MTH4100 Calculus I · 2015. 7. 28. · Calculus provides powerful techniques for solving problems...
Transcript of 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
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
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.
Bill Jackson Calculus I
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.
Bill Jackson Calculus I
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 . . .
Bill Jackson Calculus I
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 . . .
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
Bill Jackson Calculus I
Properties of the real numbers
The real numbers have three types of fundamental properties:
Bill Jackson Calculus I
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
Bill Jackson Calculus I
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
Bill Jackson Calculus I
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
completeness: “there are no gaps on the real line”
Bill Jackson Calculus I
Algebraic properties - Addition
The first five algebraic properties involve addition:
Bill Jackson Calculus I
Algebraic properties - Addition
The first five algebraic properties involve addition:
(A0) For all a, b ∈ R we have a + b ∈ R. closure
Bill Jackson Calculus I
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
Bill Jackson Calculus I
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
Bill Jackson Calculus I
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
Bill Jackson Calculus I
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
(A4) For all a ∈ R there is an element −a ∈ R such thata + (−a) = 0. inverse
Bill Jackson Calculus I
Algebraic properties - Multiplication
There are five analogous algebraic properties for multiplication:
Bill Jackson Calculus I
Algebraic properties - Multiplication
There are five analogous algebraic properties for multiplication:
(M0) For all a, b ∈ R we have ab ∈ R. closure
Bill Jackson Calculus I
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
Bill Jackson Calculus I
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
Bill Jackson Calculus I
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
Bill Jackson Calculus I
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
(M4) For all a ∈ R with a 6= 0, there is an element a−1 ∈ R suchthat a a−1 = 1. inverse
Bill Jackson Calculus I
Algebraic properties - Distributivity
One last algebraic properties links addition and multiplication:
Bill Jackson Calculus I
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
Bill Jackson Calculus I
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.
Bill Jackson Calculus I
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.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}.
Bill Jackson Calculus I
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.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 ...
Bill Jackson Calculus I
Order properties
For all a, b, c ∈ R we have:
Bill Jackson Calculus I
Order properties
For all a, b, c ∈ R we have:
(O1) either a ≤ b or b ≤ a totality of ordering I
Bill Jackson Calculus I
Order properties
For all a, b, c ∈ R we have:
(O1) either a ≤ b or b ≤ a totality of ordering I
(O2) if a ≤ b and b ≤ a then a = b totality of ordering II
Bill Jackson Calculus I
Order properties
For all a, b, c ∈ R we have:
(O1) either a ≤ b or b ≤ a totality of ordering I
(O2) if a ≤ b and b ≤ a then a = b totality of ordering II
(O3) if a ≤ b and b ≤ c then a ≤ c transitivity
Bill Jackson Calculus I
Order properties
For all a, b, c ∈ R we have:
(O1) either a ≤ b or b ≤ a totality of ordering I
(O2) if a ≤ b and b ≤ a then a = b totality of ordering II
(O3) if a ≤ b and b ≤ c then a ≤ c transitivity
(O4) if a ≤ b then a + c ≤ b + c order under addition
Bill Jackson Calculus I
Order properties
For all a, b, c ∈ R we have:
(O1) either a ≤ b or b ≤ a totality of ordering I
(O2) if a ≤ b and b ≤ a then a = b totality of ordering II
(O3) if a ≤ b and b ≤ c then a ≤ c transitivity
(O4) if a ≤ b then a + c ≤ b + c order under addition
(O5) if a ≤ b and 0 ≤ c then a c ≤ b c order under multiplication
Bill Jackson Calculus I
Order properties
For all a, b, c ∈ R we have:
(O1) either a ≤ b or b ≤ a totality of ordering I
(O2) if a ≤ b and b ≤ a then a = b totality of ordering II
(O3) if a ≤ b and b ≤ c then a ≤ c transitivity
(O4) if a ≤ b then a + c ≤ b + c order under addition
(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.
Bill Jackson Calculus I
Other rules for working with inequalities
The order properties (O1)-(O5) have many consequences:
Bill Jackson Calculus I
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.
Bill Jackson Calculus I
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.
We can prove that these rules are valid by using properties(O1)-(O5).
Bill Jackson Calculus I
The completeness property
Intuitively this means “there are no gaps in the real numbers”.More precisely it says:
Bill Jackson Calculus I
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 .
Bill Jackson Calculus I
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 .
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 ....
Bill Jackson Calculus I
Intervals
Definition An interval is a subset I of R of one of the followingtwo types:
Bill Jackson Calculus I
Intervals
Definition An interval is a subset I of R of one of the followingtwo types:
(a) all real numbers which lie between two given real numbers;
Bill Jackson Calculus I
Intervals
Definition An interval is a subset I of R of one of the followingtwo types:
(a) all real numbers which lie between two given real numbers;
(b) all real numbers which are either above or below a given realnumber.
Bill Jackson Calculus I
Intervals
Definition An interval is a subset I of R of one of the followingtwo types:
(a) all real numbers which lie between two given real numbers;
(b) all real numbers which are either above or below a given realnumber.
Type (a) intervals are said to be bounded (or finite). Type (b)intervals are said to be unbounded (or infinite).
Bill Jackson Calculus I
Intervals
Definition An interval is a subset I of R of one of the followingtwo types:
(a) all real numbers which lie between two given real numbers;
(b) all real numbers which are either above or below a given realnumber.
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.
Bill Jackson Calculus 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 .
Bill Jackson Calculus 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,∞).
Bill Jackson Calculus I
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...
Bill Jackson Calculus I
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.
Bill Jackson Calculus I
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.
Bill Jackson Calculus I
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 + 3.6
x−1≥ 5.
Bill Jackson Calculus I
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 + 3.6
x−1≥ 5.
x2 − 2x − 1 > 2.
Bill Jackson Calculus I
Absolute Value
Definition The absolute value (or modulus) of a real number x isdefined as:
|x | ={
x if x ≥ 0−x if x < 0.
Bill Jackson Calculus I
Absolute Value
Definition The absolute value (or modulus) of a real number x isdefined as:
|x | ={
x if x ≥ 0−x if x < 0.
Geometrically, |x | is the distance on the real line between x and 0.example:
Bill Jackson Calculus I
Absolute Value
Definition The absolute value (or modulus) of a real number x isdefined as:
|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:
Bill Jackson Calculus I
Properties of Absolute Value
Lemma (Properties of Absolute Value) Suppose a, b ∈ R.Then:
1 |a| =√a2;
Bill Jackson Calculus I
Properties of Absolute Value
Lemma (Properties of Absolute Value) Suppose a, b ∈ R.Then:
1 |a| =√a2;
2 | − a| = |a|;
Bill Jackson Calculus I
Properties of Absolute Value
Lemma (Properties of Absolute Value) Suppose a, b ∈ R.Then:
1 |a| =√a2;
2 | − a| = |a|;3 |ab| = |a| |b|;
Bill Jackson Calculus I
Properties of Absolute Value
Lemma (Properties of Absolute Value) Suppose a, b ∈ R.Then:
1 |a| =√a2;
2 | − a| = |a|;3 |ab| = |a| |b|;4 | a
b| = |a|
|b| when b 6= 0;
Bill Jackson Calculus I
Properties of Absolute Value
Lemma (Properties of Absolute Value) Suppose a, b ∈ R.Then:
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.
Bill Jackson Calculus I
Properties of Absolute Value
Lemma (Properties of Absolute Value) Suppose a, b ∈ R.Then:
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).
Bill Jackson Calculus I
Absolute Value and Intervals
We can express the set of all solutions to inequalities involvingabsolute values as unions of one or more disjoint intervals.
Bill Jackson Calculus I
Absolute Value and 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,∞);
Bill Jackson Calculus I
Absolute Value and 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,∞);
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.
Bill Jackson Calculus I
Examples
Determine the intervals that contain the values of x which satisfythe following inequalities.
(a) |2x − 3| ≤ 1.
Bill Jackson Calculus I
Examples
Determine the intervals that contain the values of x which satisfythe following inequalities.
(a) |2x − 3| ≤ 1.
(b) |x − 1| ≤ x2 − 1.
Bill Jackson Calculus I
Examples
Determine the intervals that contain the values of x which satisfythe following inequalities.
(a) |2x − 3| ≤ 1.
(b) |x − 1| ≤ x2 − 1.
Bill Jackson Calculus I
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.
Bill Jackson Calculus I
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.
Note that functions have a uniqueness property - there is only onevalue f (x) ∈ Y assigned to each x ∈ D.
Bill Jackson Calculus I
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.
Note that functions have a uniqueness property - there is only onevalue f (x) ∈ Y assigned to each x ∈ D.
The set D of all possible input values is called the domain off .
Bill Jackson Calculus I
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.
Note that functions have a uniqueness property - there is only onevalue f (x) ∈ Y assigned to each x ∈ D.
The set D of all possible input values is called the domain off .
The set Y which contains all possible output values is calledthe codomain of f .
Bill Jackson Calculus I
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.
Note that functions have a uniqueness property - there is only onevalue f (x) ∈ Y assigned to each x ∈ D.
The set D of all possible input values is called the domain off .
The set Y which contains all possible output values is calledthe codomain of f .
The set R consisting of all possible output values of f (x) as xvaries throughout D is called the range of f .
Bill Jackson Calculus I
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.
Note that functions have a uniqueness property - there is only onevalue f (x) ∈ Y assigned to each x ∈ D.
The set D of all possible input values is called the domain off .
The set Y which contains all possible output values is calledthe codomain of f .
The set R consisting of all possible output values of f (x) as xvaries throughout D is called the range of f .
We write f maps D to Y symbolically as f : D → Y .
Bill Jackson Calculus I
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.
Note that functions have a uniqueness property - there is only onevalue f (x) ∈ Y assigned to each x ∈ D.
The set D of all possible input values is called the domain off .
The set Y which contains all possible output values is calledthe codomain of f .
The set R consisting of all possible output values of f (x) as xvaries throughout D is called the range of f .
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).
Bill Jackson Calculus I
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).
Bill Jackson Calculus I
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 .
Bill Jackson Calculus I
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.
Bill Jackson Calculus I
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 .
Bill Jackson Calculus I
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
Bill Jackson Calculus I
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.
Bill Jackson Calculus I
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.
Bill Jackson Calculus I
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.
Bill Jackson Calculus I
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 .
Bill Jackson Calculus I
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 .
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:
Bill Jackson Calculus I
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 .
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.
Bill Jackson Calculus I
Example
(a) x2 + y2 = 1
The curve shown in (a) is not the graph of a function since it failsthe vertical line test.
Bill Jackson Calculus I
Example continued
The curves in (b) and (c) are graphs of functions.
Bill Jackson Calculus I
Reading Assignment
Thomas’ Calculus, Appendix 3:Lines, Circles, and Parabolas
Bill Jackson Calculus I