Mathematics Course Companion SL
Mathematics Course Companion SL
Ioannis. E. Asimakis
2
3
Contents
Section I (Common SL and HL)………………….4
Topic 1-Numbers and algebra……………………5
Topic 2-Functions…………………………………..26
Topic 3-Trigonometry and geometry…………...54
Topic 4-Statistics and probabilities…………….83
Topic 5-Calculus……………………………………107
Bibiliography………………………………………..142
4
Section I
Common SL and HL
Topics
5
Topic 1
Numbers and algebra
6
Numbers
Rounding numbers
Look to the next smallest place value, the digit to the right of
the place value you're rounding to. For example, if you want to
round to the nearest ten you'd look at the ones place.
If the digit in the next smallest place value is less than five
(0, 1, 2, 3, or 4), you leave the digit you want to round to as-is.
Any digits after that number (including the next smallest place
value you just looked at) become zeros, or drop-off if they're
located after the decimal point. This is called rounding down.
Scientific notation
Scientific notation (also referred to as scientific form or standard
index form, or standard form in the UK) is a way of expressing
numbers that are too big or too small to be conveniently written
in decimal form.
In scientific notation all numbers are written in the form
Significant figures
The significant figures (also known as the significant digits) of a
number are digits that carry meaning contributing to its
measurement resolution.
All non-zero digits are significant: 1, 2, 3, 4, 5, 6, 7, 8, 9.
Examples
Rounded to Rounded to
Precision significant figures decimal places
6 12.3450 12.345000
5 12.345 12.34500
4 12.34 or 12.35 12.3450
3 12.3 12.345
2 12 12.34 or 12.35
1 10 12.3
0 N/A 12
The number
Rounded to Rounded to
Precision significant figures decimal places
7 0.01234500 0.0123450
6 0.0123450 0.012345
5 0.012345 0.01234 or 0.01235
4 0.01234 or 0.01235 0.0123
3 0.0123 0.012
2 0.012 0.01
1 0.01 0.0
0 N/A 0
9
Exercises
Exponents
Exponentiation rules
o ( )
o ( ) ( )
o ( )
o . /
o √
If then
If then
If then .
If then .
11
Exercises
7. If show that :
a. is a multiple of 7.
b. is a multiple of 5.
8. Prove that .
12
Logarithms
If and then:
Logarithm properties:
o ( )
o . /
If then
If then
where . /
13
Exercises
a. √
√
b. √
c.
√
a.
√
b.
a.
b.
c.
4. Show that:
a.
b.
14
( )
( ) ( )
a. {
b. {
a.
b.
c. . / ( )
Sigma notation
By sigma notation we mean the symbol, Σ, which is the Greek upper case
letter, S. The summation sign, Σ, instructs us to sum the elements of a
sequence. A typical element of the sequence which is being summed
appears to the right of the summation sign.
The starting point for the summation or the lower limit of the
summation
For example :
Using this notation we can write large sums, for example the sum of the first
50 positive even numbers:
∑( )
16
∑ ∑ ∑
∑ ∑ ∑
∑
17
Exercises
a. ∑
b. ∑
c. ∑
a.
b. 4+0.4+0.004+…+0.00000000004
c.
d.
e.
∑( )
( ) ∑( )
18
( )
∑ ( ) , ( ) -
∑
19
Tip:
20
Exercises
a. ∑ ( )
b. ∑ ( )
c. ∑ . /
7. Find the sum of all integers, between 10 and 200, which are
divisible by 7.
12. The yearly income of a family is shown below for four consecutive
years:
Financial applications
of geometric series
. /
. /
. /
. /
23
Exercises
1. How much should be invested now to yield 25,000$ in 3
years time , if the money can be invested ar a fixed rate of
4.2% compounded quarterly?
Binomial Theorem
( ) ∑. / ∑. /
Exercises
6. Given that
( ) ( )
find the values of and .
Topic 2
Functions
27
Functions - basics
Exercises
a. ( )
b. ( )
c. ( )
d. ( )
a. ( ) √
b. ( ) √
c. ( ) √
d. ( ) ( )
e. ( ) (√ )
f. ( ) √ ( )
g. ( ) √
29
a. ( ) √
b. ( )
a. ( ) ( )
b. ( )
c. ( )
d. ( )
a. ( ) √
c. ( )
c. ( ) (√ )
a. ( ) and ( )
b. ( ) and ( )
31
Lines
The gradient of the line is the tangent of the angle which has
initial side the x axis and terminal side the line, counting
counterclockwise that is
32
If two lines
Exercises
2. Find the equation of the line which contains the points (1,1),
(12,8)
5. Find the equation of the line which parallel to the line with
equation:
7. Find the equation of the line which contains the point M(2,1)
and intersects the lines:
The discriminant is
.
( ) .
Vietta’ s formulas:
o
o
36
Exercises
a. ( )
b. ( )
c. ( )
a. ( )
b. ( )
c. ( )
a.
b.
c.
d.
37
( )
12. Find the values of for which the roots of the equation
are closest together.
39
Rational functions
( )
A rational function is a function of the form ( ) ( )
where ( ) ( ) are polynomials.
( )
Exercises
a. ( )
b. ( )
c. ( )
d. ( )
e. ( )
f. ( )
a. ( )
b. ( )
42
Inverse functions
( )( ) ( )( ) .
Exercises
2. Let ( ) ( ) .
a. Find the domain of .
b. Find the range of by finding its inverse.
c. Verify that ( )( ) .
3. Let ( ) and ( ) .
a. Find ( ) ( )
b. Find ( ) ( )
c.
4. Let ( ) for .
a. Find the smallest value of such that has an inverse.
b. Find the inverse function and its domain and range.
Exponential function: ( ) ( )
Logarithmic function: ( ) ( )
Exercises
1. Let ( ) .
2. Let ( ) ( )
( )
5. Let ( ) ( ) ( ).
( ) ( )
is an increasing function.
7. Let ( ) .
Transformations of graphs
Let
Vertical shifts:
( ) ( ) ( ) is a shift units up.
( ) ( ) ( ) is a shift units down.
Horizontal shifts:
( ) ( ) ( ) is a shift units to the right.
( ) ( ) ( ) is a shift units to the left.
Vertical dilation:
Horizontal dilation:
Reflections:
( ) ( ) ( ) is a reflection with respect to the axis.
( ) ( ) ( ) is a reflection with respect to the axis.
Exercises
a. ( ) ( )
b. ( ) ( )
c. ( ) ( )
d. ( )
e. ( )
2. Write ( ) in terms of ( )
( ) ( ) to ( ) .
52
a. ( ) ( )
b. ( ) ( )
a. ( ) ( ( )) .
b. ( ) ( ) .
Topic 3
Τrigonometry and Geometry
55
√( ) ( ) ( )
( )
56
Exercises
Solids
Prisms
A prism is a three dimensional solid which has two bases that are
congruent shapes lying on parallel planes.
A very important feature of solids is their volume which can be
found by the following formula:
Rectangular solid
√
Cube
√
58
Cylinder
Pyramid
Cone
59
Sphere
60
Exercises
1. A cone has volume . If its radius is doubled and its
height is tripled find the new volume of the resulting cone.
Trigonometric numbers
62
Exercises
6. Two people are standing on two sides of a hill. The top of the
hill is exactly between them. Both people are watching the
top. Line of sight of a man who is 1.8 m high makes with
the horizontal, while line of sight of a woman 1.6 m high
makes with the horizontal. The distance between the
man and the woman is 18 metres. Find the height
of the hill.
Sine rule:
Cosine rule
Since two angles that have sum of have the same sin
the sine rule can give two possible solutions. This is called
the ambiguous case. Given two possible solutions you
should check in both possible triangle that the sum of their
angles is indeed .
66
Exercises
Circumference of a circle:
Exercises
Pythagorean identity
( )
( )
( )
( )
( )
( )
( )
. /
. /
Exercises
1. Simplify in terms of or
a. . /
b. ( )
c. ( )
d. . /
2. Show that:
a. ( ) ( )
b. ( ). /
c.
a. ( )
b. ( )
8. If write in terms of
a.
b.
a.
b.
75
Trigonometric functions
( )
, -
( )
, -
76
( )
| |
77
Exercises
a. ( ) . / , -
b. ( ) . / , -
c. ( ) . / , -
( ) ( ) , -
Trigonometric equations
where
or where .
where
or where .
where
where .
81
Exercises
a.
b. √
c. √
a. √ , -
b. , -
a. , ]
b. , -
a. . /
b. ( ) √
82
a.
b.
c.
d.
e.
a. . /
b.
a. , -
b. , -
a.
b. √
a. √
b. for
83
Topic 4
Statistics and probabilities
84
Sampling methods
Sampling methods:
Exercises
Measures
of central tendency and spread
∑
̅
or
∑
̅
Measures of spread
o Variance:
∑( ̅)
( )
or
∑ ( ̅)
( )
∑( ̅)
√
or
∑ ( ̅)
√
o Interquartile range:
Median
Mode
Variance
Standard | |
deviation
90
Exercises
1. For the two following sets of data calculate the mean, the
median, the mode, the variance, the standard deviation, the
interquartile range and the range.
a. * +
b. * +
2. For the following data calculate the mean, the median, the
mode, the standard deviation, the interquartile range and
the range:
1 24
2 70
3 26
4 50
5 30
a. Find the mean, the median, the mode, the range and
the upper
and lower quartiles.
b. Draw an appropriate box and whisker plot for this
data.
11. A sample of 70 batteries was tested to see how they last. The
data in grouped form are given below:
Probability basics
More strictly:
( )
( )
( )
where ( )
and ( ) .
( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( )
( | )
( )
( ) ( ) ( )
94
Exercises
7. If ( ) ( ) and ( | ) find ( )
95
10. A bag contains 5 red balls, 3 blue balls and 2 green balls.
A ball is selected without replacement and then a second
ball is chosen. Find the probability of selecting one blue
ball and one green ball in any order.
16. The probability that it rains on any given day is and the
probability that it is cold is 0.6. The probability that is
neither cold or raining is 0.25.
where ( )
( ) ∑
∑ ∑
( ) ∑( ) ( )
98
Exercises
0 1 2 3
( ) 0.2 0.3 0.1
0 1 2
( )
( )
0 1 2 3
( ) 0.2 0.25
Given that ( ) .
1 2 3 4
1/5 2/5 1/10 x
( ) ( )
( ) ( )
Binomial distribution
( ) ( )
where ( ) ( )
Expected value:
( )
Variance:
( ) ( )
102
Exercises
1. A box contains three red balls, two green balls and five blue
balls. We chose one ball at random and then replace it at the
box. Find the probability that if we draw 10 times we will
draw a red ball exactly four times.
Normal Distribution
. /
( )
√
If ( ) then ( )
105
Exercises
Topic 5
Calculus
108
Limits
( )
Categorizing limits
For example:
The non existence of the above limit is clear if you see the
graph of the function . If approaches zero from the
positive numbers the function tends to positive infinity.
If approaches zero from the negative numbers the
function tends to negative infinity.
110
o if
Then the graph increases to positive infinity to the
right.
o if
Then the graph increases to negative infinity to the
right.
o if and is even.
o if and is even
Then the graph decreases to negative infinity to the
left.
o if and is odd
Then the graph decreases to negative infinity to the
left.
o if and is odd
Then the graph increases to positive infinity to the
left.
111
Let ( )
Then
( )
Continuity
A function is continuous if its graph on the intervals of the
domain can be drawn without lifting your hand from the
paper. Or more formally if:
( ) ( )
112
Exercises
a.
b.
c.
√
a.
√
b.
c.
√
a.
b.
c.
d.
e.
113
Differentiation basics
( ) ( )
( )
( ) ( )( )
( ) ( )
( )
114
( ) ( )
√
115
Exercises
( )
Differentiation rules
( ( ) ( )) ( ) ( )
( ( ) ( )) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
. ( )
/ ( )
. ( ( ))/ ( ) ( ( ))
or in Leibnitz notation:
118
( ( )) ( ) ( )
( ) ( )
( ) ( )
( )
( ( ( ))
( )
( ( )) ( ( )) ( )
( ( )) ( ( )) ( )
( )
( ( ))
√ ( )
( )
( ( ))
√ ( )
( )
( ( ))
( )
( ) ( ( ))
( ) ( )
( ) . ( )/
119
Exercises
a. ( )
b. ( )
c. ( ) √ √
a. ( )
b. ( )
c. ( )
d. ( )
e. ( )
a. ( )
b. ( )
c. ( )
a. ( ) ( )
b. ( ) ( )
( ) ( )
c.
d. ( ) ( ( ))
120
a. ( ) √
b. ( )
c. ( ) ( )
Extrema
Monotony
Fermat’s theorem:
If is a differentiable function and has a minimum or a
maximum point at ( ( )) then ( )
Exercises
a. ( )
b. ( )
c. ( )
d. ( )
Concavity
Exercises
a. ( )
b. ( )
c. ( )
d. ( ) ( )
e. ( )
Antidifferentiation-Indefinite integral
∫ ( ) ( )
Rules of integration
o ∫ ( ) ( ) ∫ ( ) ∫ ( )
o ∫ ( ) ∫ ( )
129
∫ | |
∫ ( ) ( ( ))
( )
so
( )
∫ ( ) ( ( )) ∫ ( )
130
Exercises
a. ∫
b. ∫
c. ∫
d. ∫ √
e. ∫
a. ∫
√
b. ∫
√
c. ∫
131
a. ∫
b. ∫ ( )
c. ∫ ( )
d. ∫
e. ∫
a. ∫ √
b. ∫
c. ∫
d. ∫( ) ( )
e. ∫
Definite integrals
∫ ( ) ∑ ( )
∫ ( ) , ( )- ( ) ( )
o ∫ ( ) ∫ ( )
o ∫ ( ) ∫ ( )
o ∫ ( ) ∫ ( ) ∫ ( )
o ∫ ( ) ( ) ∫ ( ) ∫ ( )
( )
∫ ( ( )) ( ) ∫ ( )
( )
133
Exercises
a. ∫
b. ∫
c. ∫
d. ∫
a. ∫
b. ∫
c. ∫
d. ∫ . /
a. ∫
b. ∫ ( )
c. ∫ ( )
134
a. ∫
b. ∫
a. ∫ ( )
b. ∫
135
∫ ( )
Tip!: If you are asked for the area between the graph of
and the axis, you should find the intercepts of and use
them as the limits of integration.
∫| ( )|
∫| ( ) ( )|
Tip!: If you are asked for the area between the graphs of the
functions and , you must find the point of their
intersection and use them as the limits of integration.
136
Exercises
6. Let ( ) √ .
Kinematics
( ) ( )
( ) ( ) ( )
( ) ∫ ( )
( ) ∫ ( )
139
∫ ( )
∫ | ( )|
Exercises
( ) {
√
Bibliography
Internet sources
[1] www.christosnikolaidis.com
[2] el.khanacademy.org
[3] www.varsitytutors.com
[4] www.revisionvillage.com
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