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MATHEMATICS
in English
2nd year
R. O’Donovan
«[email protected]
”Take some more tea”, the March Hare said to Alice, very earnestly.
”I’ve had nothing yet”, Alice replied in an offended tone, ”so I can’t take more”.
”You mean you can’t take less”, said the Hatter: ”it’s very easy to take more than nothing”.
Lewis Carroll, Alice’s Adventures In Wonderland
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2
How Things Are Said
Sign
Example
=
Spoken form
equal or
equals or
are
+
1+2
one plus two
−
3−1
three minus one
×
2×3
two multiplied by three or
six)
: or ÷
six divided by two
6=
6:2
6
2
2 6= 3
>
3>2
3 is greater than 2
≥
4≥3
4 is greater than or equal to 3
<
2<3
2 is less than 3
≤
2≤5
2 is less than or equal to 5
±
±4
plus or minus 4
−
−4
minus 4
a
ba
b to the power of a
2
32
√
4
3 to the two or
23
√
3
2 to the three or
2 cubed
the third root or
the cubic root of 8
√
3
√
3
8
or
is or
or
three take away one
two times three
or
or
or
two into six (goes three)
or
2 is not equal to 3
or
makes
or
2 does not equal 3
one from three
two threes (are
six over two
negative 4
or
b to the a
3 squared
the square root of 4
2 to the power 4 or
10−4
10 to the minus 4
.
0.1
zero point one
.
.2
point two (= 0.2)
3.105
three point one oh five
−4
make
one and two
24 = 16
4
or
or
2 power 4 or
2 to the 4 = 16
(American) naught point one
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From Symbols to Words
∠
angle
c
abc
the angle defined by the three points a, b, c with b as vertex
⊥
is perpendicular to
//
is parallel to
±
plus or minus
=
is equal to
≡
is identically equal to
≈
is approximately equal to
'
is ultraclose to
6=
is not equal to
<
is less than
≤
is less than or equal to
>
is greater than
≥
is greater than or equal to
⇒
implies
⇐
is implied by
⇔
implies and is implied by; if and only if; is equivalent to
iff
if and only if
(informal arrow) leads to
{a, b, c . . . }
the set with elements a, b, c, . . .
∈
is element of; belongs to; in
B = {x ∈ A | . . . }
the set B defined by all x in the set A such that . . . (where the dots
are a given property that the elements must satisfy to be in set B.)
∅
the empty set
| A | or #A
the number of elements of the set A
A
the complement of A (with respect to a universe)
¬A
the negation of A; non A;
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∪
union
∩
intersection
⊂
is a subset of
⊆
is a subset of and maybe equal to
N
the set of natural numbers (positive integers)
Z
the set of integers (positive or negative or zero)
Z∗
the set of integers without zero
Q
the set of rational numbers
R
the set of real numbers
R+
the set of positive reals
R−
the set of negative reals
R∗
the set of reals without zero
C
the set of complex numbers
f, g, h, u, v
functions (unless otherwise specified)
w, x, y, z
unknowns or variables (unless otherwise specified)
a, b, c, d
constants (unless otherwise specified)
p, q, r
parameters (unless otherwise specified)
y = f (x)
the rule of the function f
f (x)
the function value for x
f −1
the inverse function
f ◦g
the composition of the function f and g; f circle g
(f ◦ g)(x)
the value of the composition of f and g at x
f (g(x))
f of g of x; the value of the composition at x
→
into; for a mapping
f :A→B
f from A into B; the function f takes all inputs from set A and has
its outputs in B
7→
maps to
f : x 7→ y
the function f maps x to y; f is the function under which x is
mapped into y
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f : A→B
x 7→ f (x)
the complete specification for the function: input, output and rule.
|x|
the modulus of x; the absolute value of x
sin
the sine function
cos
the cosine function
tan
the tangent function; the tan function
arcsin or sin−1
the arcsine function; the inverse function of sine
arccos or cos−1
the arccosine function; the inverse function of cosine
arctan or tan−1
the arctan function; the inverse function of tan
loga
the logarithm in base a
log
the logarithm in base 10
ln
the logarithm in base e
y0
y prime; the derivative of y – as function of an independent variable,
usually unstated x
dy
df
or
(x)
dx
Rdx
b
a f (x)dx
the derivative of y = f (x) depending on x
R
f (x)dx
the definite integral of y between a and b; the integral of y between
a and b
the indefinite integral of y with respect to x; the antiderivative of y
E(X)
expectation of (random variable) X
V ar(X)
variance of (random variable) X
µ or x
population mean
σ2
population variance
σ
a
b
population standard deviation
binomial coefficient
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GREEK ALPHABET
A
α
alpha
B
β
beta
Γ γ
gamma
∆
delta
δ
E epsilon
Z ζ
zeta
H η
eta
Θ θ
theta
I ι
iota
K κ
kappa
Λ λ
lambda
M µ
mu
N ν
nu
Ξ ξ
xi
O
omicron
o
Π π
pi
P ρ
rho
Σ
σ
sigma
T
τ
tau
Y
υ
upsilon
Φ
φ
phi
X
χ
chi
Ψ ψ
psi
Ω
omega
ω
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Mathematics
1
FUNDAMENTALS
Mathematics is yet another language. When spoken orally, it can be pronounced using English,
French, German or any other natural language. It can also be seen as a specific dialect of these
languages.
Mathematics is a language such that absolute certainty can be achieved within.
This is fundamental: in order to achieve this, mathematics is a strongly restricted language.
Words are redefined as to become univocal (i.e: one and only one non ambiguous meaning
accepted by all)
“Pink ice creams taste better” is not a mathematical statement because too many words have
ambiguous interpretations.
Symbols
“The signs customarily employed [. . . ] are of two kinds. The first consists of letters, of which
each represents either a numbers left indeterminate or a function left indeterminate. This indeterminacy makes it possible to use letters to express the universal validity of propositions, as
in
(a + b)c = ac + bc
√
The other kind consists of signs such as +, −,
, 0, 1 and 2, of which each has a particular
meaning.” (G. Frege 1879)
“A variable is a symbol which is to have one of a certain set of values, without it being
decided which one. It does not have first one value of a set and then another; it has at all times
some value of the set, where so long as we do not replace the variable by a constant, the “some”
remains unspecified.” (B. Russel 1908)
“A variable is one of a certain set of values, without its being decided which one. [. . . ] A
variable is represented by a symbol.” (G. Frege 1910)
Context
Definition 1 (Statement)
A statement is a declarative sentence that is either true or false.
No statement is meaningful without a context.
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Examples:
(1) Lausanne is a Swiss town.
(3) It is raining.
(2) 25 is the square of 5 and -5.
(4) 20 is the square of 10.
Counter-examples:
(1) The train will probably be late.
(3) Some pretty red roses.
(2) He may have forgotten his key.
(4) What time is it?
Usually, when not stated, the unknowns are of the required kind.
Traditionally:
• a, b, c, . . . are fixed values (known or not), constants.
• i, k, j are natural numbers.
• x, y, z, w are unknowns or variables.
• n is an unknown integer.
• p, q, r are parameters.
• Uppercase letters are used for sets.
But in geometry, traditionally:
• a, b, c . . . are sides.
• A, B, C . . . are points.
• h is the altitude (or height).
• b is the base of a triangle (in formulae) – which means that it cannot be used to indicate
another side...
• r is for the radius of a circle.
Communication, convenience and good manners are why the context is not always explicitly
given. If for justifiable reasons the context is changed, it must be done explicitly.
“For what values of a is a · n + 1 a even number?” implies that the answer is supposed to
be independent from n, that a is unknown and “even numbers” implies that the context is that
of integers.
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2
LOGIC
2.1
True/False
The two fundamental rules of classical logic, which will constitute the permanent context in
mathematics (unless explicitly specified)
1 - Consistency : A statement cannot be simultaneously True and False.
2 - The Excluded Middle Principle : A statement must be either True or False (no
third or middle possibility).
The two rules above define what we accept as a statement in the context of classical logic.
We will talk of the Truth value of a statement. A truth value is 0 or 1 if considered numerically,
otherwise: True or False.
Examples
“2 + 3 = 5” is a true statement (truth value is True).
“2 + 4 = 3” is a false statement (truth value is False).
A Statement is a sentence which is either true or false.
This does not mean that we actually know whether the statement is true or false. It means
that the answer would only be one of the two: for instance “The billion billionth decimal of π is
5” is a statement, but no human nor machine has the answer yet...
2 + 3 = 5 is a true statement (truth value is True), 2 + 4 = 3 is a false statement (truth value
is False).
The words right and wrong are for answers, thus 3 is the wrong answer for the computation
of 2 + 4. These words have nothing to do with logic.
Exercise 1
Which of the following are statements?
(1) The sky is blue.
(4) All the King’s horses and all the King’s
men.
(2) The rainbow has fifty colours.
(5) This statement is True.
(3) A dog.
(6) This statement is False.
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2.2
Truth Tables
A truth table is a simple way to solve simple logical problems.
For the truth table of “A AND B” put one column for A, one column for B and put all
possible combinations of truth for A and B. Then “A AND B” is true if and only if A is true
and B is true.
Why? You must think of “A AND B” as a single new statement (also written A&B). Remember that a statement is true if it is not false. What observation would prove that it is
false?
A B A&B
1 1
1
1 0
0
0 1
0
0 0
0
2.3
Logic operations
Apart from consistency and the excluded middle principle, the basic logic that we will use has
the following properties.
(1) ∧ meaning and: both A and B are True
(2) ∨ meaning or: either A or B (or both) are True
(3) ⇒ meaning implies: if A ⇒ B, and if A is True, then B is also True. This defines
implication as a relation between two statements A and B.
!
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A ⇒ B is not the same as B ⇒ A
(4) Transitivity: if A ⇒ B and B ⇒ C then A ⇒ C
(5) Negation: if A is True then ¬A is False and if A is False then ¬A is True.
Exercise 2
Write the truth tables for “A OR B” and for “A XOR B” – where XOR stands for “exclusive OR”
which is used when both cannot be simultaneously true.
Exercise 3
Write the truth table for ¬A.
Exercise 4
Write the truth tables for “(¬A) ∨ (¬B)” and for ¬(A ∧ B). Compare them.
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Exercise 5
For the table of A ⇒ B, first find when is the statement false? All other cases must be true.
Exercise 6
Write the truth table for A ⇒ B and the truth table for ¬B ⇒ ¬A. Observe that they are
similar. This is a simple proof for the contrapositive equivalence.
Exercise 7
Use a truth table to answer: If A ⇒ B and C ⇒ A are both true, and if A and C are true, is
B necessarily true?
Exercise 8
Give an example of a statement of the form A ⇒ B which is true but for which B ⇒ A is false.
Give an example of a statement of the form A ⇒ B which is true and for which B ⇒ A is
also true.
(1) If A ⇒ B has been proved, and if A is True, then B is also True.
Assume that the statement “If it rains in Geneva the roads are wet in Geneva” is true.
Then it is true even if it is sunny and even if you are in Australia. But given the truth if
this statement, if you observe that it is raining in Geneva, then you can assert – without
checking – that in Geneva the roads are wet.
(2) If a statement is True, then its negation is False and if a statement is False, its negation
is True. This may seem obvious because of the excluded middle principle, but in fact it
defines what a negation is!
Other rules of logic appear during a mathematical education. We stress here on the implication because its misunderstanding is one of the main causes of mistakes.
Exercise 9
In the following four situations, the two statements must be accepted. The question is the same
for each: “What is the conclusion” (From Lewis Carrol 1896)
(1) All cats understand French.
(3) All hard working students succeed.
All ignorant students fail.
Some chicken are cats.
(2) None of my sons is dishonest.
(4) All lions are fierce.
Honest men are always respected.
Some lions don’t drink coffee.
Exercise 10
(1) Knowing that if my neighbour is late in the morning she runs to catch the bus, and that
this morning my neighbour was running to catch the bus, is it possible to conclude that
she was late?
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(2) When it rains the roads are wet. What is the conclusion if the road is wet? What is
the conclusion if the road is dry? What is the conclusion if it is raining? What is the
conclusion if it is not raining?
Exercise 11
The negation of a statement is a statement which is the description of the simplest possible fact
which would make the initial statement wrong.
Give the negation of the following statements:
(1) The night sky is black.
(3) The decimal form of a rational number is
of finite length or it has a repeating sequence of digits.
(2) I am young and handsome.
Exercise 12
Let A be the statement: ’I am tired’ and B the the statement: ’I am happy’.
Translate in English the following statements:
(1) ¬A
(5) ¬(A ∧ B)
(2) ¬¬A
(6) ¬A ∨ ¬B
(3) ¬A ∨ B
(7) ¬(A ∨ B)
(4) ¬(¬A ∨ B)
(8) ¬A ∧ ¬B
2.4
Converse and equivalence
The converse of A ⇒ B is B ⇒ A.
In general, the converse of an implication is not True. In the exceptional case when two
statements imply each other, we say that they are equivalent, and symbolize this with a twoway arrow.
In other words, A ⇔ B means A ⇒ B and B ⇒ A.
2.5
Contrapositive
The contrapositive is the equivalence between A ⇒ B and ¬B ⇒ ¬A.
Equivalence means that proving one of the two is also a proof for the other one.
Exercise 13
Show that if
“if circles have angles then a square is a circle”
is true then
“if a square is not a circle then circles have no angles”
is also true.
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Exercise 14
Show that if the statement “if x > 1 then x2 > 1” is true, then the statement “if x2 < 1 then
x < 1” is also true. (Written [x > 1 ⇒ x2 > 1] ⇒ [x2 < 1 ⇒ x < 1])
Do not prove the second one directly! Use that “if x > 1 then x2 > 1” is true.
2.6
Proving/Disproving
To prove a statement means to prove that it is true.
To prove that a statement is not true is sometimes said: to disprove.
There are two kinds of statements: those that state that under certain conditions a given
property can be satisfied and those that state that under certain conditions a given property is
satisfied.
In the context of a zoo:
• All polar bears have perfect teeth.
• Some elephants are white.
To prove the first statement, it will be necessary to check all the teeth of all the polar bears.
To find a single polar bear with a faulty tooth would be enough to disprove it.
To prove the second statement, it would be enough to find a white elephant (even if there
are more than one, to find one is enough.) To disprove it would need checking all the elephants.
In the context of geometry:
• All rectangular triangles with sides of length a, b, c – with c being the longest side –
satisfy a2 + b2 = c2
• Some multiples of 7 can be divided by 3.
Exercise 15
For each of the following, state what would be necessary to prove it and what would be necessary
to disprove it. (It is not necessary to know if the statements are true or not.)
(1) No dog is smaller than a cat.
(2) Some dogs are smaller than some cats.
(3) There is a peace Nobel prize every year.
(4) It never rains when the temperature is over 30◦ C
Exercise 16
In an A ⇒ B argument, if A is false, then anything can happen.
Explain which steps are wrong in the following “proof” that 1 = −1
If 1 = −1
then 12 = (−1)2
as 12 = 1 and (−1)2 = 1
therefore 1 = 1
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as the conclusion is correct, this proves that 1 = −1
There are 3 main ways to prove a statement of the form A ⇒ B
(1) Direct proof: assume A is true and use only that to prove that in that case B is necessarily
true.
(2) Proof by contrapositive: assume ¬B is true and use only that to prove that in that case
¬A is necessarily true.
(3) Proof by contradiction (also called reductio ad absurdum): Assume A and ¬B to be both
true and deduce a contradiction (of the from 1 = 0 for instance).
The two last methods rely on the excluded middle principle.
Exercise 17
The following example seems to shows that 2 = 1! What went wrong and where?
a−a
=
a−a
a(a − a)
=
a(a − a)
multiply both sides by a
expand the left hand side
a2
−
a2
=
a(a − a)
factorise the left hand side using
a2 − b2 = (a + b)(a − b) [Here, a = b]
(a + a)(a − a)
=
a(a − a)
simplify both sides by (a − a)
a+a
=
a
collect similar terms
2a
=
a
2
=
1
divide both sides by a
2.7
Axioms
In any logical system, some statements have to be accepted without possible proof. They are
called axioms. Euclid (300 BC) was the first to introduce this concept, such as: ’any two points
can be joined by a straight line’. He took his axioms to be self evident.
It is important to understand that it is not possible to get away without axioms when building
a language in which certainty can be achieved. The axioms which are chosen define the context
of a particular logical system. Once the axioms and the rules of logic are accepted, theorems
can be proved and they are absolutely certain within this system.
In a more abstract sense, axioms are accepted without interpreting them in the real world.
Then proofs are deductions for that case “if the axioms are true, then...”
Exercise 18
Consider the following axioms. (They are not meant to be “self evident”!)
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Iceberg Mathematics
Axiom 1: Between any two polar bears on an iceberg is a unique penguin on the same
iceberg. We say that the penguin is related to these bears.
Axiom 2: Every penguin on an iceberg is between two polar bears on that iceberg.
Axiom 3: On the iceberg there exist three distinct polar bears which are not related to the
same penguin.
Using only these axioms and rules of logic, prove the following propositions (or theorems).
Proposition 1: If two penguins are related to more than one polar bear, then they are in
fact the same penguin.
Proposition 2: For any two distinct penguins, either they are related to no commmon polar
bear or they are related to exactly one polar bear.
Proposition 3: For every penguin there is at least one polar bear to which it does not relate.
2.8
Equality
The equal sign “=” relates something on the left hand side with some other thing on the right
hand side by asserting that they are the same. Obvious?
The hidden assumption is that sometimes you have to perform several computations before
the truth value is obvious.
An equal sign does not mean that the two are equal, but the whole object becomes a statement
– which is either true or false.
2 + 4 = 5 + 1 is true because if you compute both sides, you get 6 = 6
An assertion is meaningless if it equates mathematical objects which are not of the same
kind. If a statement is incomplete in such a manner that its truth value cannot be decided, then,
formally, it is not a statement!
Exercise 19
Some of the following “statements” are true, some are false, some are meaningless therefore not
really statements. Which are which and why?
(1) 3 + 3 + 3 + 3 = 3 · 4
m
(2) 3 · 6s = 18m
s
(3) 3kg · 8
(9) 1 = 0
(10) a = a
√
(11) x = −5
m
kg
= 24
s2
m
(12) a, b and c may be negative integers.
(4) Let A =< 1; 2 >, B =< 0; −5 > and
C =< −1; 2 > then ∆ABC = 5
(13) Let 2x − 3 = 0 then S =
(5) x = 2
(14) c + 1 = d
(6) b > a
(15) x2 = x2
(7) 26 = 1
(16) a1 , a2 , a3 are natural numbers.
(8) Let 2x + 1 = 0 then S = {− 21 }
(17) −a is positive.
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3
2
(18) a2 = x2 because the index are the same.
2.9
(19) a1 + a2 = a3
Solving Equations
An equation is a conditional statement, it is an equality where one or more letter appear as
unknown values. It is a statement which gives a condition to be satisfied by the unknown(s).
For equations with one unknown:
To solve an equation (unknown written as x) is to find values such that if these values are
substituted for x they produce a true statement.
We conclude that an equation has no solution if we can prove that there is no way to make
the statement true.
Whether true or false, the truth value of a statement is not changed if the same operation
is performed on each side of the equal sign – with two strong restrictions: It is not possible
to divide both sides by 0 because the division by zero is not defined, and it is not possible to
multiply both sides by zero because the truth value is not preserved.
Exercise 20
Why is the division by zero undefined?
Could it be possible to “invent” a value for, say 5/0?
Exercise 21
Why is the multiplication by zero not truth preserving?
An equation is satisfiable if there is (at least) one value which substituted
for the unknown makes it a true statement.
The set of these values are is the solution set.
An equation is unsatisfiable if there is no value which substituted for the
unknown can make it a true statement.
It has no solution. The solution set is empty: { } or ∅.
An equation is an identity if is true for any value substituted to the
unknown.
Because an equation contains at least one unknown value, it is neither True not False. All
depends on the value taken by the unknown.
Exercise 22
State whether the following equations are satisfiable, unsatisfiable or identities. (Prove your
answers.)
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(1) (x + y)2 = x2 + 2xy + y 2
(9) 3 − 5 = x
√
(10) v = 16
√
(11) w = 5
√
(12) u = −5
√
(13) −x = 5
(2) (x + y)2 = x2 + y 2
(3) x + 1 = 2x
(4) x + 1 = x
√
(5) x2 = x
√
(6) x2 = −x
√
(7) ( x)2 = x
(14) x = 2
(15) y = y + 1
(8) 5 − 3 = x
(16) x = y
The equation 2x = 6 does not mean that x must take the value 3. It states that if x = 3
then the statement is True and if x 6= 3 then the statement is False. Both are important!
Quantifiers
∀ means “for all”.
If P is a statement about a property of mathematical objects,
∀x ∈ A
P (x)
reads “for all x in A, x has the property P ” or: “all x in A have property P ” or “all x in A
satisfy P
∀x ∈ N
2·x∈N
means that any natural number multiplied by 2 is a natural number.
∃ means “there exists (at least one)”.
If P is a statement about a property of mathematical objects,
∃x ∈ A
P (x)
reads “there is an x in A such that x has the property P ” or: “there is an x in A which has
property P ” or: “there exists an x in A which satisfies P ”.
∃x ∈ N
x:2∈N
means that there is a natural number which if divided by 2 yields a natural number. ∃ does
not mean that there is only one.
∃ and ∀ are called “quantifiers”.
To disprove
∀x ∈ A
it is sufficient to show
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P (x)
∃x ∈ A
¬P (x)
where ¬P (x) is the negation of P , this means “there is an x which does not satisfy P ”
To disprove
∃x ∈ A
P (x)
it is necessary to show
∀x ∈ A
P”
¬P (x)
“All x satisfy non-P ” which by the excluded middle principle is equivalent to “No x satisfies
Exercise 23
Let x and y ∈ R. Prove or disprove:
(1) ∀x
∃y
(x + y = 3)
(3) ∀x
∀y
(x + y = 3)
(2) ∃y
∀x
(x + y = 3)
(4) ∃x
∃y
(x + y = 3)
Exercise 24
Prove or disprove the following statements. State whether you are using a direct or indirect
proof.
Make the context explicit.
(1) All prime numbers are odd numbers.
(2) All prime numbers greater than 9 are odd numbers.
(3) All prime numbers greater than 123 are odd numbers.
(4) The square of an even number is an even number.
(5) The square of an odd number is an odd number.
(6) If x2 is even and if x is a natural number, then x is even.
(7) If x2 is odd and if x is a natural number, then x is odd.
p √
p2
= 2 then 2 = 2 and thus p is an even number.
q
q
p √
(9) If
= 2 and if p is an even number then q is also an even number.
q
√
(10) No fraction can be equal to 2
(8) If
(11) All rational numbers are either of finite length or have a repeating sequence of decimals.
√
(12) Their exists a number other than 2 which is not a rational.
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Exercise 25
(1) Consider the statement “x < 5 and y = 2”. What is its negation?
(2) Consider the statement “∀x ∈ N
f (x) = 0”. What is its negation?
Exercise 26
True of false? Justify!
(1) ∃x ∈ N such that x2 + 4x = 5
(2) ∃x ∈ N such that 3x + 2 = 4x − 3
(3) ∃x ∈ R such that 5/0 = x
(4) ∃x ∈ R such that 5/x = 0
(5) ∀x such that if x is a multiple of 3 and a multiple of 5, then x is a multiple of 20.
(6) ∀x ∈ N, then 2x + 1 is an odd number.
(7) ∃x ∈ N, ∃y ∈ N such that x2 + 3xy = 4y
(8) ∃x, ∃y such that if x 6= y, then x2 = y 2
Exercise 27
Write the negation of the statements in the previous exercise.
2.10
Theorems
A theorem has three parts:
(1) A context – not necessarily given explicitly in the theorem. It is about geometry, or
functions or numbers, etc.
(2) Initial assumptions; the hypothesis: these must be given.
(3) A conclusion.
A theorem is a statement which has been proved which relates the hypothesis and the
conclusion in the following manner:
Theorem 1 (Example Theorem)
For x satisfying conditions C
P (x)
If x satisfies conditions C then it satisfies the property P . This does not give any information
about x if it does not satisfy the initial condition.
21
Theorem 2 (Sum of Angles)
The sum of the internal angles of a triangle is 180◦
Context: Plane (Euclidean) geometry.
Hypothesis: the figure we are talking about is a triangle.
Conclusion: its angles sum up to 180◦
If the figure is not a triangle, we do not say that the sum of internal angles is not 180◦
The theorem could be written:
Theorem 3 (Sum of Angles)
If A is a triangle, then the sum of the internal angles of A is 180◦
or
A is a triangle ⇒ the sum of internal angles of A is 180◦
{z
}
|
{z
}
|
hypothesis
conclusion
Theorem 4 (Example Theorem)
If x > 5 then x2 > 25
Theorem 5 (Converse of the Example Theorem - False!)
If x2 > 25 then x > 5
It can be proved to be false with just one counter-example: x = −10.
The contrapositive, however, is True.
Theorem 6 (Contrapositive of the Example Theorem)
If x2 ≤ 25 then x ≤ 5
A theorem is true (because it has a proof) – not sometimes true, nor always true: just True.
The only thing that could make it seem not to be true would be a change of context where it
might have no meaning at all. But the unsaid context is really part of the theorem.
Reasoning and using intelligence are more important than finding the actual answer! This
is why, throughout this mathematics class
calculators are banned !
But computers can help...
22
3
LETTERS
Expressions using letters are called literal expressions.
Exercise 28
(1) Write the list of the five successive numbers after n, and the three successive numbers
before u6
(2) Write the first four natural numbers.
(3) Write the first four integers.
(4) Write the first three fractions after
1
4
(5) Write a list of 5 numbers which are not fractions.
1
2
and the difference between each is
3
3
Add all positive integers from 1 to 10 is the sum 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10. A
shorter writing can be used with the concept of a variable: n, i, j or k are often used when the
variable takes only integer values. Add all values of n from 1 to 10 by steps of 1. This is written
(6) Write a list of four fractions, the first being
10
X
n
n=1
Exercise 29
Expand each of the following sums and give the result:
(1)
4
X
(5)
j
j=1
(2)
4
X
(4)
k(k − 1)
k=0
(6)
2
i
2
X
2 · 3i
i=0
i=1
(3)
5
X
0
X
(7)
2j
2
X
j=−5
i=0
4
X
4
X
(8)
u
i=0
2xi
ak · xk
k=0
Exercise 30
Show that:
a·
10
X
j=0
bj =
10
X
a · bj = a · b0 +
j=0
10
X
j=1
23
a · bj
3
Exercise 31
X
Which of the following statements about the sum
ak are true? (Explain why).
k=0
(1) The sum has three terms.
(2) The first term is 0.
(3) The last term is three.
Exercise 32
Calculate
3
X
i=1


4
X

i · j
j=2
24
4
FUNCTIONS
Exercise 33
(1) What is a function? (definition)
(2) Give examples of functions – from the mathematical world and from the “real world”. Try to
give examples which are very different from one another: difference in the type of function,
difference in the presentation.
(3) Do functions and equations have anything in common? (what? why? where? justify
everything!) give examples or counter-examples.
(4) Give examples of things which “look like functions” but are not – and explain why they
are not.
(5) How can functions be represented? Give examples of each.
(6) On a graphical representation of a curve, which observations would contradict that it can
represent a function?
(7) What is the difference between variable and unknown?
(8) Which of these are functions?
(a) temperature with respect to time.
(b) time with respect to temperature.
(c) y = 2x2 + 5
(d)
R −→ R
x 7→ 1 − x
f:
(e) u : x 7→ 3x − 3
(f) g(x) = 1
(g) 3x2 + 1
Exercise 34
Specify the domain of the following functions:
(1) f : x 7→
(2) f : x 7→
(3) f : x 7→
(4) f : x 7→
(5) f : x 7→
1
x
√
√
√
√
(6) f : x 7→
x
x−2
x+3
12 − x
√
20 + x
1
x+3
√
x−2
(8) f : x 7→
x−4
√
x−2
(9) f : x 7→
x+1
√
4
(10) f : x 7→
(7) f : x 7→
25
x2
x−2
− 3x − 4
√
3
(11) f : x 7→
x−2
x2 − 3x − 4
√
x−1
(12) f : x 7→ √
1−x
Exercise 35
Which of the following graphs represent functions?
Exercise 36
Do functions and equations have anything in common? How can you distinguish them? (give
examples, counter-examples, justify everything!!!)
Exercise 37
The absolute value, denoted |a| is defined by:
(
a if a ≥ 0
|a| =
−a if a < 0
Sketch the graph of and specify the range of
f : [−1; 4] −→ R
x
7→
|2x − 3|
(start by writing the above function without the absolute value)
Exercise 38
A piecewise-defined function is a function defined by more than one expression:


−4 if x < −1


x − 2 if − 1 ≤ x < 2
g : x 7→


 1 x if x ≥ 2
2
Sketch the graph of g(x).
26
Exercise 39
A step function is a function defined by steps: The cost of parking at an airport is 1CHF for the
first hour or any part of the first hour, and 2CHF for each additional hour or any part of the
additional hour. Sketch its graph.
Exercise 40
Sketch the graph of
(1) f : x 7→ |x + 2| − 3
(2) g : x 7→ |x2 − 4|
(3) h : x 7→ |x − 2| + |x + 3|
4.1
Lines
Exercise 41
Sketch the graph of
f : [−2; 4] −→ R
7→
x
3x − 2
Exercise 42
Sketch the graph of
g : [0; 10] −→ [−5; −4]
7→
x
2x + 5
Exercise 43
Sketch the graph of
h : [−2; 4] −→ R
7→
x
3x − 200
Exercise 44
Sketch the graph of
h : [2000; 2010] −→ R
x
7→
3x − 200
Exercise 45
Let x 7→ 23 x + 1 be a rule. Sketch the graph of the corresponding function such that the the
point < 2; 4 > is the centre of the drawing and the horizontal interval is 4 units. The vertical
interval must be the corresponding output interval. The axes must be drawn passing through
this point.
27
Exercise 46
If < 1; −4 > and < 2; 3 > are two points of a line, what is the slope of the line?
Give the equation of the line.
Exercise 47
Let < 3; 5 > be a point on a line of slope 21 Give the equation of a point < x; y > on the line.
(This is the equation of a line with respect to < 3; 5 >).
Exercise 48
Give the equation of the line of slope -2 through A < 100, −650 > Sketch this line, centred on
A, horizontal interval of length 0.01
Exercise 49
What kind of function is f :
R
x
Is it the same as f :
→R
n
X
7
→
ak · xk
k=0
R
→R
x
7→ a0 +
n
X
ak · xk
k=1
28
5
POLYNOMIALS
Exercise 50
Do the following expressions represent polynomials? If so, indicate the degree and coefficients.
Specify which is the leading coefficient.
(1)
2x2 − 5x + 1
√
3
(4) 1000
(5) (t + 5)2 − (t2 + 25) − (10t + 1)
3x7 + 5x − 4
x
√
(3) x2 + 1
(2)
(6)
√
3x2 +
√
5x +
√
7
Exercise 51
Evaluate the following polynomials for the given value of x:
(1) P (x) = x3 + 3x2 + 5x + 6 when x = −2
(2) Q(x) = 2x3 + 5x2 − 7 when x = 1
Exercise 52
Expand the following polynomials:
(6) (x + 1)(x2 − 1) − x(1 − x2 )
(1) (x2 + 1)(5 − 2x)
(2) (x2 + 1)(5 − 2x) + 2x3
(7)
x2
+ (x − 1)
2
(4) (4x + 1)2 − (4x − 1)2
(8)
2
(t − 1) − (2t + 6) − 4(t − 5)
3
(5) (x + 1)3 − x(x + 1)2 − 2x2
(9) x2 (x + 2)(2x − 3)
(3) (4t + 1)2 + (4t − 1)2
Exercise 53
Let A(x) = −x4 + x3 + 3x2 + x; B(x) = x4 − x2 + 2; C(x) = 2x2 + x − 3.
(1) Determine, without expanding, the coefficients of x3 and x4 in A(x) · B(x). Same question
for B(x) · C(x).
(2) Determine, without expanding, the degree of:
(a) [A(x) + B(x)]
(b) [A(x) − B(x)]
(c) C(x)[A(x) + B(x)]
(d) [A(x)]2
(e) [B(x)]3 + 3[C(x)]
(3) If n = deg[P ] and m = deg[Q], what can you say about deg[P · Q], deg[P + Q] and
deg[P − Q]?
29
Exercise 54
Can you find two polynomials A and B satisfying the following conditions? If so, specify A and
B and if not, justify why.
(1) The degree of A · B is 10 and the degree of A + B is 3.
(2) The degree of A · B is 11 and the degree of A + B is 3.
(3) The degree of A and the degree of B is 4 and the degree of A ± B is 4.
(4) The degree of A and the degree of B is 3 and the degree of A + B is zero.
6
POLYNOMIAL FUNCTIONS
Exercise 55
What is the difference between a polynomial, a polynomial equation and a polynomial function?
Exercise 56
Consider a square piece of cardboard 12cm× 12cm. Cut a small square of xcm out of each
angle. A box without top can then be made.
(1) Express the volume V of the box with respect to x. Is it a polynomial function?
(2) Compute the zeroes of V . What do these zeroes represent?
(3) Compute others images then sketch the plot of V
(4) Use the plot to show that there are 2 ways of making a box of 100cm2
(5) Estimate the value of x which makes a box of maximal volume.
Exercise 57
Same box as above.
(1) This time express the outside area A of the cardboard with respect to x. Is it a polynomial?
(2) For what values of x is there a maximal area? a minimal area?
Exercise 58
A box is made from a piece of wood 12 meters longs and 6 × 6cm
The ends of the box must be a square with a side of x meters. The total volume is 1m3 . For
which values of x is this possible?
Exercise 59
Chebyshev Polynomials
The polynomial function of Chebyshev of order 4 is f : x 7→ 8x4 − 8x2 + 1 and is used in
statistics. For what values of x does it yield f (x) > 0?
Hint: write z = x2 and solve for second order in z)
30
Exercise 60
A tank is made to stock gas. It is a cylinder with two hemispherical ends. The length of the
cylindrical part is 10m. Find the radius such that the volume is 27m3
6.1
Polynomial Equations and Roots
Exercise 61
For each of the following polynomials, determine the degree, number of distinct roots and their
multiplicity.
(1) P (x) = (2x + 4)(x + 2)(x − 1/2)(x − 1)(2x − 2)
(2) Q(x) = −3(x + 3)(x + 3)(x + 3)(x − 3)
(3) R(x) = (x − 1)2 (x − 2)2
(4) S(x) = (x2 − 2)(x − 2)2
(5) T (x) = (x2 + 2)(x2 + 4)
(6) U (x) = (x2 + 2) + (x2 − 4)
Exercise 62
Write a polynomial function (in factored form) which has zeros at x = 2 and x = 1
Exercise 63
Determine a polynomial P (x) satisfying the following conditions, where S is the solution set of
P (x) = 0, i.e, the set of roots of P :
2
(1) degree 2; S = −3;
7
(2) degree 3; S = {−2; 0; 12}
(3) degree 1000; S = {1; 2; 3; ...; 998; 999; 1000}
(4) degree 2; S = {1; 2} and P (0) = 10
(5) degree 3; S = {−3; 1; 2} and P (3) = 48
(6) degree 4 having 2 distinct roots of multiplicity 2.
(7) degree 5 having 1 distinct root. This root is 3 of multiplicity 3.
Exercise 64
True or False? Justify.
(1) Two polynomials having the same roots are equal.
(2) A polynomial which has exactly two distinct roots is of degree 2.
31
(3) A polynomial of degree ≥ 2 can always be factored in factors of degree one.
(4) A polynomial whose roots set is S = {1, 2, 3, 4, 5} cannot be of degree 4.
(5) A polynomial whose roots set is S = {1, 2, 3, 4, 5} is of degree 5.
(6) Two polynomials having the same roots and the same multiplicities are equal.
(7) A polynomial of an odd degree has at least one root.
6.2
Dividing polynomials
The division of a polynomial by another polynomial can be made by supposing that the result
exists, that its order of the resulting polynomial is no higher than the difference of the two initial
orders (why?) and writing it in general form. Then multiply this polynomial with the divisor and
work out the coefficients.
Exercise 65
Divide 5x3 − 8x2 − 6x + 4 by x − 2
Exercise 66
Divide 2x3 + 4x − 6 by 2x − 2
Exercise 67
Divide x4 − 5x3 − 2x2 + 20x − 50 by x − 5
Exercise 68
Give the quotient of the following polynomials:
(1) 3x3 + 7x2 + 3x + 7 by x2 + 1
(2) x2 − x5 by x3 − x2
(3) x2 − 9 by x + 3
(4) x4 − 1 by x − 1
Exercise 69
As with the division of integers by integers, the long division will stop when the remainder is
less than the divisor. Here, it will be when the order of the remainder is less than the divisor.
For A and B in the following, give the result of the long division (Division with remainder)
of A by B (C is the quotient and r is the remainder.)
32
(1) A = 3x4 + 2x3 − 8x2 − x + 8
(3) A = 2x3 + x + 2
B = x2 − 1
B = x2 + x − 1
(2) A = x4 − 3x3 − 11x2 + 15
(4) A = 2x3 − 3x2 − 8x + 12
B = 2x2 − 5x2 − x + 6
B =x−5
For each of the above, rewrite the rational function P (x) =
A(x)
r(x)
as C(x) +
B(x)
B(x)
Exercise 70
P (x)
Show that for any polynomial P , the value P (a) is the remainder of the division
.
(x − a)
Show that this implies that P can be factored.
Show that this implies that if P is of degree n, then P has at most n factors (counting
multiplicity).
Show that this implies that if P is of degree n, then P has at most n roots (counting
multiplicity)
Exercise 71
Factorising a polynomial of higher orders: Find a root by trial and error guessing. When you
have found a root x0 , divide the polynomial by x − x0 . Do the same thing with the quotient.
When the quotient is down to order 2, find the roots in the usual way (maybe there are none).
(If you cannot guess an initial root, there is nothing that can be done...)
Factorise completely the following polynomials:
(1) 2x3 − 3x2 − 8x + 12
(3) 2x5 + 3x4 − 3x3 − 2x2
(2) x4 − 9x2 + 4x + 12
(4) x4 + x3 − x − 1
Exercise 72
Solve the following equations.
x3
4
+ 2x = x2 +
6
3
(1) 35t2 + 7t = 0
(7) x3 − 1 = 0
(2) 121x2 − 25 = 0
(8) x3 + 8 = 0
(3) x2 − 4x − 21 = 0
(9) 3x3 = 12x2 + 63x
(4) 2x2 + 2x − 40 = 0
(10) t2 (t − 14) + 13t = 0
(5) 2y 2 + 11y + 9 = 0
(11) 2y 3 + 11y 2 + 9y = 0
(16) x4 + 4x + 12 = 9x2
(6) x3 + 1 = 0
(12) x3 + 3x2 + 3x + 1 = 0
(17) x4 − x2 − 6 = 0
33
(13)
(14) x4 − 1 = 0
(15) x4 + 1 = 0
6.3
Polynomial Inequalities
Exercise 73
Fill in the dots with < or >.
(1) a < b
⇔
a + 1......b + 1
(2) a < b
⇔
a − 1......b − 1
(3) a < b
⇔
(−a) . . . . . . (−b)
(4) a < b
⇔
(5) a < b
⇔
1
1
......
a
b
1
1
−
...... −
a
b
Exercise 74
Draw a sign table for the polynomial for each of the following polynomials (Series 4, exercise 4
should help with factoring)
(1) x2 − 4x − 21
(5) x3 + 3x2 + 3x + 1
(2) 2y 2 + 11y + 9
(3)
−2u2
(6) x4 − 1
+ 12u + 14
(4) 2y 3 + 11y 2 + 9y
(7) x4 + 4x + 12 − 9x2
Exercise 75
Solve the following inequalities.
(1) x2 − 4x − 21 < 0
(5) x3 + 3x2 + 3x + 1 > 0
(2) 2y 2 + 11y + 9 ≥ 0
(3)
−2u2
(6) x4 − 1 < 0
+ 12u + 14 < 0
(4) 2y 3 + 11y 2 + 9y ≤ 0
(7) x4 + 4x + 12 − 9x2 ≥ 0
Exercise 76
Solve the following inequalities and write the solution set using interval notation:
x x
x 1
+ ≥ +
2
3
4 2
x−4
7x
<
− (3x − 2)
(2)
2
2
(5) 16x2 ≥ 9x
(1)
(3)
(x2
− 1) − (x +
1)2
(6) x3 > x
(7) −5 ≤ 1 − 3x ≤ 10
≤0
(4) (x − 7)(x − 3) + 1 ≥ (x −
(8) −6 < 2x − 4 < 2
2)2
(9) −x + 2 < 2x − 1 < −2x + 1
34
Exercise 77
A certain experiment requires that the temperature be between 30 and 40 degree Celsius. If
Farenheit and Celsius degrees are related by the formula
5
C = (F − 32)
9
what are the permissible temperatures in Farenheit?
Exercise 78
A construction company needs to purchase a new crane and is hesitating between two models.
Model A costs 50’000CHF and has a yearly maintenance fee of 4’000 CHF. The model B costs
40’000 CHF and has a yearly maintenance fee of 5’500 CHF. For how many years will the
company have to use model A before its yearly cost is cheaper than model B?
7
RATIONAL EXPRESSIONS
Exercise 79
Simplify the expressions and specify the domain:
4
2
+ 2
− 3x + 2 x − 1
(1)
2
x2 − 4
·
x + 2 4x − 2x2
(8)
(2)
2y + 6y y 2 − 2y
·
y−2
8
(9)
(3)
a2 a2 − 25 4a2 + a
·
16a3 − a a2 + 5
(10)
3
8z
2
+
−
2z − 1 4z 2 − 1 2z + 1
(4)
25x3 − x x − 1
10x
·
·
5x2
5x − 1 x2 − 1
(11)
a2 − 4 a2 − 6
−
a2 + 6 a2 + 4
(5)
a2 − a a + 2
÷
(4a)2
2a
(12)
2 − 3x 2 + 3x
−
2 + 3x 2 − 3x
(6)
9x2 − 1
3x + 1
÷ 3
2
x −x
x − x4
(13)
z+3 z+1
−
z+2
z
(7)
a2 + a − 2
a2 + 7a + 10
÷
a2 + 2a − 15 a2 + 10a + 25
(14)
x2
4x
2x
+
− 1 2x + 1
4x2
t2
t
1
−
− 25 2t + 10
Exercise 80
Simplify the expressions and specify the domain:
(1)
−1
1
1−x
(2)
1
1
1−x
(3)
1
−1
1−x
1
1−
x
35
x
−1
1
(4) x − x
x−1
−1
(5)
x−1
1−
x
1
x
x−1
(6)
(7)
1
x
1+x
1−x
(10)
1+x
1−
1−x
1+
1
−1
x
1
x
(11)
1
1+
x
1−
(8)
1
x−1
x
(9) 1 −
x−1
x
Exercise 81
Solve the following equations.
(1)
3
13 + 2x
=
4x + 1
4
(2)
3
9
=
7x − 2
3x + 1
(3) 2 −
5
=2
3x − 7
(4)
2
4
7
+
=
5 10x + 5
2x + 1
(5)
2x + 3 1 − 2x
4
2x + 3
+
=
+ 2
x
x+1
x+1 x +x
(6)
1
x2 − 6
3x + 2
−
= 2
+2
x
x−3
x − 3x
(7)
x2
x−1
x+1
(12)
x−1
1−
x+1
1+
1
5
4
+ 2
= 2
−x x +x
x −1
36
8
ASYMPTOTES
Consider the curve of f : x 7→
x2
x+2
y
x
-2
Exercise 82
For values of x close to −2 the function has large positive or negative values. Why?
For large positive or negative values, the function is close to the oblique dashed straight
line. Why?
Exercise 83
x−7
Let f : x 7→
.
x+7
(1) What is the domain of the function f ?
(2) Determine the y-intercept and the x-intercept.
(3) Draw a sign table for the function f .
(4) What happens to f (x) when x becomes very close to -7?
(5) What happens to f (x) when x becomes very large (in absolute value)?
(6) Sketch a graph of function f .
37
Exercise 84
For each of the following functions:
(1) Determine its domain.
(4) Determine the equations and the nature
of all the asymptotes, if any.
(2) Determine the y and x-intercepts.
(3) Draw its sign table.
• f1 : x 7→
3x + 5
4−x
• f2 : x 7→
x+4
(x + 1)(x − 3)
• f3 : x 7→
2x + 1
x−2
(5) Draw its graph.
(x + 1)(x − 2)
3x + 1
x−1
• f5 : x 7→ 2
x −4
• f4 : x 7→
• f6 : x 7→
x2 − 6x + 9
x−3
38
• f7 : x 7→
x−2
(x − 3)(x − 5)
• f8 : x 7→
x−2
(x − 3)(5 − x)
9
COMPOSITE FUNCTIONS
f (g(x)) or f ◦ g(x) (f circle g) is a composite function.
First g then f on the result. If f and g have domain R, we have:
g
f
R −→ R −→ R
Exercise 85
g : x 7→ x − 1 and f : x 7→ x2 . Write f ◦ g(x) and g ◦ f (x)
Exercise 86
Let f : x 7→ x2
g : x 7→
1
x+1
h : x 7→
√
x
(1) Find the domain of each function.
(2) Give the expanded form of f ◦ g, f ◦ h, g ◦ f , g ◦ h, h ◦ g, f ◦ f and give their domains.
(3) Find F, G, H such that
(a) f ◦ F (x) = x
[= F ◦ f (x)]
(b) g ◦ G(x) = x
[= G ◦ g(x)]
(c) h ◦ H(x) = x
[= H ◦ h(x)]
with their respective domains.
Exercise 87
Write the following as composites of elementary functions:
√
(4) j : x →
7 sin2 (x + 1)
(1) f : x 7→ 1 − x2
(2) g : x 7→ (x + 5)3 + 1
(3) h : x 7→
9.1
x3
(5) k : x 7→ (5 +
2
−1
(6) l : x 7→
√
p
3
(x + 2)2 )2 + 18
x+
√
4
x+1
Recursive Composite Functions
It is sometimes possible to define a function by using the function itself in the definition. This
will not lead to a contradiction or impossibility to solve if, after finitely many steps, a defined
value of the function is reached and if the definition of the function enables to calculate back
to the initial value. An object which is described by using its own description is recursive.
For example, when a video camera films the screen on which the camera output appears, its
generates a recursive picture: a picture of the screen on which there is a picture of the screen
on which there is... etc.
39
Exercise 88
The Fibonacci Sequence:
f : N −→ N
∀n ≥ 2
f :1
7→
1
f :2
7→
1
f :n
7→
f (n − 1) + f (n − 2)
Calculate f (10)
Exercise 89
Using the Fibonacci sequence:
g : N −→ N
g:n
7→
f (n)
f (n − 1)
Calculate g(10)
Exercise 90
More tricky:
h : N −→ R
(
x − 10
h : x 7→
h(h(x + 11))
if x > 100
if x ≤ 100
Give fullest possible description of the values taken by the function.
By stating that the output set is R we are sure that this set is big enough. What is the
smallest possible output set; i.e: the set of all outputs: the range of h.
9.2
Absolute value and composite functions
For the next 4 exercises, f (x) = |x|, g(x) = x − 1, h(x) =
Exercise 91
(1) Find the domain of f and k.
(2) Determine the y and x-intercepts of f and k.
(3) Draw their sign tables.
(4) Find all the asymptotes of f and k, if any.
(5) Sketch the graphs of f and k.
(6) Same questions for f ◦ k and k ◦ f .
40
x2 + 2x + 1
1
and k(x) =
x
x+1
Exercise 92
(1) Determine f ◦ g and g ◦ f , as well as their domains.
(2) Draw sign tables for f ◦ g and g ◦ f
(3) Draw sketches of graphs for f ◦ g and g ◦ f
Exercise 93
(1) Determine h ◦ f well as its domain.
(2) Draw the sign table for h ◦ f
(3) Find the asymptotes of h ◦ f .
(4) Draw a sketch of the graph of h ◦ f
Exercise 94
(1) Determine h ◦ g ◦ f as well as its domain.
(2) Draw the sign table for h ◦ g ◦ f
(3) Find the asymptotes of h ◦ g ◦ f .
(4) Draw a sketch of the graph of h ◦ g ◦ f .
9.3
Shifting
Exercise 95
Given the function f and its graph. Draw approximately, the graph of
(1) f (x) + 1
(4) 2f (x)
(2) f (x + 1)
(5) f (−x)
(3) f (2x)
(6) −f (x)
f
f
1
1
1
1
41
f
f
1
1
1
1
f
f
1
1
1
1
42
10
INVERSE FUNCTIONS
Exercise 96
Let f : x 7→
3x + 2
.
2x − 5
(1) Determine the domain, intercepts, asymptotes, draw the sign table and sketch the graph
of f . Specify its range.
(2) Determine f −1 (x) and its domain.
(3) Show that (f ◦ f −1 )(x) = (f −1 ◦ f )(x) = x
Exercise 97
Let f be defined as follows:
f:
[0; +∞[ −→ B
x 7→ −x2 + 3
Specify the largest possible set B so that f is one-to-one and determine f −1 .
Sketch the graph of f and f −1 on the same yx−plane.
Exercise 98
For each of the following functions, find the largest possible restriction of its domain where it is
one-to-one, therefore invertible, and find its inverse for that restricted input set. The restricted
function and its inverse will have their input and output sets (restricted domain and range)
defined explicitly.
(1) a : x 7→ 2x + 5
(2) b : x 7→
5x + 1
2x − 3
(3) c : x 7→ 6x − 4
(6) f : x 7→
1
x−2
(7) g : x 7→
3−x
2x + 1
(4) d : x 7→ x4
(8) h : x 7→ x2 − 5x + 6
(5) e : x 7→ x5
(9) j : x 7→ 10x
43
Exercise 99
Here is the graph of a one-to-one function.
(1) Sketch the graph of its inverse.
(2) Give the domain and range of f .
(3) Give the domain and range of f −1 .
44
11
Exponential and logarithm
Exercise 100
Consider a population of bacteria which doubles every hour. At t = 0 (initial time), the population
contains 1000 bacteria.
(1) Give the formula expressing the population P as a function of time t.
(2) Sketch the graph this function from[−5; 5].
(3) After how many hours will the population be multiplied by 4, 8, and 16?
(4) After how many hours will the population be multiplied by 3? (give your best approximation).
(5) Write the equation corresponding to the above question. Can you solve it? If so, do it. If
not, why?
(6) After how many hours will the population be multiplied by 5? (give your best approximation).
(7) Write the equation corresponding to the above question. Can you solve it? If so, do it. If
not, why?
(8) At what time did the population contain 500 bacteria? 250? 150?
(9) Write the equation corresponding to the above question (number of bacteria = 150).
(10) Looking at the graph, specify the domain and the range, draw a sign table and specify the
y- and x-intercepts.
(11) Is this function one-to-one? If so, sketch its inverse.
(12) Looking at the graph of its inverse, specify the domain and the range, draw a sign table
and specify the y- and x-intercepts.
The inverse function of the exponential function ax = y is the logarithm loga (y) = x. It
reads: logarithm in base a of y.
ax = y
⇔
loga (y) = x
Bases for logarithms are the base 10, written log (with no subscript) and base 2, written Log
Exercise 101
Compute the following.
(1) log2 (8)
(3) log3 (81)
(5) log2 (512)
1
(6) log3
9
(4) log5 (125)
(7) log10 (1)
(2) log(100)
45
(8) log10 (105 )
(9) log10 (10n )
Exercise 102
Using the properties of exponents
ax · ay = ax+y
ax
= ax−y
ay
(ax )c = acx
prove the three laws of logarithms:
Theorem 7 (Logarithms)
For any positive a, x and y:
(1) loga (x) + loga (y) = loga (x · y)
x
(2) loga (x) − loga (y) = loga
y
(3) n · loga (x) = loga (xn )
Exercise 103
Solve the following equation
5x = 17
Exercise 104
Evaluate the following expressions:
(1) log5 (25) − log5 (125) =
(3) log
105
·
√
3
10 + 3 log
(2) log8 (4) + log8 (16) =
Exercise 105
Let f : x 7→ log(x − 3) be a function.
(1) Specify its domain and range
(2) Find f −1 and specify its domain and range.
(3) Verify that (f ◦ f −1 )(x) = (f −1 ◦ f )(x) = x
(4) Sketch f and f −1 on the same plane.
Exercise 106
Let f : x 7→ 10x , g : x 7→ log(x), and h : x 7→ x2 − 1 be three functions.
Compute: (f ◦ g ◦ h)(x)
46
1
102
=
Exercise 107
Find the base of an exponential function passing through the point (3; 343)
Exercise 108
Find the exact solutions of the following equations
1
8
(1) 1000x = 1018
(3) 2x =
(2) 2x = 10243
(4) 81x = 3
(5) 125x = 25
(6) 27x =
1
9
How would you compute the solution of 4x = 17 using your calculator?
Exercise 109
Without using a calculator, compute
1
(1) log2
16
(4) log15 (12) + log15 (35) − log15 (28)
(2) log(log(10))
(5) log5 (log7 (7))
(3) log8 (16)
(6) logπ (1)
Exercise 110
(1) Show that loga (b) · logb (a) = 1
(2) Show that
loga (x)
= 1 + loga (n)
logna (x)
(3) Solve the following system:
(
log(x) + log(y) = 2
x + y = 25
Exercise 111
Simplify the following expressions using the properties of the logarithm.
(1) − log(1/x)
(3) logx ((xy )z )
(2) log√x (x2 )
(4) logx (2y) + logx (1/y)
Exercise 112
On the same graph paper, plot the graphs of
47
(5) logx (xy /xz )
√ x
(6) logx
x
• id : x 7→ x
• f −1 : x 7→
√
x
• f : x 7→ x2
• g −1 : x 7→ log2 (x)
• g : x 7→ 2x
Suggested scale: include values for x, y ∈ [−8; +8]. Compute values of y for all integer
values of x, as well as for a few more values between -1 and +1.
Exercise 113
Solve the following equations
(1) logx
1
16
(10) logx (256) = 4
=2
(2) log6 (x) = 2
(11) logx (125) = 3
(3) logx (81) = 4
(12) logx (1000) = 3
1
(13) logx 16
= −2
(4) log4 (x) = 0
(5) log5 (2x − 1) = 2
(14) log(x + 3) + log(3) = log(12)
(6) log(3 − x) = 2
(15) log(x) − log(4) + log(5) = log(25)
(7) log4 (3x + 1) = 5
(8) log2 (x) = 4
(16) log(x + 7) + log(3) = log(6)
(9) log4 (x) = 3
(17) 2 log(x − 2) = log(3x − 8)
Exercise 114
Compare both expressions and decide which one is greater than the other
√
(1) 3
√
(2) 4
3
and 9
2
· 4 and 16
√
10 5
(3)
and 1
1000
Exercise 115
Solve the following equations:
(1) 5x = 53
(8) 0, 1x = 100x+4
(2) 3x+2 = 3
(9) 9 · 33x+2 = 271−3x
(3) 7x+5 − 73 = 0
(10) 9x = 3 x
(4) 52x+4 = 25x
(11) 3 · 2x+1,5 = 6 · 4x−1
2
(5) 0, 0014x−2 = 0, 1x+3
1
3
(6) (53x−1 ) = (5x−2 )
1
(12) 5 · 72x+4 = 35 · 7x−5
1
2
(13)
1
(7) (53x−1 ) 3 · (5x−2 ) 2 = 0
48
8x − 1
= 150 000
0, 03
Exercise 116
The population in a town doubles every ten years. There were 1’000’000 people in 1990.
(1) Express the population as a fonction of time.
(2) What was the population in 1985?
(3) What will the population be in 2015?
(4) In how many years will the population be multiplied by 3?
√
√
(Use 2 ≈ 1.4, 3 ≈ 1.7 and sqrt5 ≈ 2.2)
Exercise 117
The Richter Scale is a system used to measure the magnitude of an earthquake. It enables one
to compare the energies which are released at the epicentre. The magnitude M is related to
the energy E by the following empirical formula:
log(E) = 4, 4 + 1, 5 × M
giving the energy E in ergs from the magnitude M. Note that E is not the total “intrinsic” energy
of the earthquake, transferred from sources such as gravitational energy or to sinks such as heat
energy. It is only the amount radiated from the earthquake as seismic waves, which ought to be
a small fraction of the total energy transfered during the earthquake process. Charles Richter
established this measure in California in 1935.
An earthquake of magnitude 3 is hardly felt and on a small range.
Small damages occur with a magnitude of 4.5.
Magnitude of the San Francisco Earthquake in 1906 was 7.8.
Magnitude of the Mexico Earthquake in 1985 was 8.1.
Magnitude of the Lisbon Earthquake in 1963 was 9 (This was the worst earthquake ever
registered.)
(1) Find the energy of an earthquake of magnitude 2.4.
(2) Find the magnitude of an earthquake which releases 1010 Joules?
1 Joules is equal to the work done when the point of application of a force of 1 Newton
moves through a distance of 1 meter. (Or approximately, the energy needed to move an
object of 100g by 1m, on earth).
(3) The formula shows that an earthquake of magnitude 8,4 releases one billion times more
energy than an earthquake of magnitude 2,4. Show why.
(4) Suppose that one earthquake releases one million times more energy than another. Find
their difference in magnitude on the Richter scale.
(5) Find the energy released with a magnitude of 1.
(6) Find the energy released with a magnitude of 0.
(7) Is it possible to give a magnitude when there is no earthquake, i.e: when there is no
energy released? Explain your answer.
49
The Richter scale is an open scale because it has no theoretical upper bound.
Summary:
For any positive real number b (b 6= 1), then
y = bx
⇐⇒
logb (y) = x
log10 (x) = log (x)
y = log (x)
⇐⇒
10y = x
Let u, w, and b be positive real numbers with b not equal to 1. Then:
logb (u · w) = logb (u) + logb (w)
logb
u
w
= logb (u) − logb (w)
logb (uc ) = c · logb (u),
loga (x) =
50
logb (x)
logb (a)
c∈R
12
Trigonometry
Angles
An angle is defined by two lines meeting at a point called the vertex of the angle. It can be
regarded as the measure of the rotation involved in moving from one line to coincide with the
other line. It has a direction represented by the sign.
The measure of the rotation is performed by drawing a circle centred at the vertex and
observing what amount of the circle is covered by the rotation.
Up till now, the measure for angles was done in degrees. One degree (1◦ ) is the angle
1
defined by 360
th of a circle. i.e: one complete turn is 360◦
!
4
This measure does not depend on the size of the circle.
Exercise 118
Another measure is based on an arc of a circle with centre at the vertex of the angle. If r is the
radius of the circle and l the length of arc subtending the angle, then the angle is rl i.e: it is the
proportion of a complete turn measured by the length along the circle compared to its radius.
This is called the radian measure.
Considering a circle of radius r:
(1) What is the measure of a complete turn (a round angle)?
(2) What is the unit?
(3) What is the measure of a right angle (a quarter turn)?
(4) What is the measure of a flat angle?
(5) Why is it that a radius equal to 1 will be used to simplify things?
Exercise 119
Find the formula for transforming a degree measure to a radian measure.
Exercise 120
Transform the following into radian measure:
(1) 60◦
(6) 120◦
(2) 22.5◦
(3)
(7) 10◦
30◦
(8) 1◦
(4) 15◦
(5) 7.5◦
(9) 452◦
Exercise 121
Find the formula for transforming a radian measure to a degree measure.
51
Exercise 122
Transform the following into degree measure:
(1) 1
(2)
(5)
π
3
7π
4
(6) 15π
(7) 16π
(3) 0.1
(8) 3
2π
(4)
3
(9) 1.5
Exercise 123
The minute hand of a clock measures 12cm. What is the distance covered by the tip of the hand
in 20 min? What is the total distance covered in one day?
(use π ≈ 3)
Exercise 124
Given that the circumference of the earth is (theoretically) 40 000km, what is the radius of the
earth?
Exercise 125
One nautical mile is the length of arc of 1 minute at the surface of the earth.
What is one minute of arc in radian measure?
What is the length of one nautical mile? 1
Exercise 126
If a bridge were built at a constant height of 10m all around the earth’s equator – with its
theoretical measure of 40 000km, how much longer than the equator would the bridge be?
Exercise 127
A circular sector is the part of a disc lying between two radii.
Find the area of a sector of angle θ (in radian measure).
Exercise 128
A circle has a radius of 2.5m. Find the area of a sector of angle
3π
4
Exercise 129
In a unit circle, what is the angle of a sector of area 1? (radian measure).
1
Because the earth is not a perfect sphere, the official length of the nautical mile is now slightly different and
does not depend on which part of the earth you are.
52
Angles are a measure of rotation. The positive direction is anticlockwise!
Because radian measures have no units and as in mathematics units are considered an
unnecessary nuisance:
!
4
From now on till the end of college, in mathematics,
only radian measure will be used.
Cicular functions
In a right-angled triangle the following ratios have been defined:
opposite
sin(α) =
hypotenuse
hypotenuse
adjacent
cos(α) =
opposite
hypotenuse
α
opposite
adjacent
tan(α) =
adjacent
We now redefine these ratios in the Trigonometric Circle which has a radius equal to 1
centred on the origin and angle direction is anti-clockwise.
The original definitions work only for positive values (lengths are always positive), i.e: only
for the first quarter circle. The definition is now extended to the whole circle: the sine is the
vertical coordinate of a point, the cosine is its horizontal coordinate, the position of the point
itself depends on the arc-length. If one considers the ration between arc-length and complete
turn, the the position of the point depends on the angle measures on the outside circle of radius
1.
y
1
θ
sin(θ)
cos(θ)
1 x
As the radius is 1, we then have
sin(α) =
opposite
= opposite
1
adjacent
= adjacent
1
opposite
sin(α)
tan(α) =
=
adjacent
cos(α)
cos(α) =
53
If you unroll the circle, the circular functions appear as in the following
The Sine curve
y
θ
-1·π
-1
0·π
1·π
2·π
3·π
The Cosine curve
x
θ
-1·π
-1
0·π
1·π
2·π
3·π
The Tangent curve
y
θ
-1·π
-1
0·π
1·π
2·π
3·π
Exercise 130
Use the definition of the trigonometric ratios to prove the following theorem (explain why it is
true).
Theorem 8 (Relation Between Sine and Cosine)
for any θ
sin2 (θ) + cos2 (θ) = 1
Exercise 131
Draw two axes, x and y at right angles. Centred on the crossing point, draw a circle with a
radius of 8cm. (Consider 8cm to be one unit)
On that circle, draw with best possible approximation, an angle of π/5.
Where are sin(π/5), cos(π/5) and tan(π/5).
Give their approximate values.
54
Exercise 132
What are the periods of the trigonometric functions?
Exercise 133
Are the trigonometric functions defined on all the interval of a period?
Exercise 134
Considering the graph of the functions (series 16) and the definitions, find the relations between
(1) sin(x) and sin(−x)
(2) cos(x) and cos(−x)
(3) tan(x) and tan(−x)
(4) x and y if sin(x) = cos(y)
Periodicity
A function is periodic if its curve repeats exactly the same pattern over and over again.
Definition 2 (Periodic Function)
A function f (x) is said to be periodic if there is a real number α such that
∀x f (x + α) = f (x)
Exercise 135
Show that for any periodic function, if ∃α
f (x) = f (x + α) then ∀n ∈ N f (x) = f (x + n · α)
Definition 3 (Period of a function)
The fundamental period or simply the period of a function, is the least possible t such
that ∀x f (x + t) = f (x)
Exercise 136
Determine the domain, the zeroes and the period of:
(1) sin(2x)
(5) sin(−x)
(2) cos(π − 2x)
(6) cos
(3) sin(x + π/2)
(7) tan
(4) cos(x − π)
Exercise 137
Solve the following equations:
55
x
3
x
π
(1) cos(x) = −
(2) sin(x) =
√
1
2
(5) cos(x) = −
2
2
√
1
2
(6) sin(x) =
(3) cos(x) = −1
√
2
(4) sin(x) =
2
3
2
(7) sin(x) = cos(x)
(8) sin(x) = tan(x)
Exercise 138
1
(1) cos(2x + π) =
2
√
x π
3
(2) sin( − ) =
2
3
2
(3) tan( πx + π) = 1
Exercise 139
Solve using the table:
(1) cos(x) = 0.4
(2) sin(x) = −0.3
Definition 4 (Arc-functions)
The inverse function of the trigonometric function are
• arcsine arcsin(x)
the arc (angle) whose sine is x
• arccosine arccos(x)
the arc (angle) whose cosine is x
• arctangent arctan(x)
the arc (angle) whose tangent is x
As sin(x) ∈ [−1; 1] (same for cos(x)) the input for arcsin(x) must be restricted to [−1; 1]
The input for arctan(x) does not need any restriction.
Exercise 140
Give the domain and range of the arcsine function. Same quesdtion for arccosine and arctangent.
Exercise 141
Use the table to calculate
(1) arcsin(x) = .8
(2) arccos(2)
56
(3) arctan(4)
Exercise 142
Observe the following drawing where the angle β has been drawn on top of the angle α.
(1) Explain why the angle right at the top is equal to α
(2) Express the lengths of a, b and c in terms of sin(α), cos(α), sin(β) and cos(β).
y
α
b
β
c
α
a
0
x
1
Exercise 143
Finish the proof of
Theorem 9 (α + β)
sin(α + β) = sin(α) cos(β) + cos(α) sin(β)
cos(α + β) = cos(α) cos(β) − sin(α) sin(β)
Exercise 144
Prove also the following theorem
Theorem 10 (tan(α + β))
tan(α + β) =
tan(α) + tan(β)
1 − tan(α) tan(β)
57
Measuring the Earth
Exercise 145
The Greeks assumed that the earth was spherical and even measured its diameter over 2500
years ago. This exercise basically reproduces this method – only we will use modern units:
metres, kilometres, radians.
In two cities 500km apart on a North/South axis, somewhere in Egypt along the Nile, a 1m
pole is planted vertically. On a given day, in each city, a person measures the length of the
shade every half hour or so. The person notes the shortest length on a piece of paper. [The
measure of the shortest shade for two cities on a North/South axis ensures that the shades are
measured at the same time.]
In city A (to the North) the shortest shade of the 1m pole measures 0.932m
In city B (to the South) the shortest shade measures 1.089m
The sun rays are assumed to be parallel.
What is the diameter of the Earth?
(Sketch the situation)
Law of Sines
Solving a triangle consists of finding each part of the triangle (all the angles and sides) given
only three pieces of information.
Exercise 146
Consider the sketch of following triangle. give:
(1) its area;
(2) the other two angles
(3) the length of the segment AC
C
h
A
3cm
π/6
5cm
B
Exercise 147
Consider a triangle with angles α, β and γ. The opposite sides are, in the same order, a, b and c.
(1) Define the area of the triangle in terms of:
58
(a) sin(α)
(b) sin(β)
(2) Show then that
sin(α)
sin(β)
sin(γ)
=
=
a
b
c
These last equalities are known as the Law of Sines.
Exercise 148
(find exact values)
π
2π
, and β =
, what is the value of γ?
5
3
π
β = , a = 5 and b = 3, what is the value of c?
2
π
π
α = , β = and c = 4, what is the value of b?
3
6
π
α = β = and a = 5, what is the value of c?
4
π
π
α = , β = and a = 10, what is the value of b?
4
6
π
a = 3, b = 2 and γ = , what is the value of c?
6
(1) If α =
(2) If
(3) If
(4) If
(5) If
(6) If
Exercise 149
(find approximate values)
π
π
and β = , what is the value of b?
5
6
π
(2) If a = 3, b = 2 and γ = , what is the value of c?
5
π
(3) If a = 4, c = 2 and β = , what is the value of b?
6
(1) If c = 4, γ =
(4) If a = 4, b = 5 and c = 6, what is the value of cos(γ)?
(5) If a = 4, b = 5 and α =
π
, what is the value of sin(β)?
3
Exercise 150
Show that
sin(α) − sin(β)
a−b
=
sin(α) + sin(β)
a+b
59
(c) sin(γ).