MTH4100 Calculus I · 2015. 7. 28. · Calculus provides powerful techniques for solving problems...

MTH4100 Calculus I

Bill JacksonSchool of Mathematical Sciences QMUL

Semester 1, 2014

Bill Jackson Calculus I

What is Calculus?

Calculus is the branch of mathematics which uses limits,derivatives and integrals to ‘measure change’. It is based on thereal numbers and the study of functions of real variables:

for one variable see Calculus I

for several variables see Calculus II


What is Calculus?

Calculus is the branch of mathematics which uses limits,derivatives and integrals to ‘measure change’. It is based on thereal numbers and the study of functions of real variables:

for one variable see Calculus I

for several variables see Calculus II

Calculus provides powerful techniques for solving problems whichhave widespread applications throughout science, economics, andengineering. It has been formalised and extended into theimportant branch of mathematics known as analysis.


Real numbers and the real line

We can think of the real numbers as the set of all infinite decimals.We denote this set by R.




examples: 2 = 2.000 . . . −3

4= −0.7500 . . . 1

3= 0.333 . . .√

2 = 1.4142 . . . π = 3.1415 . . .




examples: 2 = 2.000 . . . −3

4= −0.7500 . . . 1

3= 0.333 . . .√

2 = 1.4142 . . . π = 3.1415 . . .

The real numbers can be represented as points on the real line.

-3 -2 -1 0 1 2 3 4-3/4 1/3 π2


Properties of the real numbers

The real numbers have three types of fundamental properties:




algebraic: the rules of calculation (addition, subtraction,multiplication, division).Example: 2(3 + 5) = 2 · 3 + 2 · 5 = 6 + 10 = 16





order: inequalities relating any two real numbers (for a geometricpicture imagine the order in which points occur on the real line).Example: −3

4< 1

3,

√2 ≤ π





order: inequalities relating any two real numbers (for a geometricpicture imagine the order in which points occur on the real line).Example: −3

4< 1

3,

√2 ≤ π

completeness: “there are no gaps on the real line”


Algebraic properties - Addition

The first five algebraic properties involve addition:




(A0) For all a, b ∈ R we have a + b ∈ R. closure





(A1) For all a, b, c ∈ R we have a + (b + c) = (a + b) + c .associativity






(A2) For all a, b ∈ R we have a + b = b + a. commutativity







(A3) There is an element 0 ∈ R such that a + 0 = a for all a ∈ R.identity







(A3) There is an element 0 ∈ R such that a + 0 = a for all a ∈ R.identity

(A4) For all a ∈ R there is an element −a ∈ R such thata + (−a) = 0. inverse


Algebraic properties - Multiplication

There are five analogous algebraic properties for multiplication:




(M0) For all a, b ∈ R we have ab ∈ R. closure





(M1) For all a, b, c ∈ R we have a(bc) = (ab)c . associativity






(M2) For all a, b ∈ R we have ab = ba. commutativity







(M3) There is an element 1 ∈ R such that a 1 = a for all a ∈ R.identity







(M3) There is an element 1 ∈ R such that a 1 = a for all a ∈ R.identity

(M4) For all a ∈ R with a 6= 0, there is an element a−1 ∈ R suchthat a a−1 = 1. inverse


Algebraic properties - Distributivity

One last algebraic properties links addition and multiplication:




(D) For all a, b, c ∈ R we have a(b + c) = ab + ac . distributivity





Properties A0-A5, M0-M5, and D define an algebraic structurecalled a field.





Properties A0-A5, M0-M5, and D define an algebraic structurecalled a field.The real numbers R are an example of a field. Another example isthe field of rational numbers:

Q = {m/n : m, n ∈ Z and n 6= 0}.





Properties A0-A5, M0-M5, and D define an algebraic structurecalled a field.The real numbers R are an example of a field. Another example isthe field of rational numbers:

Q = {m/n : m, n ∈ Z and n 6= 0}.

The integers Z are NOT an example of a field because ...


Order properties

For all a, b, c ∈ R we have:


Order properties


(O1) either a ≤ b or b ≤ a totality of ordering I


Order properties



(O2) if a ≤ b and b ≤ a then a = b totality of ordering II


Order properties




(O3) if a ≤ b and b ≤ c then a ≤ c transitivity


Order properties





(O4) if a ≤ b then a + c ≤ b + c order under addition


Order properties






(O5) if a ≤ b and 0 ≤ c then a c ≤ b c order under multiplication


Order properties






(O5) if a ≤ b and 0 ≤ c then a c ≤ b c order under multiplication

A field satisfying (O1)-(O5) is called an ordered field. The realnumbers R and the rational numbers Q are both examples ofordered fields.


Other rules for working with inequalities

The order properties (O1)-(O5) have many consequences:




Lemma

For all a, b, c ∈ R we have:(a) if a ≥ 0 then −a ≤ 0;(b) if a ≤ b and c ≤ 0 then a c ≥ b c;(c) a2 ≥ 0;(d) if a > 0 then 1/a > 0.




Lemma

For all a, b, c ∈ R we have:(a) if a ≥ 0 then −a ≤ 0;(b) if a ≤ b and c ≤ 0 then a c ≥ b c;(c) a2 ≥ 0;(d) if a > 0 then 1/a > 0.

We can prove that these rules are valid by using properties(O1)-(O5).


The completeness property

Intuitively this means “there are no gaps in the real numbers”.More precisely it says:




If a set of real numbers S has an upper bound i.e. there exists anumber c ∈ R such that x ≤ c for all x ∈ S , then S has a leastupper bound i.e. there exists a number c0 ∈ R such that c0 is anupper bound for S , and c ≥ c0 for all other upper bounds c of S .




If a set of real numbers S has an upper bound i.e. there exists anumber c ∈ R such that x ≤ c for all x ∈ S , then S has a leastupper bound i.e. there exists a number c0 ∈ R such that c0 is anupper bound for S , and c ≥ c0 for all other upper bounds c of S .

The rational numbers Q do NOT have the completeness property.If we let S = {q ∈ Q : q2 ≤ 2} then S does not have a leastupper bound in Q because ....


Intervals

Definition An interval is a subset I of R of one of the followingtwo types:


Intervals


(a) all real numbers which lie between two given real numbers;


Intervals



(b) all real numbers which are either above or below a given realnumber.


Intervals




Type (a) intervals are said to be bounded (or finite). Type (b)intervals are said to be unbounded (or infinite).


Intervals




Type (a) intervals are said to be bounded (or finite). Type (b)intervals are said to be unbounded (or infinite).

The completeness property tells us that an interval which isbounded above has a least upper bound. Similarly an intervalwhich is bounded below has a greatest lower bound. We refer tothese values as end-points of the interval.


Examples

I = {x ∈ R : 3 < x ≤ 6} defines a bounded interval.Geometrically, it corresponds to a line segment on the realline. It has two end-points 3 and 6. We can describe it usingthe notation I = (3, 6], where the round bracket on the lefttells us that 3 6∈ I and the square bracket on the right tells usthat 6 ∈ I .


Examples

I = {x ∈ R : 3 < x ≤ 6} defines a bounded interval.Geometrically, it corresponds to a line segment on the realline. It has two end-points 3 and 6. We can describe it usingthe notation I = (3, 6], where the round bracket on the lefttells us that 3 6∈ I and the square bracket on the right tells usthat 6 ∈ I .

I = {x ∈ R : x > −2} defines an unbounded interval.Geometrically, it corresponds to a ray i.e. a line which extendsto infinity in one direction. It has one end-point −2. We candescribe it using the notation I = (−2,∞).


Open and Closed intervals

We can distinguish between intervals which are bounded orunbounded. We can also distinguish between intervals byconsidering whether or not they contain their end points: intervalswhich contain all their end-points are closed; intervals whichcontain none of their end-points are open; intervals which have twoend points and contain exactly one of them are half-open (orhalf-closed).

Examples...


Solving inequalities

We can represent the set of all solutions to one or more inequalitiesas an interval or, more generally, as a collection of disjoint intervals.

Examples: Find the set of all solutions to the followinginequalities.





2x − 1 < x + 3.





2x − 1 < x + 3.6

x−1≥ 5.





2x − 1 < x + 3.6

x−1≥ 5.

x2 − 2x − 1 > 2.


Absolute Value

Definition The absolute value (or modulus) of a real number x isdefined as:

|x | ={

x if x ≥ 0−x if x < 0.


Absolute Value


|x | ={

x if x ≥ 0−x if x < 0.

Geometrically, |x | is the distance on the real line between x and 0.example:


Absolute Value


|x | ={

x if x ≥ 0−x if x < 0.

Geometrically, |x | is the distance on the real line between x and 0.example:

Similarly, for any x , y ∈ R, |x − y | is the distance between x and y .example:


Properties of Absolute Value

Lemma (Properties of Absolute Value) Suppose a, b ∈ R.Then:

1 |a| =√a2;




1 |a| =√a2;

2 | − a| = |a|;




1 |a| =√a2;

2 | − a| = |a|;3 |ab| = |a| |b|;




1 |a| =√a2;

2 | − a| = |a|;3 |ab| = |a| |b|;4 | a

b| = |a|

|b| when b 6= 0;




1 |a| =√a2;

2 | − a| = |a|;3 |ab| = |a| |b|;4 | a

b| = |a|

|b| when b 6= 0;

5 |a + b| ≤ |a|+ |b|. the triangle inequality.




1 |a| =√a2;

2 | − a| = |a|;3 |ab| = |a| |b|;4 | a

b| = |a|

|b| when b 6= 0;

5 |a + b| ≤ |a|+ |b|. the triangle inequality.

Proof of (1) and (5).


Absolute Value and Intervals

We can express the set of all solutions to inequalities involvingabsolute values as unions of one or more disjoint intervals.



We can express the set of all solutions to inequalities involvingabsolute values as unions of one or more disjoint intervals.Lemma (Absolute values and Intervals) Suppose a is a positivereal number. Then:

1 |x | = a ⇔ x = ±a;

2 |x | < a ⇔ −a < x < a ⇔ x ∈ (−a, a);

3 |x | > a ⇔ x < −a or x > a ⇔ x ∈ (−∞,−a) ∪ (a,∞);

4 |x | ≤ a ⇔ −a ≤ x ≤ a ⇔ x ∈ [−a, a];

5 |x | ≥ a ⇔ x ≤ −a or x ≥ a ⇔ x ∈ (−∞,−a] ∪ [a,∞);



We can express the set of all solutions to inequalities involvingabsolute values as unions of one or more disjoint intervals.Lemma (Absolute values and Intervals) Suppose a is a positivereal number. Then:

1 |x | = a ⇔ x = ±a;

2 |x | < a ⇔ −a < x < a ⇔ x ∈ (−a, a);

3 |x | > a ⇔ x < −a or x > a ⇔ x ∈ (−∞,−a) ∪ (a,∞);

4 |x | ≤ a ⇔ −a ≤ x ≤ a ⇔ x ∈ [−a, a];

5 |x | ≥ a ⇔ x ≤ −a or x ≥ a ⇔ x ∈ (−∞,−a] ∪ [a,∞);

Proof of (4). This follows because the distance from x to 0 is lessthan or equal to a if and only if x lies between a and −a.


Examples

Determine the intervals that contain the values of x which satisfythe following inequalities.

(a) |2x − 3| ≤ 1.


Examples

Determine the intervals that contain the values of x which satisfythe following inequalities.

(a) |2x − 3| ≤ 1.

(b) |x − 1| ≤ x2 − 1.


FUNCTIONS

Definition A function f from a set D to a set Y is a rule thatassigns an element f (x) of Y to each element x of D.


FUNCTIONS


Note that functions have a uniqueness property - there is only onevalue f (x) ∈ Y assigned to each x ∈ D.


FUNCTIONS



The set D of all possible input values is called the domain off .


FUNCTIONS




The set Y which contains all possible output values is calledthe codomain of f .


FUNCTIONS





The set R consisting of all possible output values of f (x) as xvaries throughout D is called the range of f .


FUNCTIONS






We write f maps D to Y symbolically as f : D → Y .


FUNCTIONS






We write f maps D to Y symbolically as f : D → Y .

We write f maps x to f (x) symbolically as f : x 7→ f (x).


Variables

We often think of the input and output values of a function asvariables. The function tells us how to determine the value of theoutput variable y from the value of the input variable x . We writey = f (x) and refer to x as the independent variable and y as thedependent variable. The function f acts like a ”black box” whichinputs x and outputs y = f (x).


Variables

We often think of the input and output values of a function asvariables. The function tells us how to determine the value of theoutput variable y from the value of the input variable x . We writey = f (x) and refer to x as the independent variable and y as thedependent variable. The function f acts like a ”black box” whichinputs x and outputs y = f (x).

Examples:y is the height of the floor of the lecture hall depending on thedistance x from the whiteboard;y is the stock market index depending on the time x ;y is the volume of a sphere depending on its radius x .


Real functions

The domain D and the codomain Y of a function f can be anysets. In this module, however, we will always take D and Y to besubsets of R.


Real functions

The domain D and the codomain Y of a function f can be anysets. In this module, however, we will always take D and Y to besubsets of R.We will often be lazy and not specify the domain and codomain off explicitly: in this case we will assume that the domain of f is thethe largest set of real numbers for which the definition of f makessense and that the codomain of f is R. We refer to this largest setof input values as the natural domain of f .


Real functions

The domain D and the codomain Y of a function f can be anysets. In this module, however, we will always take D and Y to besubsets of R.We will often be lazy and not specify the domain and codomain off explicitly: in this case we will assume that the domain of f is thethe largest set of real numbers for which the definition of f makessense and that the codomain of f is R. We refer to this largest setof input values as the natural domain of f .

Examples:Function Natural Domain Codomain Range

y = x2

y = 1/xy =

√x

y =√1− x2


Remark

A function is fully specified by not only giving the rule f , but alsogiving its domain D, and its codomain Y . Thus

f : R → R defined by f : x 7→ x2

andg : [0,∞) → R defined by g : x 7→ x2

are different functions since they have different domains.


The graph of a function

Definition A graph is the set of all points in the cartesian planesatisfying one or more equations and/or inequalities involving thecoordinates. In particular, the graph of a function f is set of allpoints (x , y) which satisfy the equation y = f (x) i.e. all pointswhose coordinates are the input-output pairs for f .Example: f : R → R is defined by f (x) = x + 2.


The graph of a function

Definition A graph is the set of all points in the cartesian planesatisfying one or more equations and/or inequalities involving thecoordinates. In particular, the graph of a function f is set of allpoints (x , y) which satisfy the equation y = f (x) i.e. all pointswhose coordinates are the input-output pairs for f .Example: f : R → R is defined by f (x) = x + 2.

Given a function f , we can sketch its graph by plotting some of itspoints (x , f (x)) in the plane and then ‘joining them up’. Calculuswill help us do this more accurately by telling us, for example,when the function is increasing or decreasing.


Curves

Definition A curve is of the set of all points (x , y) in the cartesianplane whose coordinates satisfy some equation involving thevariables x , y .


Curves


The graph of a function f is a special kind of curve since it isdefined by the equation y = f (x). However some curves are notgraphs of any function:


Curves


The graph of a function f is a special kind of curve since it isdefined by the equation y = f (x). However some curves are notgraphs of any function:

Recall that a function f can have only one value f (x) assigned toeach x in its domain. This leads to the vertical line test:

No vertical line can intersect the graph of a function more than once.


Example

(a) x2 + y2 = 1

The curve shown in (a) is not the graph of a function since it failsthe vertical line test.


Example continued

The curves in (b) and (c) are graphs of functions.


Reading Assignment

Thomas’ Calculus, Appendix 3:Lines, Circles, and Parabolas


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