Your First Basic CALCULUS Problem Let’s Do It Together….
Summary
TLDRIn 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
📚 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.
📈 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.
🔍 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.
🎢 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.
📖 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.
Mindmap
Keywords
💡Calculus
💡Derivative
💡Integration
💡Algebra
💡Parabola
💡Slope
💡Vertex
💡Tangent Line
💡Maximum Altitude
💡Math Notes
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
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