Calculus AB/BC – 1.4 Estimating Limit Values from Tables
Summary
TLDRIn this calculus lesson, Mr. Bean introduces the concept of finding limits using a table of values and a graphing calculator, specifically the TI-84 Plus CE. He demonstrates how to approximate limits by inputting values close to a given point and discusses the importance of using a calculator for accuracy. The lesson covers the use of table setup for precise values, function notation for more accurate calculations, and emphasizes the need for rounding or truncating to three decimal places for AP exam standards. Mr. Bean also touches on the continuity and increasing nature of functions, concluding with a practical example of finding the cosine of a limit.
Takeaways
- 📚 The lesson is about finding limits in calculus, emphasizing the importance of using a calculator, especially a graphing calculator, for accuracy and efficiency.
- ⚠️ A warning is given that a calculator will be necessary for the lesson, with a suggestion to use a graphing calculator for the best results.
- 📈 The concept of limits is illustrated through a table of values, showing how Y-values approach a certain number as X gets closer to a specific value.
- 📉 The graphical representation of limits is discussed, with an example of how the limit is determined when X approaches -4, resulting in a Y-value approaching 2.5.
- 🔢 The use of a TI-84 Plus CE calculator is demonstrated to fill in a table of values for evaluating limits, with the function plugged into the calculator for quick computation.
- 🛠 Two methods are shown for using the calculator: the table of values and function notation, with the latter providing more accurate results due to less rounding error.
- 📉 The function notation method is highlighted as particularly useful for future problems, allowing for quick and accurate input of values into the function.
- 📌 The importance of setting the calculator to radian mode for trigonometric functions in calculus is emphasized, as opposed to degree mode.
- 📝 The lesson concludes with an example of how to calculate the limit of a function as X approaches a certain value, using the cosine function and the calculator for precise results.
- ✂️ The importance of truncating or rounding to three decimal places for exam answers is stressed, as this is a requirement for the AP exam.
- 🎓 The lesson ends with a reminder to practice the skills learned for the mastery check, highlighting the significance of precision and accuracy in calculus.
Q & A
What is the main topic of the calculus lesson presented by Mr. Bean?
-The main topic of the lesson is finding limits in calculus using a table of values and a graphing calculator.
Why is a calculator recommended for this lesson?
-A calculator is recommended because it helps in evaluating limits by plugging in values close to the point of interest, and a graphing calculator is particularly useful for its table of values feature.
What does the limit represent in the context of the lesson?
-The limit represents the value that the function's output (Y-value) approaches as the input (X) gets arbitrarily close to a certain point.
How does the table of values help in understanding limits?
-The table of values helps by showing the Y-values for X-values that are very close to the point of interest, illustrating what the Y-value is approaching as X approaches that point.
What is the significance of the graphical representation in the lesson?
-The graphical representation visually shows the behavior of the function as X approaches a certain value, helping to understand the concept of limits and how they are approached.
Why is it necessary to use function notation on a calculator for more accuracy?
-Function notation on a calculator allows for more precise input of values into the function, providing more accurate results with additional decimal places compared to the table of values.
What is the purpose of setting the calculator to radian mode when dealing with trigonometric functions in calculus?
-Radian mode is necessary because calculus and physics typically use radians for trigonometric functions, as opposed to degrees.
How does the statement about the function being continuous and increasing affect the interpretation of the limit?
-The statement ensures that there are no breaks or decreases in the graph, which confirms that the Y-values are consistently approaching a certain number from both sides of X.
What is the importance of using three decimal places when solving problems for the 8P exam?
-Using three decimal places is a requirement for the 8P exam to ensure precision in the answers, and it helps the graders to quickly assess the accuracy of the solutions.
What are the two methods mentioned in the script for dealing with decimal places in answers?
-The two methods are rounding the answer to three decimal places or truncating the answer, which involves writing out three decimal places and stopping.
How does the script emphasize the importance of understanding the concept of limits in calculus?
-The script emphasizes the importance by providing a step-by-step approach to finding limits using tables and calculators, and by explaining the significance of each step in the process.
Outlines
📚 Introduction to Calculus Limits and Tools
In this calculus lesson, Mr. Bean introduces the concept of finding limits. He emphasizes the necessity of a calculator, particularly a graphing calculator, for this lesson. The instructor provides a brief review of limits by explaining how to approach a value from both sides and what the limit represents. The lesson also includes a visual representation of limits through a table of values and a graph, illustrating how the limit is determined by values approaching a specific point. Mr. Bean also previews the use of a TI-84 Plus CE calculator for more efficient calculations in later parts of the lesson.
🔍 Exploring Limits with Calculator Techniques
The second paragraph delves into the practical application of finding limits using a calculator. Mr. Bean demonstrates how to input a function into a graphing calculator and use it to generate a table of values for evaluating limits as x approaches a certain value. He points out the limitations of the table's accuracy due to rounding errors and introduces an alternative method using function notation for more precise results. This method involves entering the function into the calculator and then inputting various x-values to find the corresponding y-values, which helps in determining the limit as x approaches a specific number. The instructor also discusses the importance of the function being continuous and increasing for the accuracy of the limit values.
📘 Understanding the Limit Process and Accuracy
In the final paragraph, Mr. Bean focuses on the process of determining the limit as x approaches a value, using the cosine function as an example. He explains the significance of the function's continuity and increasing nature, which ensures that the graph does not dip unexpectedly. The instructor then demonstrates how to use a calculator to find the cosine of a specific angle, emphasizing the importance of using radians instead of degrees in calculus. He also addresses the requirement for answers to be rounded or truncated to three decimal places, as per the AP exam standards, and provides guidance on how to present these answers correctly.
Mindmap
Keywords
💡Calculus
💡Limits
💡Calculator
💡Graphing Calculator
💡Table of Values
💡Function
💡Continuous Function
💡Increasing Function
💡Function Notation
💡Trigonometric Functions
💡Radians
💡Rounding and Truncating
Highlights
Introduction to the lesson on finding limits in calculus with Mr. Bean.
Emphasis on the necessity of a calculator for this lesson, with a preference for graphing calculators.
Explanation of limits using a table of numbers and the concept of X approaching a value.
Demonstration of how to represent limits graphically and through a table of values.
Example of determining the limit as X approaches negative 4 from a table of values.
Introduction of a TI-84 Plus CE calculator for solving calculus problems.
Instructions on how to use the calculator to fill in a table of values for evaluating limits.
Explanation of the table setup in the calculator to avoid undefined values.
Technique for using the calculator to input exact values for X to find corresponding Y values.
Discussion on the limitations of the table of values due to rounding errors.
Introduction of function notation as an alternative method for more accurate calculations.
Demonstration of using function notation in the calculator for precise limit evaluation.
Clarification on the importance of function continuity and increasing nature for limit evaluation.
Example problem using function notation to find the limit as X approaches negative two.
Emphasis on the accuracy of the Y values approaching the limit from both sides.
Explanation of how to use the calculator to find the cosine of a value for a limit problem.
Instruction on the importance of using radians instead of degrees in calculus calculations.
Guidance on how to round or truncate answers to three decimal places for AP exam requirements.
Conclusion of the lesson with a reminder to practice for the mastery check.
Transcripts
hello and welcome back to another lesson
in calculus this is mr. bean and today
we're gonna talk about finding limits
but just looking at a bunch of numbers
in side a table I need to warn you that
you will need a calculator for this
lesson so if you don't have one you
might want to grab one I'll warn you
again when we get to that point but
graphing calculators are the best for
this but I'll talk about that in just a
little bit so let's remember what we
were doing before when we were talking
about limits we would just have X
approach a value of 3 from both sides
and then the limit would be a 4 ok not
very difficult that's what we've been
doing and I want to show you here that
is a table if we just had a bunch of
numbers so we had this function and we
were plugging in X values really really
close to 3 you can see here that if this
in between here was the 3 and again a
limit doesn't matter what the y-value is
at x equals 3 we're just talking about
getting really close to it so I could
plug in something that's very close to 3
on both sides of so you can see this
table it's 2.9 get even closer to 0.99
we could even get closer than that 2
point 9 9 9 9 9 really really close to 3
plug that into the function and it would
spit out a y-value that is what the Y
value is approaching on this limit so
you can see this is the graphical
representation and this would be a table
of values represent representing the
exact same thing so here is a table if
we're answer the question as X
approaches negative 4 what is the limit
so here X is getting really really close
to negative 4 so negative 4 would be in
between these and you can see the Y
value here it is approaching 2.5 so that
is the limit as X approaches negative 4
just from this table of values that the
Y value is approaching okay pretty
simple not earth-shattering stuff here
that we're doing now I'm going to show
you some tricks with a calculator I'm
going to use a ti-84 plus C E and it's
not required that you personally have a
ti-84 but that's just what I'm going to
be using so anything that we don't would
that we do in our lessons if you don't
know how to do it you're going to be
expected to look it up whether you go on
Google or YouTube or something and find
somebody giving examples of how to use
your calculator or you just read
the user manual you can't expect your
teachers to know how to do every single
type of calculator so here are is our
first problem we're gonna use calculator
for so we have our function or this
crazy thing and we're gonna fill in a
table of values to help us evaluate at
the limit as X approaches negative 2 so
the first thing I'll do is in your table
let's just type in a negative 2 right
here and then we'll do some values that
are close to it so how about negative
two point one and then negative two
point zero zero one and then as we go on
the other side of negative two that
would be really close to negative two
would be negative one point nine nine
nine and then not quite as close below
negative one point nine okay so to save
ourselves the trouble of typing this
thing into the function into this x over
and over again a Kraft graphing
calculator can help us do this a little
faster so I'm going to show you two ways
to do this the first is with a table of
values so we pull my calculator over
here there we go so some of you may have
seen this before we're gonna first go to
y equals right here and you plug in the
function so I've already done that I've
got it pre-loaded so go ahead and pause
the video now and type that in if you
don't have it then we're going to look
at the table so there's two things about
the table this is the table here or we
have table set up if we just go straight
to the table you might have some numbers
here different than mine and you can see
all these X values I could even go up
and you've got a whole bunch of them oh
look at that negative two is an error so
I know that line right there will be an
error undefined which makes sense cuz if
you plug in the negative two that
doesn't work but we want to have these
exact values so the way you do that is
you go to table set up and that's in
blue let me show you that again it's a
table set up is in blue for this window
button so you have to hit the second
blue button first and then the table set
up we want this independent is our X's
instead of it being on auto we're going
to shift it over to ask that's all you
have to change now when you go back to
second and then the table here above the
graph button now it lets us plug in
anything we want negative two point one
hit enter and voila the Y value appears
twenty two point zero one
now let's go ahead and do the next value
for a table negative two point zero zero
one
there we go hit enter and there's our
next y value negative two hit enter
it'll give us an error message that's
what we wanted
the next number was negative one point
nine nine nine hit enter and then a
negative one point nine hit enter okay
so I'm gonna grab this screenshot and
drag it over here so I can have that on
my screen and remember it alright so
then I can go ahead and take these
values and plug them into the table and
there's my table all nice and pretty so
now you've got these here I have on here
that the table values are not as
accurate that was nice though I didn't
have to manually plug in all these X's
over and over again to this crazy
fraction in a calculator and hit enter
each time man that would just take
forever so this really did speed it up
but this has a problem this table of
values you see how wide this column is
your numbers can only be as white as the
column so it's going to have a rounding
error for some of them it won't go very
many decimal places like if your number
was really large you wouldn't even see
the decimal so that's the problem with
the table so I'm gonna show you another
way takes just a little longer but it is
more accurate and that is let's get rid
of this that is with the function
notation so let's go back to our
calculator and we're gonna get out of
this screen so and see this quit button
I'll hit second quit and if you'd have
anything on there just take clear and so
you clear everything off okay so here's
how the function notation works I love
this trick variables button see this
right here
variables we're gonna use this a lot
this year if you go over to the Y
variables and then the function the very
first option so number one or just enter
we entered our function into y1 so I'm
gonna hit enter y1 and now I just open
my parentheses type a negative two point
one because over here that was my first
one close the parentheses and this is
function notation it's just saying take
the function y1 and just use negative
two point one as your input value and
boom it spits that out now instead of
having to retype that you can go second
watch this trick second enter see in
blue it says entry second enter just
brings it up again and then I just
scroll over here and change it so let's
change it to negative two point zero
zero-one close my parentheses hit enter
and now you can see this is more
accurate than what I had from my table
it went out a few more decimal places
and what wouldn't matter for the problem
that we're doing here but for other
problems that might make a difference in
fact it will make a difference for later
in the year and then let's just do one
more so you can see so I'm gonna hit
second enter what if I just did the
number negative two so I'm gonna delete
these things here if I sit negative to
see it says undefined for my table error
dividing by zero why am i dividing by
zero because it says X plus 2 on bottom
and if back you can say go to and it
brings you right back to where you
entered it it doesn't like plugging in a
negative two to eight it nominator okay
so that's function notation very useful
for what we'll be doing later this year
all right back to our problem then so
the limit as X approaches negative two
if you look here at the Y values what is
the Y value getting closer and closer to
when as we surround a negative two that
is a y-value of twenty-one so that is
the limit as X approaches negative two
and notice I had both left side of
negative two and the right side of
negative two but they are both
approaching twenty-one so that's how I
can confirm since they're both
approaching the same number okay now the
last problem for this one it says the
function f is continuous and increasing
continuous just means there's never a
break in the graph it's all connected
increasing just meaning it's always
going up and that might not be a
straight line whoops
it might not be a straight line it might
be curved but the thing is always going
up increasing for all X values that are
larger greater than or equal to one okay
so why is that important it's just so
that I can see here my Y values they're
always going up it's never going to dip
down that's an important thing because
what if between four point eight five
and four point nine nine to the graph
like went down and then back up to four
point nine nine we don't know that so
this little statement just helps us know
that it's always going up in Y values
okay so what does this crazy thing mean
it just means the limit as X approaches
two is basically saying what is the
cosine of f of X what's f of X
approaching well as X gets really really
close to two f of X is getting really
really
two-five so this is just saying what is
cosine of five and that's going to use a
calculator so when you plug that in a
calculator and make sure this is
important to make sure when you type the
words let me grab my code and bring it
back over sine cosine or tangent just as
a reminder the mode if I click on mode
should always be radians right they're
not degree you want radiance in calculus
physics often those degrees we want
radians so let's quit out of that so I'm
gonna do cosine of the number five and
hit enter
drag that over to like how far do I go
when do I stop do I have to write the
whole thing out I mean that would take
forever if I keep having to write that
whole thing out so there's my answer
let's get rid of this now so what in the
world do we do with that the 8p exam
requires you to have three decimals let
me say that again because I'm gonna
repeat that a hundred times this year
and kids are still gonna miss problems
on tests because they don't go three
decimals three decimals please please
three decimals so here's how you can do
this you can do one of two things you
can round it around an answer so if we
round the answer would be zero point two
eight four okay that's simple enough or
you can truncate and truncating that's
besides the fact that that just looks
like a cool word truncating is just
writing three decimals so I'm gonna go
zero point two eight three and then you
stop okay we're not talking about
significant digits from physik excuse me
from science classes we're just talking
about a truncating is just right three
decimals and stop so basically what an
AP reader what the graders will do when
they're looking at your problems if
you've written this whole long thing
zero point two eight three six six six
six two six they just are trained to
look at three decimals they go one two
three and they ignore everything after
that they just read the three decimal
places so there's no reason to write
more so just get used to either
truncating 0.283 or rounding point two
eight four okay so one or the other it
will work for this problem all right and
then that's everything that's it for
this lesson
good luck on the lesson and packet walk
that mastery check and I'll see you back
in the next lesson
[Music]
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