Calculus (Math 1A)Lecture 1

Vivek Shende

August 23, 2017

Hello and welcome to class!

I am Vivek ShendeI will be teaching you this semester.

My office hours

Starting next week: 1-3 pm on tuesdays; 2-3 pm fridays873 Evans hall.Come ask questions!

Your GSIsKathleen KirschKenneth HungKubrat DanilovIzaak MecklerIsabelle Shankar

Some administrative details:

Enrolling in the class/sections:

enrollment@math.berkeley.edu

The bookJames Stewart, Single Variable Calculus:Early Transcendentals for UC Berkeley, 8th edition

Prerequisites

Trigonometry, coordinate geometry, plus a satisfactory grade in oneof the following: CEEB MAT test, an AP test, the UC/CSU mathdiagnostic test, or 32.

Prerequisites: test yourself

Compute (x + 4)(x + 3).

Compute 1x+1 + 1

x−1 .

Sketch y = x2 + 4x + 4.

After class, go outside. Measure the angle to the top of the belltower (you may need a protractor) and then the distance to it (e.g.walk there and count your steps). Estimate the height.

Prerequisites: test yourself

Compute (x + 4)(x + 3).

Compute 1x+1 + 1

x−1 .

Sketch y = x2 + 4x + 4.

After class, go outside. Measure the angle to the top of the belltower (you may need a protractor) and then the distance to it (e.g.walk there and count your steps). Estimate the height.

Prerequisites: test yourself

Compute (x + 4)(x + 3).

Compute 1x+1 + 1

x−1 .

Sketch y = x2 + 4x + 4.

After class, go outside. Measure the angle to the top of the belltower (you may need a protractor) and then the distance to it (e.g.walk there and count your steps). Estimate the height.

Prerequisites: test yourself

Compute (x + 4)(x + 3).

Compute 1x+1 + 1

x−1 .

Sketch y = x2 + 4x + 4.

After class, go outside. Measure the angle to the top of the belltower (you may need a protractor) and then the distance to it (e.g.walk there and count your steps). Estimate the height.

Prerequisites: test yourself

Compute (x + 4)(x + 3).

Compute 1x+1 + 1

x−1 .

Sketch y = x2 + 4x + 4.

After class, go outside. Measure the angle to the top of the belltower (you may need a protractor) and then the distance to it (e.g.walk there and count your steps). Estimate the height.

Prerequisites: test yourself

Can you compute ∫dx√

a2 − x2

If so, you should take a more advanced class.

Prerequisites: test yourself

Can you compute ∫dx√

a2 − x2

If so, you should take a more advanced class.

Grading

Your grade is determined by the homework/quizzes (15%), threemidterms (15% each), and final (40%).

HomeworkOne online homework assignment per week; due one minute beforemidnight on friday; starting next week.

QuizzesEvery thursday, in section: one problem from a list of problemsfrom the book.

ExamsThree in-class midterms (Sep. 18, Oct. 11, Nov. 8), and the finalexam (Dec 12).

Grade distribution

I intend to follow the same grade distribution as this course hashistorically had; very roughly 40% A’s, 30% B’s, 20% C’s, and10% D’s and F’s.

You can find detailed statistics at www.berkeleytime.com.

Makeup policy

There are no makeups for any reason

Instead,

I The two lowest quiz grades will be dropped.

I The lowest (curved) midterm grade can be replaced by your(curved) final exam grade.

Website

http://math.berkeley.edu/~vivek/1A.html

The website has a full syllabus, including all of the above

I will also post the slides on the website after each class.

We will also use bcourses and piazza.

What do we study in calculus?

Topic Objects What you do

Arithmetic Numbers Add, subtract, multiply, divide

Algebra Indeterminates Solve equations

Geometry Shapes Draw lines, circles, ...

Calculus Functions Limit, derivative, integral

What do we study in calculus?

Topic Objects What you do

Arithmetic Numbers Add, subtract, multiply, divide

Algebra Indeterminates Solve equations

Geometry Shapes Draw lines, circles, ...

Calculus Functions Limit, derivative, integral

What do we study in calculus?

Topic Objects What you do

Arithmetic Numbers Add, subtract, multiply, divide

Algebra Indeterminates Solve equations

Geometry Shapes Draw lines, circles, ...

Calculus Functions Limit, derivative, integral

What do we study in calculus?

Topic Objects What you do

Arithmetic Numbers Add, subtract, multiply, divide

Algebra Indeterminates Solve equations

Geometry Shapes Draw lines, circles, ...

Calculus Functions Limit, derivative, integral

What do we study in calculus?

Topic Objects What you do

Arithmetic Numbers Add, subtract, multiply, divide

Algebra Indeterminates Solve equations

Geometry Shapes Draw lines, circles, ...

Calculus Functions Limit, derivative, integral

What do we study in calculus?

Topic Objects What you do

Arithmetic Numbers Add, subtract, multiply, divide

Algebra Indeterminates Solve equations

Geometry Shapes Draw lines, circles, ...

Calculus Functions Limit, derivative, integral

What do we study in calculus?

To emphasize: in calculus, the basic objects of study are functions,

which you may not be accustomed to thinking about.

More precisely, you have seen functions everywhere, but may notbe used to the abstract notion of function, or the formalmanipulations of them.

Becoming comfortable with functions is one of the largestconceptual steps in learning calculus. We will spend the first weekon this.

What do we study in calculus?

To emphasize: in calculus, the basic objects of study are functions,which you may not be accustomed to thinking about.

More precisely, you have seen functions everywhere, but may notbe used to the abstract notion of function, or the formalmanipulations of them.

Becoming comfortable with functions is one of the largestconceptual steps in learning calculus. We will spend the first weekon this.

What do we study in calculus?

To emphasize: in calculus, the basic objects of study are functions,which you may not be accustomed to thinking about.

More precisely, you have seen functions everywhere,

but may notbe used to the abstract notion of function, or the formalmanipulations of them.

Becoming comfortable with functions is one of the largestconceptual steps in learning calculus. We will spend the first weekon this.

What do we study in calculus?

To emphasize: in calculus, the basic objects of study are functions,which you may not be accustomed to thinking about.

More precisely, you have seen functions everywhere, but may notbe used to the abstract notion of function,

or the formalmanipulations of them.

Becoming comfortable with functions is one of the largestconceptual steps in learning calculus. We will spend the first weekon this.

What do we study in calculus?

To emphasize: in calculus, the basic objects of study are functions,which you may not be accustomed to thinking about.

More precisely, you have seen functions everywhere, but may notbe used to the abstract notion of function, or the formalmanipulations of them.

Becoming comfortable with functions is one of the largestconceptual steps in learning calculus. We will spend the first weekon this.

What do we study in calculus?

To emphasize: in calculus, the basic objects of study are functions,which you may not be accustomed to thinking about.

More precisely, you have seen functions everywhere, but may notbe used to the abstract notion of function, or the formalmanipulations of them.

Becoming comfortable with functions is one of the largestconceptual steps in learning calculus.

We will spend the first weekon this.

What do we study in calculus?

To emphasize: in calculus, the basic objects of study are functions,which you may not be accustomed to thinking about.

More precisely, you have seen functions everywhere, but may notbe used to the abstract notion of function, or the formalmanipulations of them.

Becoming comfortable with functions is one of the largestconceptual steps in learning calculus. We will spend the first weekon this.

Functions of time

At each time t, the earth is at some distance d(t) from the sun.

Functions of time

At each time t, the earth is at some distance d(t) from the sun.

Functions of time

At each time t, the earth is at some distance d(t) from the sun.

Functions of time

At each time t, the earth is at some distance d(t) from the sun.The picture gets more interesting when you look closer:

Functions of time

At each time t, there is some number of living humans l(t).

Functions of time

At each time t, there is some number of living humans l(t).

Functions of time

At each time t, a stock has some price p(t).

Functions of time

At each time t, a stock has some price p(t).

Functions of time

A sample of carbon contains a certain amount of the unstableisotope carbon 14, which decays:

Functions of time

A sample of carbon contains a certain amount of the unstableisotope carbon 14, which decays:

Functions of time

A sample of carbon contains a certain amount of the unstableisotope carbon 14, which decays, hence takes different values c(t)over time.

Functions of space

At each point of space there is a temperature, T (x , y , z); youcould measure it with a thermometer.

Each point of space has some distance D(x , y , z) to the center ofthe sun.

In this class we will not see this kind of function, because we arestudying functions of one real variable.

We will see analogous things in which ‘space’ is constrained to beone dimensional; e.g. if we throw a ball straight up, or are divingin a well.

Functions of space

At each point of space there is a temperature, T (x , y , z); youcould measure it with a thermometer.

Each point of space has some distance D(x , y , z) to the center ofthe sun.

In this class we will not see this kind of function, because we arestudying functions of one real variable.

We will see analogous things in which ‘space’ is constrained to beone dimensional; e.g. if we throw a ball straight up, or are divingin a well.

Functions of space

At each point of space there is a temperature, T (x , y , z); youcould measure it with a thermometer.

Each point of space has some distance D(x , y , z) to the center ofthe sun.

In this class we will not see this kind of function, because we arestudying functions of one real variable.

We will see analogous things in which ‘space’ is constrained to beone dimensional; e.g. if we throw a ball straight up, or are divingin a well.

Functions of space

At each point of space there is a temperature, T (x , y , z); youcould measure it with a thermometer.

Each point of space has some distance D(x , y , z) to the center ofthe sun.

In this class we will not see this kind of function, because we arestudying functions of one real variable.

We will see analogous things in which ‘space’ is constrained to beone dimensional; e.g. if we throw a ball straight up, or are divingin a well.

Functions of space

At each point of space there is a temperature, T (x , y , z); youcould measure it with a thermometer.

Each point of space has some distance D(x , y , z) to the center ofthe sun.

In this class we will not see this kind of function, because we arestudying functions of one real variable.

We will see analogous things in which ‘space’ is constrained to beone dimensional; e.g. if we throw a ball straight up, or are divingin a well.

Another function

A function from {A+,A,A−,B+,B,B−,C+,C ,C−,D,F} topercents:

It is the historical grade distribution for this class.

Another function

A function from {A+,A,A−,B+,B,B−,C+,C ,C−,D,F} topercents:

It is the historical grade distribution for this class.

Another function

A function from {A+,A,A−,B+,B,B−,C+,C ,C−,D,F} topercents:

It is the historical grade distribution for this class.

Functions from formulas

f (x) = 0

Functions from formulas

f (x) = 0

Functions from formulas

f (x) = 0

Functions from formulas

f (x) = x

Functions from formulas

f (x) = x

Functions from formulas

f (x) = 2x

Functions from formulas

f (x) = 2x

Functions from formulas

f (x) = x/2

Functions from formulas

f (x) = x/2

Functions from formulas

f (x) = x2

Functions from formulas

f (x) = x2

Functions from formulas

f (x) = x2 + 4x + 4

Functions from formulas

f (x) = x2 + 4x + 4

Functions from formulas

f (x) = 1/x

Functions from formulas

f (x) = 1/x

To make these graphs

https://graphsketch.com/

Functions from formulas

f (x) = (x + 3)/(x + 4)

Functions from formulas

f (x) = (x + 3)/(x + 4)

What is a function?

In your book, you will find:

A function f is a rule that assigns to each element x in a set Dexactly one element, called f (x), in a set E .

What is a function?

In your book, you will find:

A function f is a rule that assigns to each element x in a set Dexactly one element, called f (x), in a set E .

What is a function?

In your book, you will find:

A function f is a rule that assigns to each element x in a set Dexactly one element, called f (x), in a set E .

What is a function?

It is somewhat better to say less:

A function f is a rule that assigns to each element x in a set Dexactly one element, called f (x), in a set E .

What is a function?

It is somewhat better to say less:

A function f is a rule that assigns to each element x in a set Dexactly one element, called f (x), in a set E .

What is a function

The difference between these arises when considering the questionof whether two functions are the same.

For instance, consider thefollowing two ‘rules’, which can be applied to positive integers:

Rule 1. Divide by 9 and consider the remainder. E.g.,421 = 46× 9 + 7, so we get 7.

Rule 2. Add the digits together. Repeat until the result has fewerthan one digit. E.g., 421→ 4 + 2 + 1 = 7.

In fact, though these look like rather different rules, in fact theyalways produce the same result. In mathematics, we say they givethe same function.

What is a function

The difference between these arises when considering the questionof whether two functions are the same. For instance, consider thefollowing two ‘rules’, which can be applied to positive integers:

Rule 1. Divide by 9 and consider the remainder. E.g.,421 = 46× 9 + 7, so we get 7.

Rule 2. Add the digits together. Repeat until the result has fewerthan one digit. E.g., 421→ 4 + 2 + 1 = 7.

In fact, though these look like rather different rules, in fact theyalways produce the same result. In mathematics, we say they givethe same function.

What is a function

The difference between these arises when considering the questionof whether two functions are the same. For instance, consider thefollowing two ‘rules’, which can be applied to positive integers:

Rule 1.

Divide by 9 and consider the remainder. E.g.,421 = 46× 9 + 7, so we get 7.

Rule 2. Add the digits together. Repeat until the result has fewerthan one digit. E.g., 421→ 4 + 2 + 1 = 7.

In fact, though these look like rather different rules, in fact theyalways produce the same result. In mathematics, we say they givethe same function.

What is a function

The difference between these arises when considering the questionof whether two functions are the same. For instance, consider thefollowing two ‘rules’, which can be applied to positive integers:

Rule 1. Divide by 9 and consider the remainder.

E.g.,421 = 46× 9 + 7, so we get 7.

Rule 2. Add the digits together. Repeat until the result has fewerthan one digit. E.g., 421→ 4 + 2 + 1 = 7.

In fact, though these look like rather different rules, in fact theyalways produce the same result. In mathematics, we say they givethe same function.

What is a function

The difference between these arises when considering the questionof whether two functions are the same. For instance, consider thefollowing two ‘rules’, which can be applied to positive integers:

Rule 1. Divide by 9 and consider the remainder. E.g.,421 = 46× 9 + 7,

so we get 7.

Rule 2. Add the digits together. Repeat until the result has fewerthan one digit. E.g., 421→ 4 + 2 + 1 = 7.

In fact, though these look like rather different rules, in fact theyalways produce the same result. In mathematics, we say they givethe same function.

What is a function

The difference between these arises when considering the questionof whether two functions are the same. For instance, consider thefollowing two ‘rules’, which can be applied to positive integers:

Rule 1. Divide by 9 and consider the remainder. E.g.,421 = 46× 9 + 7, so we get 7.

Rule 2. Add the digits together. Repeat until the result has fewerthan one digit. E.g., 421→ 4 + 2 + 1 = 7.

In fact, though these look like rather different rules, in fact theyalways produce the same result. In mathematics, we say they givethe same function.

What is a function

The difference between these arises when considering the questionof whether two functions are the same. For instance, consider thefollowing two ‘rules’, which can be applied to positive integers:

Rule 1. Divide by 9 and consider the remainder. E.g.,421 = 46× 9 + 7, so we get 7.

Rule 2.

Add the digits together. Repeat until the result has fewerthan one digit. E.g., 421→ 4 + 2 + 1 = 7.

In fact, though these look like rather different rules, in fact theyalways produce the same result. In mathematics, we say they givethe same function.

What is a function

The difference between these arises when considering the questionof whether two functions are the same. For instance, consider thefollowing two ‘rules’, which can be applied to positive integers:

Rule 1. Divide by 9 and consider the remainder. E.g.,421 = 46× 9 + 7, so we get 7.

Rule 2. Add the digits together. Repeat until the result has fewerthan one digit.

E.g., 421→ 4 + 2 + 1 = 7.

In fact, though these look like rather different rules, in fact theyalways produce the same result. In mathematics, we say they givethe same function.

What is a function

The difference between these arises when considering the questionof whether two functions are the same. For instance, consider thefollowing two ‘rules’, which can be applied to positive integers:

Rule 1. Divide by 9 and consider the remainder. E.g.,421 = 46× 9 + 7, so we get 7.

Rule 2. Add the digits together. Repeat until the result has fewerthan one digit. E.g., 421→ 4 + 2 + 1 = 7.

In fact, though these look like rather different rules, in fact theyalways produce the same result. In mathematics, we say they givethe same function.

What is a function

The difference between these arises when considering the questionof whether two functions are the same. For instance, consider thefollowing two ‘rules’, which can be applied to positive integers:

Rule 1. Divide by 9 and consider the remainder. E.g.,421 = 46× 9 + 7, so we get 7.

Rule 2. Add the digits together. Repeat until the result has fewerthan one digit. E.g., 421→ 4 + 2 + 1 = 7.

In fact, though these look like rather different rules, in fact theyalways produce the same result. In mathematics, we say they givethe same function.

What is a function?

A function f assigns to each element x in a set D exactly oneelement, called f (x), in a set E .

The set D is called the domain, and the set E is called thecodomain. One says f is a function from D to E .

In particular, to precisely specify a function, one is strictly speakingsupposed to say what the domain D and codomain E are.

What is a function?

A function f assigns to each element x in a set D exactly oneelement, called f (x), in a set E .

The set D is called the domain, and the set E is called thecodomain. One says f is a function from D to E .

In particular, to precisely specify a function, one is strictly speakingsupposed to say what the domain D and codomain E are.

What is a function?

A function f assigns to each element x in a set D exactly oneelement, called f (x), in a set E .

The set D is called the domain, and the set E is called thecodomain. One says f is a function from D to E .

In particular, to precisely specify a function, one is strictly speakingsupposed to say what the domain D and codomain E are.

What is a function?

A function f assigns to each element x in a set D exactly oneelement, called f (x), in a set E .

The set D is called the domain, and the set E is called thecodomain. One says f is a function from D to E .

In particular, to precisely specify a function, one is strictly speakingsupposed to say what the domain D and codomain E are.

Domain

In particular, to precisely specify a function, one is strictly speakingsupposed to say what the domain D and codomain E are.

In practice this is rarely done explicitly. Indeed in all the examplespreviously, we did not specify either the domain or the codomain.

In fact, you will often encounter questions like:

What is the domain of the function√x

Domain

In particular, to precisely specify a function, one is strictly speakingsupposed to say what the domain D and codomain E are.

In practice this is rarely done explicitly. Indeed in all the examplespreviously, we did not specify either the domain or the codomain.

In fact, you will often encounter questions like:

What is the domain of the function√x

Domain

In particular, to precisely specify a function, one is strictly speakingsupposed to say what the domain D and codomain E are.

In practice this is rarely done explicitly.

Indeed in all the examplespreviously, we did not specify either the domain or the codomain.

In fact, you will often encounter questions like:

What is the domain of the function√x

Domain

In particular, to precisely specify a function, one is strictly speakingsupposed to say what the domain D and codomain E are.

In practice this is rarely done explicitly. Indeed in all the examplespreviously, we did not specify either the domain or the codomain.

In fact, you will often encounter questions like:

What is the domain of the function√x

Domain

In particular, to precisely specify a function, one is strictly speakingsupposed to say what the domain D and codomain E are.

In practice this is rarely done explicitly. Indeed in all the examplespreviously, we did not specify either the domain or the codomain.

In fact, you will often encounter questions like:

What is the domain of the function√x

Domain

What is the domain of the function√x

Strictly speaking, this question is somewhat ambiguous.

Indeed, togive√x as a function, I should have told you what its domain was.

You should interpret this question as asking “what is the largestsubset of the real numbers on which the formula

√x makes sense

and defines a function”.

The answer is: [0,∞).

Domain

What is the domain of the function√x

Strictly speaking, this question is somewhat ambiguous. Indeed, togive√x as a function,

I should have told you what its domain was.

You should interpret this question as asking “what is the largestsubset of the real numbers on which the formula

√x makes sense

and defines a function”.

The answer is: [0,∞).

Domain

What is the domain of the function√x

Strictly speaking, this question is somewhat ambiguous. Indeed, togive√x as a function, I should have told you what its domain was.

You should interpret this question as asking “what is the largestsubset of the real numbers on which the formula

√x makes sense

and defines a function”.

The answer is: [0,∞).

Domain

What is the domain of the function√x

Strictly speaking, this question is somewhat ambiguous. Indeed, togive√x as a function, I should have told you what its domain was.

You should interpret this question as asking “what is the largestsubset of the real numbers on which the formula

√x makes sense

and defines a function”.

The answer is: [0,∞).

Domain

What is the domain of the function√x

Strictly speaking, this question is somewhat ambiguous. Indeed, togive√x as a function, I should have told you what its domain was.

You should interpret this question as asking “what is the largestsubset of the real numbers on which the formula

√x makes sense

and defines a function”.

The answer is: [0,∞).

Domain

To belabor the point, I can define a function f (x) from say [1,∞)to the real numbers, given by the formula f (x) =

√x .

The domain of this function would be [1,∞), because that’s whatI said the domain was.

The domain is part of the data included in the function.

However, when a function is given by a formula and the domain isnot explicitly specified, we understand the domain to be the largestpossible such that the formula makes sense.

Domain

To belabor the point, I can define a function f (x) from say [1,∞)to the real numbers, given by the formula f (x) =

√x .

The domain of this function would be [1,∞),

because that’s whatI said the domain was.

The domain is part of the data included in the function.

However, when a function is given by a formula and the domain isnot explicitly specified, we understand the domain to be the largestpossible such that the formula makes sense.

Domain

To belabor the point, I can define a function f (x) from say [1,∞)to the real numbers, given by the formula f (x) =

√x .

The domain of this function would be [1,∞), because that’s whatI said the domain was.

The domain is part of the data included in the function.

However, when a function is given by a formula and the domain isnot explicitly specified, we understand the domain to be the largestpossible such that the formula makes sense.

Domain

To belabor the point, I can define a function f (x) from say [1,∞)to the real numbers, given by the formula f (x) =

√x .

The domain of this function would be [1,∞), because that’s whatI said the domain was.

The domain is part of the data included in the function.

However, when a function is given by a formula and the domain isnot explicitly specified, we understand the domain to be the largestpossible such that the formula makes sense.

Domain

To belabor the point, I can define a function f (x) from say [1,∞)to the real numbers, given by the formula f (x) =

√x .

The domain of this function would be [1,∞), because that’s whatI said the domain was.

The domain is part of the data included in the function.

However, when a function is given by a formula and the domain isnot explicitly specified,

we understand the domain to be the largestpossible such that the formula makes sense.

Domain

To belabor the point, I can define a function f (x) from say [1,∞)to the real numbers, given by the formula f (x) =

√x .

The domain of this function would be [1,∞), because that’s whatI said the domain was.

The domain is part of the data included in the function.

However, when a function is given by a formula and the domain isnot explicitly specified, we understand the domain to be the largestpossible such that the formula makes sense.

What is a function?

A function f assigns to each element x in a set D exactly oneelement, called f (x), in a set E .

The set D is called the domain, and the set E is called thecodomain. One says f is a function from D to E .

In particular, to precisely specify a function, one is strictly speakingsupposed to say what the domain D and codomain E are.

Codomain

What is the codomain is√x?

Is it (−∞,∞)? Or [0,∞)?

Or maybe (−1001,−23) ∪ (−5.5249682,∞)?

The answer is that I have not given you enough information toknow, or in other words, strictly speaking I have not specified

√x

as a function.

We will avoid thinking about this by making the convention thatall functions in this class have codomain (−∞,∞), and neverusing the word codomain again.

Codomain

What is the codomain is√x?

Is it (−∞,∞)? Or [0,∞)?

Or maybe (−1001,−23) ∪ (−5.5249682,∞)?

The answer is that I have not given you enough information toknow, or in other words, strictly speaking I have not specified

√x

as a function.

We will avoid thinking about this by making the convention thatall functions in this class have codomain (−∞,∞), and neverusing the word codomain again.

Codomain

What is the codomain is√x?

Is it (−∞,∞)?

Or [0,∞)?

Or maybe (−1001,−23) ∪ (−5.5249682,∞)?

The answer is that I have not given you enough information toknow, or in other words, strictly speaking I have not specified

√x

as a function.

We will avoid thinking about this by making the convention thatall functions in this class have codomain (−∞,∞), and neverusing the word codomain again.

Codomain

What is the codomain is√x?

Is it (−∞,∞)? Or [0,∞)?

Or maybe (−1001,−23) ∪ (−5.5249682,∞)?

The answer is that I have not given you enough information toknow, or in other words, strictly speaking I have not specified

√x

as a function.

We will avoid thinking about this by making the convention thatall functions in this class have codomain (−∞,∞), and neverusing the word codomain again.

Codomain

What is the codomain is√x?

Is it (−∞,∞)? Or [0,∞)?

Or maybe (−1001,−23) ∪ (−5.5249682,∞)?

The answer is that I have not given you enough information toknow, or in other words, strictly speaking I have not specified

√x

as a function.

We will avoid thinking about this by making the convention thatall functions in this class have codomain (−∞,∞), and neverusing the word codomain again.

Codomain

What is the codomain is√x?

Is it (−∞,∞)? Or [0,∞)?

Or maybe (−1001,−23) ∪ (−5.5249682,∞)?

The answer is that I have not given you enough information toknow, or in other words, strictly speaking I have not specified

√x

as a function.

We will avoid thinking about this by making the convention thatall functions in this class have codomain (−∞,∞), and neverusing the word codomain again.

Codomain

What is the codomain is√x?

Is it (−∞,∞)? Or [0,∞)?

Or maybe (−1001,−23) ∪ (−5.5249682,∞)?

The answer is that I have not given you enough information toknow, or in other words, strictly speaking I have not specified

√x

as a function.

We will avoid thinking about this by making the convention thatall functions in this class have codomain (−∞,∞), and neverusing the word codomain again.

Range

The range of a function is the set of values it takes.

The range of f (x) = 0 is {0}.

The range of f (x) = x is (−∞,∞).

The range of f (x) = 2x is (−∞,∞).

The range of f (x) = x2 is [0,∞).

The range of f (x) = x2 + 2 is [2,∞).

Range

The range of a function is the set of values it takes.

The range of f (x) = 0 is {0}.

The range of f (x) = x is (−∞,∞).

The range of f (x) = 2x is (−∞,∞).

The range of f (x) = x2 is [0,∞).

The range of f (x) = x2 + 2 is [2,∞).

Range

The range of a function is the set of values it takes.

The range of f (x) = 0 is

{0}.

The range of f (x) = x is (−∞,∞).

The range of f (x) = 2x is (−∞,∞).

The range of f (x) = x2 is [0,∞).

The range of f (x) = x2 + 2 is [2,∞).

Range

The range of a function is the set of values it takes.

The range of f (x) = 0 is {0}.

The range of f (x) = x is (−∞,∞).

The range of f (x) = 2x is (−∞,∞).

The range of f (x) = x2 is [0,∞).

The range of f (x) = x2 + 2 is [2,∞).

Range

The range of a function is the set of values it takes.

The range of f (x) = 0 is {0}.

The range of f (x) = x is

(−∞,∞).

The range of f (x) = 2x is (−∞,∞).

The range of f (x) = x2 is [0,∞).

The range of f (x) = x2 + 2 is [2,∞).

Range

The range of a function is the set of values it takes.

The range of f (x) = 0 is {0}.

The range of f (x) = x is (−∞,∞).

The range of f (x) = 2x is (−∞,∞).

The range of f (x) = x2 is [0,∞).

The range of f (x) = x2 + 2 is [2,∞).

Range

The range of a function is the set of values it takes.

The range of f (x) = 0 is {0}.

The range of f (x) = x is (−∞,∞).

The range of f (x) = 2x is

(−∞,∞).

The range of f (x) = x2 is [0,∞).

The range of f (x) = x2 + 2 is [2,∞).

Range

The range of a function is the set of values it takes.

The range of f (x) = 0 is {0}.

The range of f (x) = x is (−∞,∞).

The range of f (x) = 2x is (−∞,∞).

The range of f (x) = x2 is [0,∞).

The range of f (x) = x2 + 2 is [2,∞).

Range

The range of a function is the set of values it takes.

The range of f (x) = 0 is {0}.

The range of f (x) = x is (−∞,∞).

The range of f (x) = 2x is (−∞,∞).

The range of f (x) = x2 is

[0,∞).

The range of f (x) = x2 + 2 is [2,∞).

Range

The range of a function is the set of values it takes.

The range of f (x) = 0 is {0}.

The range of f (x) = x is (−∞,∞).

The range of f (x) = 2x is (−∞,∞).

The range of f (x) = x2 is [0,∞).

The range of f (x) = x2 + 2 is [2,∞).

Range

The range of a function is the set of values it takes.

The range of f (x) = 0 is {0}.

The range of f (x) = x is (−∞,∞).

The range of f (x) = 2x is (−∞,∞).

The range of f (x) = x2 is [0,∞).

The range of f (x) = x2 + 2 is

[2,∞).

Range

The range of a function is the set of values it takes.

The range of f (x) = 0 is {0}.

The range of f (x) = x is (−∞,∞).

The range of f (x) = 2x is (−∞,∞).

The range of f (x) = x2 is [0,∞).

The range of f (x) = x2 + 2 is [2,∞).

Range

The range of a function is the set of values it takes.

The range of f (x) =√x is [0,∞).

Consider the function f (x) with domain [1,∞) given by theformula f (x) =

√x . Its range is [1,∞).

Range

The range of a function is the set of values it takes.

The range of f (x) =√x is

[0,∞).

Consider the function f (x) with domain [1,∞) given by theformula f (x) =

√x . Its range is [1,∞).

Range

The range of a function is the set of values it takes.

The range of f (x) =√x is [0,∞).

Consider the function f (x) with domain [1,∞) given by theformula f (x) =

√x . Its range is [1,∞).

Range

The range of a function is the set of values it takes.

The range of f (x) =√x is [0,∞).

Consider the function f (x) with domain [1,∞) given by theformula f (x) =

√x .

Its range is [1,∞).

Range

The range of a function is the set of values it takes.

The range of f (x) =√x is [0,∞).

Consider the function f (x) with domain [1,∞) given by theformula f (x) =

√x . Its range is

[1,∞).

Range

The range of a function is the set of values it takes.

The range of f (x) =√x is [0,∞).

Consider the function f (x) with domain [1,∞) given by theformula f (x) =

√x . Its range is [1,∞).

Domain and Range

Functions and graphs

Given a function f , its graph is the collection of pairs (x , f (x)).

When the domain and codomain are subsets of the real numbers(as will always be the case in this class), this collection can beplotted in the plane. We already saw many examples.

One can do this procedure in reverse: a graph defines a function.More precisely, a curve C in the plane is the graph of a functionexactly when it intersects each vertical line at most once. Thedomain of the function will be the x-values for which the verticalline intersects exactly once.

The function is given by sending x0 to the y -coordinate of theintersection of the vertical line x = x0 with C .

Functions and graphs

Given a function f , its graph is the collection of pairs (x , f (x)).When the domain and codomain are subsets of the real numbers(as will always be the case in this class), this collection can beplotted in the plane.

We already saw many examples.

One can do this procedure in reverse: a graph defines a function.More precisely, a curve C in the plane is the graph of a functionexactly when it intersects each vertical line at most once. Thedomain of the function will be the x-values for which the verticalline intersects exactly once.

The function is given by sending x0 to the y -coordinate of theintersection of the vertical line x = x0 with C .

Functions and graphs

Given a function f , its graph is the collection of pairs (x , f (x)).When the domain and codomain are subsets of the real numbers(as will always be the case in this class), this collection can beplotted in the plane. We already saw many examples.

One can do this procedure in reverse: a graph defines a function.More precisely, a curve C in the plane is the graph of a functionexactly when it intersects each vertical line at most once. Thedomain of the function will be the x-values for which the verticalline intersects exactly once.

The function is given by sending x0 to the y -coordinate of theintersection of the vertical line x = x0 with C .

Functions and graphs

Given a function f , its graph is the collection of pairs (x , f (x)).When the domain and codomain are subsets of the real numbers(as will always be the case in this class), this collection can beplotted in the plane. We already saw many examples.

One can do this procedure in reverse: a graph defines a function.

More precisely, a curve C in the plane is the graph of a functionexactly when it intersects each vertical line at most once. Thedomain of the function will be the x-values for which the verticalline intersects exactly once.

The function is given by sending x0 to the y -coordinate of theintersection of the vertical line x = x0 with C .

Functions and graphs

Given a function f , its graph is the collection of pairs (x , f (x)).When the domain and codomain are subsets of the real numbers(as will always be the case in this class), this collection can beplotted in the plane. We already saw many examples.

One can do this procedure in reverse: a graph defines a function.More precisely, a curve C in the plane is the graph of a functionexactly when it intersects each vertical line at most once.

Thedomain of the function will be the x-values for which the verticalline intersects exactly once.

The function is given by sending x0 to the y -coordinate of theintersection of the vertical line x = x0 with C .

Functions and graphs

Given a function f , its graph is the collection of pairs (x , f (x)).When the domain and codomain are subsets of the real numbers(as will always be the case in this class), this collection can beplotted in the plane. We already saw many examples.

One can do this procedure in reverse: a graph defines a function.More precisely, a curve C in the plane is the graph of a functionexactly when it intersects each vertical line at most once. Thedomain of the function will be the x-values for which the verticalline intersects exactly once.

The function is given by sending x0 to the y -coordinate of theintersection of the vertical line x = x0 with C .

Functions and graphs

Given a function f , its graph is the collection of pairs (x , f (x)).When the domain and codomain are subsets of the real numbers(as will always be the case in this class), this collection can beplotted in the plane. We already saw many examples.

One can do this procedure in reverse: a graph defines a function.More precisely, a curve C in the plane is the graph of a functionexactly when it intersects each vertical line at most once. Thedomain of the function will be the x-values for which the verticalline intersects exactly once.

The function is given by sending x0 to the y -coordinate of theintersection of the vertical line x = x0 with C .

Functions and graphs

(It is more correct to say “this is/is not the graph of a function”.)

Functions and graphs

(It is more correct to say “this is/is not the graph of a function”.)

Increasing and decreasing

The function f is increasing on the interval [a, b]. To express thisin a formula, note that for a ≤ x < x ′ ≤ b, one has f (x) < f (x ′).

The function is also increasing on [c , d ], and decreasing on [b, c].

Increasing and decreasing

The function f is increasing on the interval [a, b]. To express thisin a formula, note that for a ≤ x < x ′ ≤ b, one has f (x) < f (x ′).

The function is also increasing on [c , d ], and decreasing on [b, c].

Increasing and decreasing

The function f is increasing on the interval [a, b]. To express thisin a formula, note that for a ≤ x < x ′ ≤ b, one has f (x) < f (x ′).

The function is also increasing on [c , d ], and decreasing on [b, c].

Even and odd

A function f is said to be even if f (x) = f (−x), and said to beodd if f (x) = −f (−x).

f (x) = xn is even if n is even, and odd if n is odd.

Sums of even functions are even; sums of odd functions are odd.

A product of two even functions, or two odd functions, is even. Aproduct of an even and an odd function is odd.

This notion seems (and maybe is) a bit silly, but turns out to beoften helpful in simplifying and sanity checking computations.

Even and odd

A function f is said to be even if f (x) = f (−x), and said to beodd if f (x) = −f (−x).

f (x) = xn is even if n is even, and odd if n is odd.

Sums of even functions are even; sums of odd functions are odd.

A product of two even functions, or two odd functions, is even. Aproduct of an even and an odd function is odd.

This notion seems (and maybe is) a bit silly, but turns out to beoften helpful in simplifying and sanity checking computations.

Even and odd

A function f is said to be even if f (x) = f (−x), and said to beodd if f (x) = −f (−x).

f (x) = xn is even if n is even, and odd if n is odd.

Sums of even functions are even; sums of odd functions are odd.

A product of two even functions, or two odd functions, is even. Aproduct of an even and an odd function is odd.

This notion seems (and maybe is) a bit silly, but turns out to beoften helpful in simplifying and sanity checking computations.

Even and odd

A function f is said to be even if f (x) = f (−x), and said to beodd if f (x) = −f (−x).

f (x) = xn is even if n is even, and odd if n is odd.

Sums of even functions are even; sums of odd functions are odd.

A product of two even functions, or two odd functions, is even. Aproduct of an even and an odd function is odd.

This notion seems (and maybe is) a bit silly, but turns out to beoften helpful in simplifying and sanity checking computations.

Even and odd

A function f is said to be even if f (x) = f (−x), and said to beodd if f (x) = −f (−x).

f (x) = xn is even if n is even, and odd if n is odd.

Sums of even functions are even; sums of odd functions are odd.

A product of two even functions, or two odd functions, is even. Aproduct of an even and an odd function is odd.

This notion seems (and maybe is) a bit silly, but turns out to beoften helpful in simplifying and sanity checking computations.

Even and odd

A function f is said to be even if f (x) = f (−x), and said to beodd if f (x) = −f (−x).

f (x) = xn is even if n is even, and odd if n is odd.

Sums of even functions are even; sums of odd functions are odd.

A product of two even functions, or two odd functions, is even. Aproduct of an even and an odd function is odd.

This notion seems (and maybe is) a bit silly, but turns out to beoften helpful in simplifying and sanity checking computations.