Your First Basic CALCULUS Problem Let’s Do It Together….

Summary

In this educational video, the host John introduces viewers to calculus, emphasizing its beauty and power despite its reputation for difficulty. He presents a basic problem involving a ball's trajectory to illustrate the concept of finding maximum altitude using calculus. John explains the importance of the derivative in calculus for determining slopes and how setting the derivative to zero helps find the vertex of a parabola, representing the maximum point. He also touches on the significance of good math notes and offers resources for further learning, concluding with an encouragement to appreciate and possibly pursue the study of calculus.

Takeaways

• 📚 The video aims to introduce the viewer to the power of calculus, a subject often considered advanced but with understandable and beautiful concepts.
• 🔍 The speaker, John, is a middle and high school math teacher and founder of tabletclassmath, offering comprehensive online math help programs.
• 📈 John emphasizes the importance of good math notes for students, as they are often correlated with better math grades.
• 🤔 The video presents a basic calculus problem involving finding the maximum altitude of a ball fired into the air, illustrating the practical application of calculus.
• 📉 The concept of the derivative in calculus is introduced, which is used to find the slope of a curve at any given point, essential for identifying maxima or minima.
• 📶 The derivative is calculated for a given quadratic function, demonstrating how to find the first derivative using algebraic manipulations.
• 📈 The process of setting the derivative equal to zero to find the vertex (maximum point) of a parabola is explained, showcasing a fundamental calculus technique.
• 📊 The video simplifies calculus concepts to make them more approachable, aiming to inspire interest in the subject and its real-world applications.
• 🚀 John highlights the importance of calculus in modern technology, asserting that without it, many of our technological advancements would not be possible.
• 🔗 Links to John's math help program and comprehensive math notes are provided in the video description for those interested in further learning.

Q & A

What is the main topic of the video?

The main topic of the video is calculus, specifically introducing viewers to the power and beauty of the subject, and demonstrating how calculus can be used to solve problems, such as finding the maximum altitude of a ball thrown into the air.

What is the reputation of calculus according to the video?

Calculus has a reputation of being advanced and difficult, but the concepts and principles are understandable and beautiful.

Who is the presenter in the video?

The presenter in the video is John, the founder of TabletClass Math and a middle and high school math teacher.

What is the purpose of the video?

The purpose of the video is to introduce viewers to the power of calculus and to solve a basic calculus problem together to illustrate the concepts of the subject.

What is the significance of math notes according to the video?

Math notes are significant because students with the best math notes almost always have the best math grades, and vice versa.

What is the primary concept used in the video to solve the problem?

The primary concept used in the video to solve the problem is the derivative, which is used to find the slope of a curve at any point.

What is the derivative in the context of calculus?

In the context of calculus, the derivative is a concept that allows us to find the slope of a curve at any given point, which is instrumental in identifying maxima, minima, and other critical points.

How does the video demonstrate the use of calculus to find the maximum point of a parabola?

The video demonstrates the use of calculus by finding the first derivative of the function representing the parabola and then setting the derivative equal to zero to find the x-coordinate where the slope is zero, indicating the maximum point.

What is the role of integration in calculus as mentioned in the video?

Integration in calculus is used to find areas under curves, which is a powerful tool for various applications, although it is not directly used in the problem demonstrated in the video.

What is the importance of calculus in modern technology according to the video?

Calculus is crucial for modern technology as it has solved numerous technical problems and is foundational to many innovations; without calculus, many of our modern conveniences like cellphones and airplanes would not exist.

How can viewers find more math help from the presenter?

Viewers can find more math help from the presenter by following the links provided in the video description, which lead to comprehensive math notes and video-based math help programs.

Outlines

00:00

📚 Introduction to Calculus

The speaker introduces the topic of calculus, acknowledging its reputation for being advanced and difficult, yet emphasizing that its concepts are understandable and beautiful. The video aims to demonstrate the power of calculus through a basic problem involving a ball's trajectory, aiming to find its maximum altitude. The speaker, John, introduces himself as a math teacher and founder of Tablet Class Math, offering online math help programs and comprehensive math notes. He stresses the importance of good math notes for students, sharing his experience that well-organized notes often correlate with better math grades.

05:03

📈 Understanding the Derivative in Calculus

This section delves into the concept of the derivative in calculus, which is fundamental for finding the slope of a curve at any given point. The speaker uses the analogy of a roller coaster to explain how the slope changes along the curve, reaching zero at the peak, which represents the maximum point or vertex of the parabola. The derivative is introduced as a method to mathematically determine this point of maximum, contrasting it with algebraic techniques that can also find the vertex of a parabola.

10:04

🔍 Calculating the First Derivative

The speaker demonstrates how to calculate the first derivative of a function, using a polynomial function as an example. The process involves applying calculus rules to manipulate the function and derive a new formula that represents the slope at any point on the curve. The speaker simplifies the process, making it accessible to viewers with some algebra background, and emphasizes that the first derivative will help identify where the slope is zero, which corresponds to the maximum point on the curve.

15:05

🎢 Finding the Maximum Point of a Parabola

The speaker applies the concept of the first derivative to find the maximum point of a parabola, which is akin to finding the peak of a roller coaster's path. By setting the derivative equal to zero and solving for the variable, the speaker determines the x-coordinate where the slope is zero. Substituting this value back into the original function yields the y-coordinate, pinpointing the vertex of the parabola. This example illustrates a practical use of derivatives in calculus to solve real-world problems, such as optimizing trajectories.

20:06

📖 Conclusion and Encouragement to Learn Calculus

In the concluding part, the speaker summarizes the video's lesson on using the first derivative to find the maximum point of a function. He reiterates the importance of calculus in various fields, emphasizing its foundational role in modern technology. The speaker encourages viewers to appreciate calculus and consider learning it, acknowledging the subject's complexity but also its significance. He invites viewers to engage with his YouTube channel for more math content and offers his best wishes for their mathematical endeavors.

Keywords

💡Calculus

Calculus is a branch of mathematics that studies how things change. In the video, calculus is introduced as a powerful and advanced subject that is fundamental to understanding many scientific and technological concepts. The video aims to demystify calculus by explaining its basic principles and showing how it can be used to solve problems, such as finding the maximum altitude of a ball thrown into the air.

💡Derivative

The derivative in calculus represents the rate at which a function is changing at any given point. It is used to find the slope of the tangent line to the graph of a function at a specific point. In the video, the derivative is used to determine the maximum point of a parabolic trajectory, illustrating how calculus can be applied to real-world physics problems.

💡Integration

Integration is the concept in calculus that deals with finding the area under a curve defined by a function. It is one of the two main operations in calculus, the other being differentiation. The video mentions integration as a powerful tool for calculating areas, which is essential in various scientific and engineering applications.

💡Algebra

Algebra is a branch of mathematics that uses symbols and rules to solve equations and manipulate expressions. In the video, the presenter mentions that some basic algebra is necessary to understand calculus concepts, particularly when dealing with the manipulation of equations to find derivatives or to solve for specific values.

💡Parabola

A parabola is a U-shaped curve that can be described by a quadratic equation. In the video, the concept of a parabola is used to illustrate how calculus can be applied to find the vertex of the curve, which represents the maximum or minimum point, depending on the context.

💡Slope

Slope in the context of calculus refers to the steepness or incline of a line, which is a fundamental concept when studying the tangent to a curve at a specific point. The video explains how the slope changes along a parabola and how the derivative can be used to find the exact slope at any point, particularly where the slope is zero, indicating the maximum or minimum.

💡Vertex

The vertex of a parabola is the point where the curve changes direction, which is the highest or lowest point on the graph. In the video, the presenter uses calculus to find the vertex of a parabola, which is a practical application of derivatives to determine the maximum altitude of a projectile.

💡Tangent Line

A tangent line is a straight line that touches a curve at a single point without crossing it. In the video, the concept of a tangent line is used to explain how the derivative provides the slope of this line at any point on the curve, which is crucial for understanding the local behavior of functions.

💡Maximum Altitude

Maximum altitude in the context of the video refers to the highest point a ball reaches when thrown into the air. The video uses this concept to demonstrate how calculus can be used to solve a physics problem by finding the vertex of a parabolic trajectory, which represents the maximum altitude.

💡Math Notes

Math notes are records of mathematical concepts, formulas, and problem-solving strategies. The video emphasizes the importance of having good math notes for students, suggesting that well-organized notes correlate with better mathematical understanding and performance. The presenter also offers comprehensive math notes for various levels of mathematics.

Highlights

Calculus is considered advanced and difficult but its concepts are understandable and beautiful.

The video aims to introduce viewers to the power of calculus.

A basic calculus problem involving a ball's trajectory will be solved.

The video includes some basic algebra, suitable for viewers unfamiliar with calculus.

Introduction of the speaker, John, founder of TabletClass Math and a middle/high school math teacher.

John offers an online video-based math help program and comprehensive math notes.

The importance of having outstanding math notes for students is emphasized.

A problem involving finding the maximum altitude of a ball is presented.

Integration and derivative are introduced as the two primary concepts of calculus.

The derivative is used to find the slope of a curve at any point.

The concept of slope in the context of a parabola is explained.

The maximum point of a parabola is analogous to the top of a roller coaster.

The process of finding the first derivative of a function is demonstrated.

The first derivative provides a formula for the slope of the function.

Setting the derivative equal to zero helps find the maximum point of the parabola.

The maximum point of the parabola is calculated to be at (-4, 7).

Calculus is essential for modern technology and has wide-ranging applications.

The video concludes with an invitation to subscribe and explore more math content.

Transcripts

okay let's talk about calculus and if

you're watching this video i assume

you're interested in the subject of

calculus maybe you're planning on taking

a subject or maybe you're just kind of

just generally interested on

the math called calculus and

calculus has kind of a reputation of

being advanced and difficult and in fact

it is but the concepts and principles

are actually um

pretty you know understandable for

anybody and they're very beautiful it's

a calculus such a powerful

mathematics so the whole point of this

video is to kind of introduce you to

some of the power of uh calculus okay

and we're going to be doing uh something

here it's gonna be kind of a little

basic interesting problem and if you've

never done

calculus before stick around because

this could definitely be your first

calculus problem and we'll kind of

solve it together but

i will say in full disclosure then i'm

going to be doing some

algebra not much okay some basic algebra

in this problem so if you're not

familiar with algebra still stick around

okay you'll learn something but if you

know some algebra you'll kind of um

relate to some of the things going to be

talking about here in a second but first

i'm going to quickly introduce myself my

name is john i'm the founder of tablet

class math i'm also a middle and high

school math teacher and over many years

i've constructed what i like to believe

is one of the most robust comprehensive

online video based math help programs

there is now of course i'll let you be

the judge of that if you want to check

it out but

you can find a link to my math help

program in the description of this video

but whether you need to take a full

comprehensive math course i can help you

or you need assistance in the course you

might be taking right now i can help you

all my

courses have full comprehensive lessons

and i teach you how to solve the most

common

type of problems you're going to be

facing

at the middle high school level or even

more advanced levels okay i literally

solved thousands of problems and i think

that's where uh my program is very

unique okay not many programs really

have that much video content but again

if you're interested you can follow the

link in the description below okay now

let's talk about math notes if you are a

math student you may or may not be but

if you are a math student you need to

have

outstanding math notes okay after

decades of teaching math one thing is

apparent to me those students with the

best math notes almost always have the

best math grade

grades and the reverse is true okay

those students with no math notes uh

sloppy disorganized math notes and that

was mean way back in the in the day i to

learn how to take math notes and that's

maybe your situation okay so if your

notes are not that great don't get down

on yourself too much just recognize that

you have to start improving okay but in

the meantime you need something to study

from so i actually offer very

comprehensive detailed math notes you

can find a link to those in the

description of this video as well those

include pre-algebra algebra 1 geometry

algebra 2 and trigonometry okay so let's

get into this problem and you might be

saying well what is this problem okay

well here

let's just imagine we had like some ball

getting fired up in the air maybe a

cannon or something and this ball was

traveling up and up and up and up and up

okay and then it kind of peaked and now

it's on a way back down right so

uh let's

try to identify at what point

does this ball reach its maximum

altitude okay that's kind of the general

idea here uh again it's not going to be

an exact this would be like an example

of a physics problem okay of course

physics uses calculus all the scientists

use calculus as the mathematics the more

you know when you study them at a higher

level but we're going to basically find

the maximum of a shape right and i'm

going to do that in just one second but

let's kind of emphasize why this is so

unique now if you've taken algebra you

might recognize this shape here

as a parabola okay it's a parabolic

shape and we can just use

some other algebraic techniques to find

what we call uh the vertex right which

is this point up here

okay which is the maximum but i'm going

to show you how we can use calculus to

find this point as well because this is

a nice illustration

basic illustration of calculus now in

calculus when you learn calculus okay

there's pretty much two big primary um

concepts that we learn okay big picture

topics let's just say the first one is

integration okay integration is a little

symbol like this you've probably seen

this before

looks kind of crazy like so

and let's suppose this was on the xy

chart right here integration uh

is

basically um the way we can find area

underneath the curve with something okay

so very very powerful i've not done a

few videos on the basics of calculus so

we learned integration then we learned

this other thing uh

called the derivative okay the

derivative kind of looks like this

symbol dx d y

or y prime there's all kinds of ways you

can uh it's expressed okay but we learn

about the derivative so there's like a

big huge topic and calculus is the

derivative and then we also learn about

integration okay in this particular

problem we're going to be using this

concept here the derivative uh and the

derivative more or less okay

let's kind of erase all this stuff here

the derivative allows us to find

the slope okay the derivative has to do

conceptually with the slope of something

so here's my little curve my little

parabola

and along this parabola let's say at

this point right here what's the slope

of this parabola well it would kind of

be like

this line like right here right

that would be the slope but what

precisely is the slope of the parabola

at this point

well that's where we need calculus okay

but let's just kind of follow this uh

parabola around and let's just study

what the slope is looking like how about

this point right here well at that point

the slope might be something like this

right

okay so we can see the slope is changing

along this parabola how about this point

right here

well the slope is like well maybe like

this and we kind of drew it kind of bad

something like that now

this

line okay is what we call a tangent line

it just this is one line that just

touches at one single point that

parabola at these uh precise points so

this

these lines are going to touch at that

one point now here this would not be the

case but i'm just kind of sketching

around here trying to make a

bigger point now

if you don't know basic algebra

when we talk about the slope we're

talking about the angle of something

right the steepness of something here

so the slope in this direction

we indicate that by this little variable

m

is going to be positive okay so slopes

that run like this way increase from

left to right is a positive slope uh

slopes that go down this way are the

negative slope so what do you think a

line that goes this way what's the angle

of this line how steep is this line okay

this is a horizontal line

you might be saying that it has no slope

it's zero and you would be correct okay

so lines that are flat okay or

horizontal okay have zero slope i'm

going to erase this here in a second

okay so these lines horizontal lines

have zero slope and vertical lines here

their slope is what we call undefined we

don't really need to get into that for

this particular um

video but just so you know

if i'm trying to find the maximum point

just think of it

as a roller coaster this is probably

good

uh representation here you are going up

a roller coaster you're increasing okay

so positive slope is like you're

increasing on your roller coaster and

then right here at the very very top

you're like flat right you're perfectly

horizontal you are like this okay the

slope is zero

then you go down the other side roller

coasters are so cool

okay

and now we have negative slope okay so

this is where the slope is negative this

is where the slope is positive and right

here on the top when everyone is just

like scared scared scared and then

they're like oh and you kind of go back

down this is this one point right there

okay

is the top okay of the roller coaster

the top or the maximum okay

of

of the

this parabola and that's what we're

trying to determine now again

if you studied algebra there's other

techniques but this is a basic simple uh

parabola okay

there could be more challenging

techniques that algebra can help us out

and we're going to have to use our

friend calculus okay so we're going to

be using the derivative to

determine the maximum of this parabola

so let's get into it now

okay

so here is our problem okay we want to

find the maximum now this particular

function here okay its graph is not

exactly this just kind of drew this to

make it look kind of interesting

so here is

this graph this graph's really more like

this on the x y plane so here's our

function

and if i was to graph this parabola it

would be like so okay but again i'm

trying to find

this maximum point right there now

that's also known as the vertex we want

to find this maximum point okay where

does this thing max out now

as i indicated we're going to use this

concept here

called the derivative and this is the

notation we want to find the first

derivative of this function okay

so this is the notation again d x d y it

looks like this and

again you don't need to know a lot about

it just know

that we're going to find the first

derivative okay now

let's this is the the function when i

find the first derivative

d x d y

okay or

f prime okay this right here this is

actually the first derivative i'm gonna

find i'm gonna show you how i got got to

this what does that mean when i find the

derivative okay this function right here

its graph is this parabola here is the

parabola this thing right here is the

graph of that parabola when i find the

first derivative okay these are all

equivalent

notations i could write it as d x d y

uh or i could write it like this now

notice i said the first derivative i

could find the second derivative that

means something in more advanced

mathematics especially like physics and

stuff but i don't want to digress too

much because i could just get overly

excited about calculus it's such a cool

topic but anyways here's the deal

when i find the first derivative of this

function

here's the function here's its graph

when i find the first derivative it's

going to give me a formula for the slope

okay

so the first derivative is a formula for

the slope of anywhere along this line

this is actually the first derivative

okay

so

i'm going to show you precisely what

this means so this is going to give me a

derivative a

formula for the slope or m okay now

remember we talked about a roller

coaster m is positive here

m is zero right there okay and then m is

negative right there so what i'm going

to want to find is where is the slope

equal to 0 okay remember we just talked

about that because that is going to be

the top of this parabola where where the

slope is 0

is going to be

where uh the maximum okay point is on

this parabola

all right so how do we find the first

derivative okay so that's what it means

so here is how we find the first

derivative so

in uh calculus we're given a bunch of

rules not that difficult just kind of

algebraic manipulations but

here is our polynomial function our

little quadratic equation if you've

taken enough algebra to understand so

here's what we're going to do okay to

find the first derivative we're going to

take this thing is written in highest

power to lowest power we take this

little exponent right there okay here

let's kind of make it bigger

negative

x

squared okay

so we're going to take the 2

i'm going to multiply by that number

that's a negative 1 okay so it's going

to be i'm sorry we're going to take i'm

sorry you're going to be take the 2 or

multiply by that negative

number which is this one okay so two

times negative one

is negative two

all right so take that little two

whatever the exponent might be if it was

three i would multiply by three so take

the two multiply

by this number that's negative one x

squared so 2 times negative 1 is

negative 2. then i'm going to take that

same variable x i'm going to write it

like so okay and then whatever this

power is it was 2 i'm sorry it is 2. i'm

going to decrease it by 1.

so that would be just to the first power

one okay

let me do that again all right so we can

understand this rule to take have a

functional polynomial function to be

able to find the first derivative it's

not that difficult so we have negative x

squared okay

i take the 2

multiply by this number which is just a

negative 1. 2 times negative 1 is

negative 2.

it's the variable x now whatever this is

i'm going to drop it down by 1. so

that's 2 to the first or just 2. okay i

could write it just like that and that

is step one okay now i'm going to do the

same thing

over here

all right so what's the power of this

it's 1

okay so it's going to be

1 times negative 8

is what

negative 8 times x 2

this is 1. i'm going to drop it down by

one

so that's x to the zero

anything to the zero power is one

anything to the zero power is one so

that's really negative eight times one

so

my first derivative is negative two x

minus eight because it's negative eight

times one this is really just one and

then

when you're taking the derivative of a

number it's just zero it goes away okay

so this here

is our first derivative again i'm using

different notation

here it's the function x okay f of x

so

all equivalent notation what we just

found is the first derivative of this

function f so it can be

written as d x d y very very common okay

or we could write this little apostrophe

that's the first derivative of the

function f or this function's name is f

i can write it like this okay all

equivalent f prime we would call this

okay f prime first derivative of this

function again this is a lovely formula

for the slope okay it's going to tell me

where the slope is at now i'm interested

in

where is the slope equal to zero okay

remember

okay

just a quick review the slope is

positive in this direction

its slope is negative in this direction

but at this one precise point at the top

of the roller coaster the slope is zero

all right so let's find that point now

okay

erase all of this

okay so

here is my

formula for my slope i know my slope

along this function along this parabola

is equal to negative two x

minus eight so i'm gonna i wanna say to

myself okay mr slope where are you equal

to zero because that's where i'm

interested in knowing okay where the

slope is equal to zero so i'm going to

set the slope

the formula

uh or the equation here for the slope

equal to zero and i'm gonna solve

that equation so negative two x

minus eight equals zero i'm going to

move the 8 over to the other side of the

equation that's negative 2x

is equal to 8 and i'm going to divide

both sides equation by negative 2

so x is equal to negative 4 okay

so right here um one two three four

negative four

along where x is negative four at this

point

that is the maximum

along the x-axis of where that's at and

if i wanted to know this exact

coordinate i would just plug in negative

4

into my original function and i actually

just did that right here

okay let's plug in negative 4 into our

original function to find out what the

y-coordinate is and when i do that

whoops

you can see here

i'm plugging in very precisely negative

4 here i do all the math and f of

negative 4 for this function is 7.

so the top of that parabola okay right

there okay it's located we have one two

three four negative four

seven okay seven this way all right not

negative seven

but seven

in this direction right there so that

vertex that maximum point for this

parabola occurs at negative 7

4. now

uh this is an illustration of using the

first derivative okay of a function

all right to solve a problem now there

is another way more direct approach we

can use algebra but this is a an easy

problem okay uh algebra can you know

help us with the easy problem but then

there's a lot of other things that

calculus just you know cannot help us

with all right we

you know but i you know chose this

because this is our first calculus

problem together i wanted this to make

sense uh to you and hopefully even if

you didn't get all the algebraic

manipulations and i think um hopefully

you can understand the concept of the

roller coaster stuff here and get an

appreciation for the basic main basic

concepts of calculus integration

and uh the derivative right now

again you know i'm oversimplifying a lot

of things and if you have you know a

pretty advanced math uh background you

might say hey you missed that you missed

this yes listen i get that right i'm not

trying to teach a full calculus course

here that's just the point

of this video the point is to teach you

a little something about the subject of

calculus so you can appreciate it and

maybe even

get motivated enough excited about

enough to be like you know what i want

to learn this uh subject okay it's

definitely

uh

a great goal you might be like yeah well

you're a math teacher you love math of

course you're going to like want

everyone to know calculus i'm like yeah

well it is

people just don't realize okay without

calculus

um

all our modern day technology would just

fall apart right calculus has solved

so many problems uh technical problems

we just would not be here without

calculus we wouldn't have our cell

phones we wouldn't have uh airplanes

yadda yadda yadda okay anyways i don't

want to

start preaching here about a calculus

importance of it but it's just a

fascinating mathematics subject and it

does take you know it is a serious

subject you do have to really build up

your math skill sets uh to be prepared

to actually take an informal way but i

think that would be a noble goal for

sure well if you found this video

entertaining interesting in some way

or helpful uh definitely appreciate you

smashing that like button that helped me

out and uh if you're new to my youtube

channel hopefully you'll consider

subscribing i actually have hundreds and

hundreds of math videos been on youtube

for a long time

obviously i love teaching math so have a

lot of uh videos already organized in

various playlists there to help you out

uh so you can just go to my channel

check things out but i'm posting stuff

all the time all types of various of uh

level of mathematics but if you want my

best help definitely follow

the links in the description of this

video but with that being said i

definitely wish you all the best in your

mathematics and adventures thank you for

your time and have a great day