Calculus AB/BC – 1.4 Estimating Limit Values from Tables

Summary

In 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

00:00

📚 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.

05:01

🔍 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.

10:02

📘 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.

Keywords

💡Calculus

Calculus is a branch of mathematics that deals with rates of change and accumulations. It is the foundation for understanding limits, derivatives, and integrals. In the video, Mr. Bean introduces a lesson on finding limits, which is a fundamental concept in calculus, essential for understanding how functions behave as their input values approach a certain point.

💡Limits

In calculus, a limit is the value that a function or sequence approaches as the input or index approaches some value. The script discusses how to find limits by looking at the behavior of a function as the variable 'x' approaches a certain number, such as 3 or -4, from both sides, which is a core concept in understanding continuity and differentiability.

💡Calculator

A calculator is an electronic device used to perform arithmetic operations and more complex calculations. In the context of the video, Mr. Bean emphasizes the importance of having a calculator for the lesson, particularly a graphing calculator, to help students find limits by evaluating the function at values close to the point of interest.

💡Graphing Calculator

A graphing calculator is a type of calculator that can be used to graph functions and equations. It is particularly useful in calculus for visualizing how functions behave as their inputs approach certain values. The script mentions that graphing calculators are the best for the lesson, indicating their utility in understanding limits visually.

💡Table of Values

A table of values is a method used to organize and display the relationship between two variables, typically the input and output of a function. In the script, Mr. Bean uses a table of values to demonstrate how the output of a function changes as the input approaches a specific value, which helps in determining the limit.

💡Function

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. The video script refers to a 'function' as the mathematical expression that students are analyzing to find its behavior as 'x' approaches certain values.

💡Continuous Function

A continuous function is one where there are no breaks or jumps in the graph, meaning it can be drawn without lifting the pen from the paper. In the script, Mr. Bean mentions that the function 'f' is continuous and increasing, which is important for understanding how the function behaves as 'x' approaches a certain value without any unexpected dips or jumps.

💡Increasing Function

An increasing function is one where the output values increase as the input values increase. In the context of the video, Mr. Bean states that the function is increasing for all 'x' values greater than or equal to one, which helps in understanding the direction of the function's behavior as 'x' approaches a certain point.

💡Function Notation

Function notation is a way of writing functions where the function name, often 'f', is followed by parentheses containing the input value. In the script, Mr. Bean demonstrates using function notation on a calculator to find the value of the function at specific points, which is a more precise method than using a table of values.

💡Trigonometric Functions

Trigonometric functions are functions of an angle, including sine, cosine, and tangent, which relate the angles of a triangle to the lengths of its sides. In the video, Mr. Bean uses the cosine function to find the limit of a function as 'x' approaches two, which involves calculating the cosine of a numerical value.

💡Radians

Radians are a unit of angular measure used in trigonometry and calculus, where one radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. The script mentions that the calculator should be in radians mode when working with trigonometric functions, which is the standard for calculus problems.

💡Rounding and Truncating

Rounding is the process of adjusting a number to a certain number of decimal places, while truncating is the process of cutting off the digits beyond a certain point without rounding. In the video, Mr. Bean instructs students to either round or truncate their answers to three decimal places for the 8p exam, which is a specific requirement for that test.

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]