AP Calculus AB — The Squeeze Theorem, Continuity, and the Intermediate Value Theorem
Corrected lesson transcript


## 0:00 — The Squeeze Theorem, Continuity, and IVT


## 0:05 — The Squeeze Theorem

[0:05] Okay, what's the first thing we always do?
[0:09] Direct substitution.
[0:14] Okay, zero squared times sine of, what does one over zero go to?
[0:30] Any number in terms of limits over zero goes to either positive or negative infinity.
[0:41] We don't know until we do it from the left or from the right.
[0:47] Let's just call it infinity.
[0:50] And what does sine of infinity or sine of a very large number do?
[0:56] Go to.
[0:58] What's that?
[1:02] Let's, uh, so this goes to one of these.
[1:12] What does sine do as x gets larger and larger and larger?
[1:17] Think of the sine function, what does it do?
[1:20] It just keeps going, oscillating.
[1:22] It does not approach a number.
[1:25] Okay?
[1:26] Since sine does not approach a number, this does not exist.
[1:34] So zero times does not exist, I have no idea what that means.
[1:39] Okay, so the function value definitely doesn't exist zero, there's a hole there, or there's
[1:43] There may be a hole or some other behavior at zero; direct substitution has not settled it.
[1:49] Okay, so anyways, direct substitution does not work.
[1:53] So,
[1:56] we need another trick.
[1:59] It's one of our limit laws. Why don't you take out that limit laws.
[2:03] Okay, the very last one is called the squeeze theorem.
[2:33] And
[2:34] has two hypotheses.
[2:38] Let me look at what a hypothesis is in math.
[2:42] So, most theorems in math are an if
[2:47] something is true, then
[2:50] something is true. So all the things that must be true are called the hypotheses.
[2:59] You don't have to write that down.
[3:01] Okay, so this has two hypotheses, two things that have to be true for the conclusion to
[3:06] be true.
[3:08] Okay, so first off, we have to have three functions.
[3:18] The middle function must be always greater than some function and always less than some
[3:22] other function.
[3:24] and the limit as x approaches some number of this one and this one must equal the same
[3:30] number.
[3:33] So to use the squeeze theorem, we have to take our function and make sure it's bound
[3:41] just oscillates back and forth, they're always bound between two other functions.
[3:49] What's the largest, no matter what x is, what's the largest sine of some angle will be?
[3:55] 1.
[3:57] Right, so here's the graph.
[3:59] Is this right?
[4:00] The maximum value here is at 1, and the minimum value is negative 1.
[4:06] So, this function, which is y equals 1 and y equals negative 1, it's bound between those.
[4:27] So we can always write this inequality.
[4:37] This is always less than or equal to one and always greater than or equal to negative one.
[4:46] The expression is at most one and at least negative one.
[4:50] You can have all kinds of functions inside the sine function.
[4:53] When we have sine of something we always know it's bound between negative one and positive one.
[4:59] And the only thing we're missing is this guy right here, okay, so we take this
[5:19] And we multiply all three sides by x squared.
[5:33] We multiply an inequality by a function or a number.
[5:37] When do we reverse the sign?
[5:42] When do we reverse the inequality?
[5:44] If it's negative.
[5:45] will x squared ever be negative?
[5:48] No, so we don't need to do that.
[5:50] If we did, if we just multiply by x, for example,
[5:54] this will still work out, but you have to do separate cases,
[5:58] but let's not worry about that.
[5:59] Okay, so now we got this inequality.
[6:01] And now this is that middle function here.
[6:07] And now we simply take the limit as x approaches zero of all three sides.
[6:15] We go the limit as x approaches zero.
[6:18] The limit as x approaches zero. The limit as x approaches zero.
[6:25] And what is the limit as x approaches zero of negative x squared?
[6:30] Zero.
[6:31] Just direct substitution.
[6:33] It's a continuous polynomial.
[6:35] That's zero.
[6:37] The limit as x approaches zero of x squared?
[6:41] Zero direct substitution.
[6:48] So this limit must be less than or equal to zero and greater than or equal to zero.
[6:54] So what must this limit equal?
[6:56] Zero.
[6:58] So we often write by the squeeze theorem.
[7:08] The limit as x approaches 0 of x squared sine of 1 over x equals 0.
[7:26] And when do we use the squeeze theorem?
[7:29] only special cases typically involving a sine or a cosine.
[7:38] Because we can always bound sine between one and negative one, same with cosine.
[7:47] Does that make sense?
[7:48] You just have to...
[7:50] How do I know to do that?
[7:51] Well, if everything else doesn't work, try it.
[7:55] And typically it involves a trig function. Let me graph these functions.
[8:02] x squared.
[8:13] Oh, geez, what did I do? To the power of sine of x?
[8:16] That's kind of cool. That's not the function.
[8:23] There it is.
[8:27] The function x squared times sine of one over x is always bounded between those two functions.
[8:33] If I zoom out.
[8:41] If you zoom in you see green functions are functions.
[8:48] You get squeezed in from both sides.
[8:51] That's why we call this squeeze theorem.
[8:53] sometimes called referred to as the sandwich theorem.
[8:56] That function's a sandwich between two functions.
[9:03] But there it is.
[9:07] Okay, so that's the squeeze theorem.

## 9:10 — Continuity at a Point and on an Interval

[9:10] Is this function continuous or over the numbers?
[9:14] No. No. That's discontinuous x equals four. Is it continuous x equals seven? Yes. How
[9:28] last year or the year before, how do we define what it meant to be continuous?
[9:40] There's no break in the function.
[9:43] Some of your teachers might have said if you can write the whole function without lifting
[9:48] up your pencil, it's continuous.
[9:51] That's not really a good definition.
[9:53] So we have a mathematical definition.
[9:56] And they didn't define it for you before because you didn't know what a limit was.
[10:03] So, the first thing we are going to do is define what it means to be continuous at a
[10:07] number.
[10:10] And here is the definition.
[10:17] Then we will all explain.
[10:18] I want you to write it down on the off-sides.
[10:19] Okay, so the definition, a function is continuous
[10:25] at some number, not on a set of numbers.
[10:29] At a single point, if, has to be defined,
[10:35] the limit must exist as x approaches that number.
[10:39] Really is all we care about.
[10:43] The limit as x approaches that number
[10:46] must equal the function value.
[10:47] When the limit equals the function value, the function is continuous at that value.
[10:57] It is continuous at that value.
[11:04] So here
[11:06] is continuous at 7 because the limit
[11:09] as X approaches 7 equals F of 7.
[11:19] It's
[11:19] discontinuous at 4 because the limit
[11:23] as X approaches 4 does not exist.
[11:27] So it definitely cannot be continuous.
[11:35] And then the next thing is it's continuous on an interval.
[11:40] What's an interval?
[11:43] It's a set of numbers between two numbers.
[11:48] We usually use interval notation.
[11:51] This case on an open interval.
[11:54] if it's continuous at every value on that interval.
[12:01] If it's continuous over all real numbers,
[12:03] we say it's continuous everywhere.
[12:09] So this function is continuous on this open interval
[12:14] and on this open interval.
[12:20] We say it's continuous on this set of numbers.
[12:24] It's continuous on every value inside this open interval, every value inside this open interval.

## 12:29 — One-Sided and Endpoint Continuity

[12:29] What is the limit as x approaches 4 from the left?
[12:36] Let's call this 5.
[12:39] The limit as x approaches 4 from the left? 5. And what is f of 4? 5.
[12:52] Okay, if these are the same numbers, the function is definitely discontinuous at 4,
[12:59] however we say it's continuous from the left, because we went from the left.
[13:08] So it is continuous from the left because the function value equals that limit from
[13:14] the left.
[13:15] It is discontinuous from the right because the limit exists from the right but is not
[13:19] equal to the function value.
[13:21] So the definition being continuous from the left to the right is what I just said.
[13:32] No, that's not it.
[13:35] The function is continuous from the left at x equals a if the limit as x approaches a from the left equals f of a.
[13:40] f of x continuous on the left if the limit as x approaches at x equals a if f of x.
[14:01] equals f of a.
[14:05] f of a.
[14:16] The function is continuous from the right at x equals a if the limit as x approaches a from the right equals f of a.
[14:22] from the right
[14:29] at x equals a if the limit as x approaches a from the right
[14:40] equals F of.
[14:42] Let's call this fx.
[15:04] What is the limit as x approaches negative 2 of f of x?
[15:19] as x approaches 2, is it equal to 1?
[15:25] No. The answer is no. Why not?
[15:29] Because this limit
[15:31] as x approaches negative 2
[15:34] from the left does not exist.
[15:39] You can't approach it from the left.
[15:42] So this does not exist.
[15:49] The limit as x approaches negative 2 from the right,
[15:54] what does that equal? One.
[15:57] Okay.
[16:03] And what is the limit as x approaches
[16:06] 4 from the left of f?
[16:11] 5. The limit as x approaches 4 from the right does not exist.
[16:20] And the limit as x approaches 4.
[16:25] Same as x.
[16:28] Does not exist.
[16:34] OK.
[16:42] So is this function discontinuous at negative 2?
[16:50] It's a tricky question.
[16:53] How about a zero?
[16:54] Is it continuously zero?
[16:56] Is it continuous on every, or on this open interval?
[17:04] For sure.
[17:06] Ok, so, it's discontinuous at negative 2 because the limit, this limit does not equal f of
[17:22] negative 2.
[17:26] However, that's an end point, right?
[17:28] We have two end points.
[17:31] So, we define what it means to be continuous.
[17:34] We actually say this is continuous on the closed interval, even though it's not continuous
[17:42] at negative two.
[17:44] But we say it's continuous on the closed interval if the left endpoint is continuous from the
[17:54] Where do we have?
[17:57] If it's continuous from the right of the left endpoint, and if it's continuous from the
[18:04] left, the right endpoint, we say it's continuous on the closed interval.
[18:13] Okay, so this is the actual, so a function is continuous on the closed interval a to
[18:23] b.
[18:25] It has to be continuous everywhere inside the interval.
[18:32] And the limit as x approaches a, the left endpoint, from the right is equal to the function
[18:39] and B, the right endpoint, continuous from the left.
[18:46] Let me say it's continuous on that closed interval.
[18:50] Why do endpoint conditions matter?
[18:57] Well...
[19:03] Are we done, right?
[19:22] This is the square root of x, right?
[19:28] Is it continuous at x equals zero?
[19:30] No, because the limit as x approaches zero from the left doesn't exist.
[19:39] However, what interval is it continuous on?
[19:51] including zero because it's a left endpoint and as x approaches zero from the right does
[20:03] equal the function value.
[20:04] So it is continuous on this closed interval which is not talking about continuous at a
[20:11] point.
[20:13] It's as long as the left endpoint is continuous from the right, the right endpoint is continuous
[20:18] from the left.
[20:20] It's continuous on this interval.

## 20:21 — Properties of Continuous Functions

[20:21] So what is the domain of this?
[20:29] Including zero, correct?
[20:32] Okay, so this is continuous on its domain.
[20:42] So in fact, every function we know
[20:52] is continuous on its domain.
[20:56] Every polynomial, every rational function,
[20:58] every radical function, every trigonometric,
[20:59] every exponential, every logarithmic function is continuous on its domain.
[21:06] If you know the domain, you know where it's continuous.
[21:17] You don't have to write, well, you can just say all functions we know are continuous on its domain.
[21:23] And then we've got a couple rules.
[21:27] Well, not rules, but properties.
[21:32] Any number times a continuous function is continuous.
[21:39] Seven times the square root of x is continuous on its domain.
[21:44] Any number times a continuous function is continuous.
[21:46] The sum or difference of continuous functions is continuous.
[21:53] Let's say g of x is sine of x plus x cubed or something.
[22:02] We know this is continuous because we know this is continuous and this is continuous.
[22:07] And the sum of continuous functions is continuous.
[22:12] The product of continuous functions is continuous.
[22:18] cosine of x times e to the x is continuous function.
[22:24] Trigonometric functions and exponential functions are continuous on their domains.
[22:27] All exponential functions are continuous.
[22:29] The product of continuous functions are continuous.
[22:35] The quotient is continuous is whenever g of any value is not equal to zero.
[22:45] And it won't be continuous there.
[22:48] But in scalar multiple any coefficient times continuous function, sum or difference, product
[22:54] of continuous functions and quotient are continuous.
[22:57] Where is this continuous?
[23:07] Is x to the fifth plus one continuous?
[23:10] No.
[23:12] It's a polynomial for sure continuous everywhere.
[23:17] Is e to the x continuous?
[23:20] Yeah, all exponential functions are continuous everywhere.
[23:25] Will e to the x ever be zero?
[23:31] All exponential functions always give you a positive value.
[23:34] since this will never be zero,
[23:38] this is definitely a continuous function
[23:40] over all real numbers.
[23:43] So we're looking for whether a function is continuous
[23:45] or not, especially with the quotients.
[23:51] You have to check, is this continuous?
[23:53] Where is this continuous?
[23:54] Where is this continuous?
[23:55] And then a special case to quotients,
[23:58] we gotta make sure it's not zero.
[24:00] Whenever the denominator is zero, it's not continuous.
[24:02] It's discontinuous.
[24:07] And then one more property.
[24:12] The composition of continuous functions are continuous.
[24:22] Just write that.
[24:25] The composition of continuous functions are continuous.
[24:27] F composed with G as long as F and G are continuous
[24:32] F of G is continuous
[24:34] Example...
[24:34] I have no idea what this function looks like, but do we know whether it is continuous or
[24:44] not?
[24:47] Is sine of x continuous over all real numbers?
[24:51] Yes, is 2 to the x continuous over all real numbers? Yes.
[24:56] So I plugged in sine of x
[24:58] into this function
[25:01] with guarantees continuous over all real numbers.
[25:05] Okay.

## 25:05 — Piecewise Continuity

[25:05] Okay, let's do some examples.
[25:08] Let's see how far we go.
[25:11] I think we did
[25:13] a similar example.
[25:15] Let's do this.
[25:25] Okay.
[25:27] Actually, let me do this plus one.
[25:33] And then, e to the x, get the x to the first one.
[25:41] So where, or on what, where is this continuous?
[25:56] So how are we going to do this?
[26:03] So, we have to check, so any piecewise functions, we have to check each piece separately.
[26:10] Okay.
[26:13] So if x is greater than or equal, let's go with greater than zero, greater than pi, then
[26:27] f of x equals sine of x, right?
[26:36] Is sine of x continuous?
[26:39] Definitely.
[26:41] OK, so we definitely know it's continuous from pi to infinity.
[26:48] For zero less than or equal to x and x less than pi, f of x equals x squared plus one.
[27:18] Then if x is less than zero, f of x equals e to the x, which is an exponential function,
[27:32] continuous everywhere.
[27:35] We know it's continuous here.
[27:39] So we check each piece. As long as they are continuous, it's continuous on the open interval.
[27:47] So really all we have to do is check at pi and at zero.
[28:03] at x equals zero
[28:09] okay this is where it changes
[28:12] from this function to this function
[28:18] we have to do the limit from the left and the limit from the right separately
[28:21] So what is the limit as x approaches...
[28:28] uh... sorry, let's go to pi first, my bad.
[28:33] pi from the left.
[28:43] So from the left,
[28:45] f of x is equal to this.
[28:48] So this is the limit
[28:49] as x approaches pi from the left of x squared plus one.
[28:56] And what does that equal? Direct substitution of pi squared plus one.
[29:02] Sum number.
[29:07] The limit as x approaches pi from the right.
[29:22] limit as x approaches pi from the right.
[29:28] So f of x is equal to sine of x.
[29:32] which equals sine of pi which equals
[29:44] zero thank you okay
[29:50] so what is the limit as x approaches pi
[29:57] It does not exist.
[30:08] Is this function continuous from the left or the right or neither at pi?
[30:19] What is f of pi?
[30:25] Pi is here, so it's sine of pi, which is zero.
[30:33] So is it continuous from the left or the right?
[30:38] At pi.
[30:40] From the right. So, final with this.
[30:50] Our function looks like this.
[30:53] So it is continuous on the right.
[30:59] What is it?
[31:00] x squared plus one. Looks like that.
[31:03] sorry, open circle here, is discontinuous on the left, and it's definitely discontinuous
[31:09] at pi.
[31:12] Now let's check at zero.
[31:15] The limit as x approaches zero from the left, the limit as x approaches zero from the left,
[31:27] From the left, f of x is right here.
[31:33] x squared plus 1.
[31:41] Direct substitution plus 1.
[31:49] Question?
[31:54] Oh, I'm sorry.
[31:55] I was looking at the wrong thing. This is 0 from the left. This is e to the x. My bad.
[32:11] Sorry, this is e to the x, which is e to the 0, which is 1. So the limit as x approaches
[32:23] 0 from the right, the limit as x approaches 0 from the right of, this was the x squared
[32:33] plus 1, which is 0 squared plus 1, which is 1. So, the limit as x approaches 0 is 1. And
[32:48] f of zero is this guy, zero squared plus one.
[33:04] So it's definitely continuous at zero.
[33:13] Ok, any questions on this?
[33:19] So for piecewise we just gotta check, the main thing we gotta do is check where the
[33:22] two functions meet.
[33:25] And then of course inside you have to check where is the continuous.
[33:27] I chose three continuous functions.

## 33:29 — The Intermediate Value Theorem

[33:29] OK, the intermediate value theorem.
[33:35] Last quick, this is a quick topic.
[33:38] Let me find it.
[33:39] Shh.
[33:40] Okay, write this down and then we'll talk later.
[33:43] Okay.
[33:44] Okay, so this is called an existence theorem.
[33:48] We're gonna have three, sort of four different existence theorems in calculus.
[33:56] All it does is tell you a value exists.
[34:01] It doesn't help us find the value at all.
[34:03] It just guarantees for it to exist.
[34:06] This is an existence theorem: it guarantees a value exists without telling us how to find it.
[34:10] Anybody remember this?
[34:13] But technically we don't really have it till this class.
[34:18] Why is that?
[34:26] Because we don't need it, somebody said.
[34:30] This is not true.
[34:31] We definitely need it.
[34:48] Here we go.
[34:55] What about this has to do with calculus?
[35:00] I don't see anything about limits in here.
[35:05] Those are good.
[35:06] Well, their sort is.
[35:10] OK.
[35:12] The only hypothesis, hypothesis C, is f has to be a continuous function on a closed interval.
[35:27] And truly we didn't define what it means to be continuous till just today.
[35:32] Before your teacher said, well you can draw it without lifting up your pencil, it's continuous.
[35:38] But truly right, it's continuous on a closed interval, means the limit as x approaches
[35:47] some other value, let me call it C,
[35:51] Continuity on the closed interval includes continuity from the right at a and from the left at b.
[35:58] for every value on the open interval.
[36:01] And it's continuous from the left at b,
[36:05] and it's continuous from the right at a.
[36:07] Okay, we now know the actual definition.
[36:10] So that's why it's part of this class, okay?
[36:14] But all this thing says is, as long as you have a continuous function, let's put in some
[36:28] numbers for class one and eight or something, two and ten.
[36:40] as long as we have a continuous function, let n be any number between f of a and f of
[36:49] b. What is f of b? 10. f of a is 2.
[37:01] n can be any number between 10 and 2. Any number here.
[37:16] And then we are guaranteed, actually as long as f of a is not equal f of b, that's the
[37:24] one thing, it's definitely not equal.
[37:27] There must exist a value c, there must be a c for any number, let's call this number
[37:35] n there must be a C in the open interval such that F of C equals n. So for example,
[37:58] 5, are we guaranteed to have a value between 1 and 8?
[38:05] Yes, we are guaranteed.
[38:06] Because 5 is between 10 and 2, we are guaranteed a value in 1 to 8, the open interval.
[38:18] such an F of C equals five.
[38:21] Our function takes on every single value
[38:24] and sometimes it can take on it twice, right?
[38:26] So in fact, there are three different C values,
[38:31] C one, C two, C three,
[38:34] but this theorem intermediate value theorem
[38:36] just guarantees there's at least one.
[38:40] When is this used?
[38:42] And typically something like this.

## 38:54 — IVT Example: Prove a Zero Exists

[38:54] Show where is a
[39:04] Show that f has a zero between zero and two.
[39:28] Now what's a zero of a function?
[39:34] A zero of a function is an x value.
[39:38] Okay, a zero is an x value that makes f of x equal to zero.
[39:57] So to show that, we're going to use the intermediate value term.
[40:01] Since f is a polynomial, it is continuous on the closed interval from zero to two.
[40:12] And why is that?
[40:20] How do we know this is continuous on the closed interval?
[40:25] because it's a polynomial, are continuous everywhere.
[40:41] And what is f of zero equal to?
[40:47] Zero cubed minus four, negative four.
[40:50] F of 2 is 2 cubed minus 4 is 4.
[41:01] By the Intermediate Value Theorem, there exists a c in the open interval from zero to two.
[41:26] the open interval such that
[41:34] f of c equals zero because zero
[41:38] is between negative four
[41:43] and four
[41:53] This function looks something like this, right?
[42:01] This was negative four.
[42:09] This was positive four.
[42:12] Our function since it's continuous it takes on every single value here.
[42:19] And this would be our value of C right there.
[42:22] Such a F of C is zero.
[42:24] Okay.
[42:25] We're only going to use it a few times but sometimes we're going to have to mention the
[42:29] intermediate value theorem.
[42:31] And so on.
[42:32] Okay we're going to stop here.