Uploaded by TheIntegralCALC on 26.03.2012

Transcript:

Hi everyone! This is the first video in a short series of videos in which I’ll be covering

the basics of calculus

If you follow along with me, you’ll learn how to find the rate of change of a function at

a point

as well as the area between the graph of a function and the x-axis

These are the two fundamental problems to which calculus provides an answer and

everything we learn will be based on these two concepts

Today we’re going to learn about functions, which, at least for now, are really the only

kinds of equations we're interested in

In algebra, we wrote equations as y equals, and then something

involving x

If an equation is a function

we can replace the y with f(x)

When we used to write coordinate points as (x,y), we can now write them as (x,f(x))

A function is a special kind of equation that tells us about the unique relationship

between two variables

x and y

The reason functions are so special and interesting to us

is because a function will give you one

and only one y

for any x you put into it

you put one value in

you get one value out

oftentimes we like to think about a function machine

that has input side and an output side

The machine itself is the equation that relates x and y

When you put an x-value into the input side of the machine, it will calculate

exactly one y-value and spit it out on the output side of the machine

Because of this kind of one-to-one relationship between inputs and outputs

you can think of a function as being made up of its entire set

of input-output pairs

each pair is one point on the graph of the function

The graph, therefore, is a picture of all the input/output pairs, and it can

help us visualize and predict the output that the function will return for any

input

Now, if you’re taking calculus, it probably means that you’ve taken lots of other

math classes to get to this point

which means that somewhere along the line you’ve probably talked about domain and range along the way

The concepts of domain and range are very closely associated with the

concepts of a function

and the one to one

input-output relationship that it models

you can think of the domain of a function as the set of all of the possible input values

depending on the function

some values will be impossible to input

for example

you remember from previous math courses that at least with real numbers you can't

take the square root of a negative number

so if you have a square in your function

you won’t be able to input any value into your function machine

that will make the value underneath the square root sign

negative

these inputs just won't work

you can't put them into your function so they're not in the domain of your

function

The range of a function is completely dependent on the domain

because the range is made up of all of the values you can get out of the

function machine

as a result of inputing all the values in the domain

For example, if your function is x squared, you can input any value you want

without breaking the laws of mathematics or your function machine

so the domain is the set of all real numbers

The range of the function, however, is only positive real numbers

Regardless of whether you input negative or positive values into the function, because

you’re squaring each value, you can only get positive numbers out of it, and so the

range of the function is only positive real numbers

Functions are NOT any equation that can return multiple output values for the same input value

circles or a great example of non-functions

If we pick any x-value inside the circle

the equation of the circle returns two output values

for the single input value

remember

by definition, an equation is only a function if it returns exactly one

output

for every input value

so we can clearly see that a circle

will never be a function

this brings us to a great way to test for functions

it's called the vertical line test

and it says that, if you draw a perfectly vertical line anywhere in the domain of a

function

it will cross the graph only once

If you can draw a perfectly vertical line

that crosses the graph more than once

then the points where the line intersects the graph

represent the multiple output values for the single input value, and the graph therefore

cannot represent a function

Keep in mind also that a function can itself be comprised of multiple

functions

This polynomial function is a great example

the entire thing represents a function

but so do each of the four terms inside it

is taking separately

each one would pass the vertical line test

this function is therefore what we call a combination

of other functions

functions can also be compositions of functions

where one function is nested inside another

For example, e to the x is a function, and x squared is a function

e to the x squared

is a composition of functions where x squared

is nested inside e to the x

Sine of x squared + 1

is another good example of a composition of functions

Next time we’re going dive into the real foundation of calculus

and talk about limits and continuity

I'll see you then

click on a video for a more in-depth look at the topics covered

review this video by clicking the links on the right

and subscribe to notified of future videos

the basics of calculus

If you follow along with me, you’ll learn how to find the rate of change of a function at

a point

as well as the area between the graph of a function and the x-axis

These are the two fundamental problems to which calculus provides an answer and

everything we learn will be based on these two concepts

Today we’re going to learn about functions, which, at least for now, are really the only

kinds of equations we're interested in

In algebra, we wrote equations as y equals, and then something

involving x

If an equation is a function

we can replace the y with f(x)

When we used to write coordinate points as (x,y), we can now write them as (x,f(x))

A function is a special kind of equation that tells us about the unique relationship

between two variables

x and y

The reason functions are so special and interesting to us

is because a function will give you one

and only one y

for any x you put into it

you put one value in

you get one value out

oftentimes we like to think about a function machine

that has input side and an output side

The machine itself is the equation that relates x and y

When you put an x-value into the input side of the machine, it will calculate

exactly one y-value and spit it out on the output side of the machine

Because of this kind of one-to-one relationship between inputs and outputs

you can think of a function as being made up of its entire set

of input-output pairs

each pair is one point on the graph of the function

The graph, therefore, is a picture of all the input/output pairs, and it can

help us visualize and predict the output that the function will return for any

input

Now, if you’re taking calculus, it probably means that you’ve taken lots of other

math classes to get to this point

which means that somewhere along the line you’ve probably talked about domain and range along the way

The concepts of domain and range are very closely associated with the

concepts of a function

and the one to one

input-output relationship that it models

you can think of the domain of a function as the set of all of the possible input values

depending on the function

some values will be impossible to input

for example

you remember from previous math courses that at least with real numbers you can't

take the square root of a negative number

so if you have a square in your function

you won’t be able to input any value into your function machine

that will make the value underneath the square root sign

negative

these inputs just won't work

you can't put them into your function so they're not in the domain of your

function

The range of a function is completely dependent on the domain

because the range is made up of all of the values you can get out of the

function machine

as a result of inputing all the values in the domain

For example, if your function is x squared, you can input any value you want

without breaking the laws of mathematics or your function machine

so the domain is the set of all real numbers

The range of the function, however, is only positive real numbers

Regardless of whether you input negative or positive values into the function, because

you’re squaring each value, you can only get positive numbers out of it, and so the

range of the function is only positive real numbers

Functions are NOT any equation that can return multiple output values for the same input value

circles or a great example of non-functions

If we pick any x-value inside the circle

the equation of the circle returns two output values

for the single input value

remember

by definition, an equation is only a function if it returns exactly one

output

for every input value

so we can clearly see that a circle

will never be a function

this brings us to a great way to test for functions

it's called the vertical line test

and it says that, if you draw a perfectly vertical line anywhere in the domain of a

function

it will cross the graph only once

If you can draw a perfectly vertical line

that crosses the graph more than once

then the points where the line intersects the graph

represent the multiple output values for the single input value, and the graph therefore

cannot represent a function

Keep in mind also that a function can itself be comprised of multiple

functions

This polynomial function is a great example

the entire thing represents a function

but so do each of the four terms inside it

is taking separately

each one would pass the vertical line test

this function is therefore what we call a combination

of other functions

functions can also be compositions of functions

where one function is nested inside another

For example, e to the x is a function, and x squared is a function

e to the x squared

is a composition of functions where x squared

is nested inside e to the x

Sine of x squared + 1

is another good example of a composition of functions

Next time we’re going dive into the real foundation of calculus

and talk about limits and continuity

I'll see you then

click on a video for a more in-depth look at the topics covered

review this video by clicking the links on the right

and subscribe to notified of future videos