P(B)
© International Baccalaureate Organization 2004
n_
---x2
---
- - - - - - - _.- - - - - -
-- - ---
---- -- - --
- P(A n B)
5
Topic 6-Core:
6.7
6.9
6.10
Statistics and probability (continued)
Conditional probability
Independent events
peA n B) = peA) PCB)
Bayes'Theorem
P(B)P(AIB)
P ( B I A) - -------;-------::--'-------'---:------::- P(B)P( A I B) + P(B,)P( A I B')
= f.L= LX
= x)
Expected value of a
discrete random
variable X
E(X)
Expected value of a
continuous random
variable X
E(X)=f.L=
Variance
Var(X)=E(X
Variance of a discrete
random variable X
Var(X)
Variance ofa continuous
random variable X
Var(X) = [(X-f.L)2
f(x)dx=
Binomial distribution
X ~ B(n, p) ~ P(X
= x) =(:
Mean
E(X) = np
Variance
Var(X) = np(l- p)
P(X
x
[xf(x)dx
= 2)x
_f.L)2 =E(X2)_[E(X)]2
- f.L)2P(X
= x) = 2>2 P(X = x) - f.L2
[X2 f(x)dx-f.L2
)pX(I_
p)"-X, x
= 0, 1, ... , n
Poisson distribution
6.11
Mean
E(X)=m
Variance
Var(X) =m
Standardized normal
variable
z=--
x - f.L
(J'
6
© Intemational Baccalaureate Organization 2004
I
I
Topic 7 -Core: Calculus
7.1
Derivative of f(x)
y=f(x)=>
dy =f'(X)=lim(f(X+h)dx
h-->O
Derivative of x"
f(x)
= x"
Derivative of sin x
f(x)
= sinx
=> f'ex)
= cosx
Derivative of cosx
f(x)
= cosx
=> f'ex)
= -sinx
Derivative of tanx
f(x)=tanx=>
Derivative of e"
f(x)
= eX => f'ex) = e"
Derivative of In x
f(x)
= Ln x=> f'ex)
=> f'ex)
f(X))
h
= nxn-1
f'(x)=sec2
x
=~
X
Derivative of secx
f(x)
= secx
=> f'ex) =secxtanx
Derivative of cscx
f(x)
= csc x
=> f'ex)
Derivative of cotx
f(x)=cotx=>
Derivative of a'
f(x)
= a' => f'ex) = a' (In a)
Derivative of log, x
f(x)
= log,x
Derivative of arcsin x
f(x)=arcsinx=>
= -csc xcotx
f'(X)=-csc2
=> f'ex)
X
= _1_
xlna
f'(X) =
h
l-x2
Derivative of arccosx
= arccosx=>
f(x)
1
= --==
f'ex)
~
7.2
I
=> f (x)
Derivative of arctan x
f (x)
Chain rule
y=g(u),
Product rule
dy
dv
y=uv=>-=u-+vdx
dx
Quotient rule
y
© Intemational Baccalaureate Organization 2004
r
= arctan x
1
= --2
l+x
dy
where u=f(x)=>-=-xdx
dy
du
du
dx
du
dx
u
dy
du
dv
v dx -u dx
v
dx
v2
= - => - = ---"''''-----'''''''-
7
Topic 7 -Core: Calculus (continued)
n+1
7.4
Standard integrals
fxndx=_x_+C,
n+!
fsinxdx
n;e-l
= -cosx+C
fcosxdx=sinx+C
fex dx=ex
+c
+C
f dx _1_
Ina
f_ldx = ~ arctan (~)
a'
a
a
aX
=
aX
+C
+X2
fhdx=arcSin(~)+c,
a -x
7.5
Area under a curve
A=
r
ydx or A
=
r
a
Ixl<a
xdy
Volume of revolution
(rotation)
7.9
8
lntegration by parts
fu:dx=uv-
fv~dx
or fudv=uv-
fVdu
© Intemational Baccalaureate Organization 2004
Name:
Math SL
1.
(a)
(b)
Given that logjx - log3(x - 5) = logy4, express A in terms of x.
Hence or otherwise, solve the equation logjx - log3(x - 5) = 1.
2a.
Find the exact solution ofthe equation
42x = 8(1-x).
b. Simplify 7ln e"
c.
Simplify
3. Let f(x)
log,(2~
= X2
J
+ 1 and g(x) = e'
b)Find [g-lof](-3)
c) Find the domain of g-I(X)
4.
Solve the equation In (2x
5.
Solve the equation log, 81 + log, ~ + log, 3 = log, x.
6.
a)
Determine the x-intercept of y
b)
Determine the equation of the vertical asymptote of y
-1) = 4 .
9
7. Let f(x)
= 3 + log, (x + 2) .
= 3 + log, (x + 2) .
= ex+3
(a) (i) Sbow tbat
8 (a) Find
f-I
(x)
= In x -
3
\og232
(b) Given tbat
log,
32X)
( 8Y
can be written as px + qy, find tbe value ofp and q.
_
x
A is part of the graph of y
= x. B is part of the graph of y = 2x. C is the reflection of graph B in line
A. Write down
(a)
b
the equation of C in the form y =j{x);
the coordinates of the oint where C cuts the x-axis.
Working:
2.
$1000 is invested at 15% per annum interest, compounded monthly. Calculate the minimum
number of months required for the value of the investment to exceed $3000.
Working:
'-3.
--'1
+:
.
The mass m kg of a radio-active substance at time t hours is given by m
(a)
Write down the initial mass.
b)
The mass is reduced to 1.5 kg. How long does this take?
Working:
Answers:
(a)
L...-
----'
=
(6 marks)
4e-02t.
.
(b)
(6 marks)
4.
Ifan account pays 7.25% compounded
$250 in an account for 5 years?
Working:
continuously,
how much you have if you deposited
Answers:
(4 marks)
5.
The diagram shows three graphs.
3
Review of SL-l
Date -----------------------
Topic: Functions and Equations
Subtopic: Quadratics & Transformations
You should be able to:
1) Move confidently between the three forms of a quadratic function.
f(x) = a(x - h)2 + k
where (h, k) is the vertex and x = h is the line of symmetry
f(x)
=
a(x - b)(x - c) where (b,O) and (c,O) are the roots or x-intercepts
f(x)
=
ax2+ bx + c
where the x-coordinate of the vertex is
b
-=--,
2a
x
-b
= -
2a
and the
y-intercept is (O,c)
2) Extract important information about the graph of quadratic function from its
equation.
3) Solve quadratic equations using a variety of techniques including factoring, the
quadratic formula, and the GDC.
4) Write equations of quadratic functions given information about their graphs.
5) Use the discriminant to determine the nature of the roots of a quadratic function.
The discriminant is the quantity (b2 - 4ac) under the square root of the quadratic formula.
6) Identify and describe the following transformations: translations (slide left, right,
up, down), reflections (flip over) and stretches (narrow, wider)
7) Find the equation of the image function following one or more transformations
8) Sketch the image of a function under a transformation.
9) Give a full geometric description of the transformation(s) that map a function or
its graph onto its image.
SL
Quadratics
Name ---------------------
Non-Calculator Part:
1.
a)
b)
c)
d)
e)
f (x) = _X2 + 2x + 8, find:
For the quadratic:
the x-intercepts
the y-intercepts
the coordinates of the vertex
the axis of symmetry
sketch a graph
Working:
Answers: a)
c)
_
_
d)
b)
_
_
....e) show below
(5 marks)
f-
,
I
I
2.
a) Determine the discriminant of
X2
+ 8x + k
= o.
b) Find the values of "k" for which the equation has
i)
no real roots
ii)
two distinct roots
Working:
Answers: a)
b(i)
----
_
b (ii) _-----''--_
(5 marks)
3. Write the equation of the parabola that has vertex (5, -12) and y-intercept 38.
Working:
Answer:
(4 marks)
4. The diagram shows the parabola y
the point B is the maximum point.
=
(7 - x)(1
+ x). The points A and C are the x-intercepts and
y
B
x
Find the coordinates of A, Band C.
Working:
Answers:
(4 marks)
5. Farmer Brown wishes to construct a rectangular chicken pen, using her
barn for one of the sides. The other three sides will be constructed out of
80 m of chicken wire. What are the dimensions that will give her the
maximum area?
Working:
Answer:
...........................................
I
( 5 marks)
6.
The diagram shows part of the graph with equation y
the x-axis at -2 and 3.
= X2
+ px + q. The graph cuts
y
6
~~-_~I~O--~~-+---X
-2
-4
Find the value of
(a)
p;
(b)
q.
Working:
Answers:
(a)
.
(b)
(4 marks)
7.Complete the square to change y = 3x2 - 4x + 1 to the form
y = a(x - h)2 + k .
Working:
Answers: (2 marks)
I
r
I
I
8.The sum of a number and twelve times its reciprocal are 7.
a. Write an equation
b. Solve the equation.
Working:
L-
~
Answers: (3 marks)
a)
_
SL
Quadratics
Name ---------------------
Calculator Part:
1. [3 points] A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet
per second at an angle of
with respect to the ground. The path of the baseball is given
by the function I(x) = -0.0032x2 + X + 3, where f(x) is the height of the baseball (in
feet) and x is the horizontal distance from home plate (in feet).
45°
a) What is the maximum height of the ball?
b) When will it reach that height?
c. If no one catches the ball, when will it reach the ground?
Working:
Answers: (3 marks)
a)
b)
2.
c)
_
[4 points] A tennis ball is thrown vertically upward and its height H, in m, above
the ground is given at one second intervals as:
l=k=(=:t)====:1=~.=2=====1=;=5.=2====1=;=4=.2====1=~=3=.2====:I =~0=.=2
====I=i=.2=====
a) Find the quadratic regression equation for this model.
b) What is the maximum height reached?
c) When does this occur?
d) When is the ball
30 m above the ground?
(4 marks)
Transformations:
3. [5 points] Given y
=
f(x ),
r~
I~
1.1
1\
Draw each of the following:
a) y
=
f(x+2)
b) y = 4 - f(x)
c) y
=
f(2x)
b) y=2f(x) - 3
d) y
=
fe-x)
Quadratics Practice:
1. Let f(x)
=
a(x - 4)2+ 8.
(a) Write down the coordinates of the vertex of the curve off.
(b) Given that f(7)
=
-10, find the value of a.
(c) Find the y-intercept of the curve off.
2. Let f(x)
=
2X2+ 4x - 6
(a) Express f(x) in the form f(x) = 2(x - h)2 + k
(b) Write down the equation of the axis of symmetry of the graph of f.
(c) Express f(x) in the form f(x) = 2(x - p)(x - q)
(d) Graph f(x)
(e) Graph y = - f(x)
(f) Let g(x)
=
f(x + 3)
(g) Find g(-3)
(h) Describe fully the transformation that maps the graph of f to the graph of g.
Topic: Trigonometry
Subgroup: Triangle formulas
You should be able to:
1) Define sine, cosine, and tangent ratios (SOH CAH TO A) for right-angled
triangles.
2) Use the Law of Sines for (ASA, AAS) or ambiguous case (SSA) for non-right
angled triangles.
3) Use the Law of Cosines for (SSS, SAS) for non-right angled triangles.
4) Use the Law of Sines or Law of Cosines to solve problems set in real life contexts
such as naviagation.
5) Calculate the area of non-right angled triangles.
Trig Practice:
1. The depth, y metres, of seawater in a bay t hours after midnight may be represented
by the function
y
=p
+ q cos ( 2:rc ) t, where p, q, and k are constants.
k
The water is at a maximum depth of 14.3 metres at midnight and noon, and is at
minimum depth of 10.3 metres at 6:00 am and 6:00 pm.
(a) Write down the value ofp.
(b) Write down the value of q.
(c) Write down the value ofk.
'L.
In triangle PQR, PR = 5 em, QR = 4 em and PQ = 6 em.
(a) Calculate the size of <PQR
(b) Calculate the area of triangle PQR
Part 2
Name:
GDC allowed - but remember to show your work! Round any decimals to three places.
I.
Let log P ~ x , log Q ~ y and log R ~ z. Express
IOg(~
J
_
in tenns of x, y. and z.
Working:
L-
~
I ":":
.
(4 marks)
2
4. Let f(t)
When t
When t
=
=
(a) (i)
(ii)
(iii)
(iv)
=
a cos b( t - c) + d. Part ofthe graph ofy
=
f(t) is given below.
3, there is a maximum value of29, at M.
9, there is a minimum value of 15.
Find the value of a.
Show that b = Jr / 6
Find the value of d.
Write down the value for c.
The transformation P is given by a horizontal stretch of a scale factor of \12, followed by a
translation of
(b)
(3-10 J
(means right
3 down 10)
Let M' be the image ofM under P. Find the coordinates ofM'
The graph of g is the image of the graph of f under P.
(c) Find get) in the form get) = 7 cos B(t - C) + D
r
_1__ "-----__