AP Calculus AB — Graphing the Derivative
Corrected lesson transcript


## 0:00 — Graphing the Derivative

[0:05] Okay, so today we're talking about, given a function, how do we graph a derivative?
[0:13] This is what we did last class, on Wednesday.
[0:18] We're given a function, and we found the derivative.
[0:23] And how do we find the derivative?
[0:27] by the definition, right, which is the limit as h goes to zero.
[0:34] I worked this one out for us and it ends up with this, okay, which is another function, okay.
[0:45] So this is a quadratic function. The derivative turned out to be a linear function.
[0:51] The derivative is the slope of the tangent line.
[0:56] We talked about this.
[0:57] A slope is a number, not a function.
[1:02] So how is this still the slope of the tangent line?
[1:10] It depends on the value of x.
[1:11] Every single x value, the function has a different slope.
[1:16] So you want any slope at any particular point.
[1:17] you plug in an x value okay so the derivative is a whole nother function
[1:26] This was a quadratic, and the derivative happens to be linear. Given a graph, we will attempt to graph its derivative.
[1:33] keep giving a graph of some function and we're gonna attempt to graph the
[1:37] derivative so here are those two functions the quadratic and then the
[1:45] derivatives in red. How are we gonna be given this and come up with this? Why don't you

## 1:56 — Start with Zeros, Signs, and Steepness

[1:57] We always start with the easiest value we know.
[2:00] Zero.
[2:01] And that's when the slope of the tangent line is zero.
[2:04] Right here.
[2:06] Anytime you have a minimum or a maximum, when it's a smooth,
[2:10] OK, the tangent line is horizontal,
[2:15] and the derivative is zero.
[2:18] So we're going to start with that.
[2:21] And then we find the zeros of the derivative.
[2:26] And then you go to the right and the left.
[2:29] Soon as we go to the right is it a positive slope or negative slope?
[2:33] Positive
[2:35] It's just gonna keep getting larger and larger and larger and larger
[2:39] So I know this is to the right of whatever number. This is one and a half or something
[2:45] It's gonna go up
[2:48] It's not obvious. It's a straight line
[2:51] And for right now it really doesn't matter we know it's positive just keep getting larger
[2:55] And then we go to the left of that zero or where the slope of the tangent line is zero.
[3:01] Look, is that a positive slope or negative?
[3:03] It's negative, it just keeps getting steeper and steeper.
[3:07] And so I know our derivative is a negative number down here, just keep it steeper.

## 3:11 — A Smooth Graph and Polynomial Degree

[3:11] If we wanted to we could estimate the values of the slope plug them in here and then plot
[3:18] it but we almost never do that.
[3:20] Okay so we just first figure out where the derivative is zero how many times is the derivative
[3:26] zero?
[3:30] So when the has a horizontal tangent line we got one right here one right here and one
[3:38] There are three horizontal tangents. Wherever we have a horizontal tangent line, the derivative is zero.
[3:45] zero, the derivative is zero, and the derivative is zero.
[4:00] And then between these two zeros of the derivative, just you can put a pencil or you can do whatever,
[4:08] The tangent line negative or positive
[4:12] The slope of the tangent line is negative throughout this interval.
[4:17] Here the slope is most negative. Then it is still negative, but becomes less steep as it approaches zero.
[4:24] so wherever the
[4:27] Slope is the steepest
[4:29] that's going to be our
[4:31] least slope here and
[4:35] Just put it some it doesn't matter how down
[4:38] If you want to estimate that, I don't know, it's about a slope of negative 2.
[4:42] Maybe I should have put it down here at negative 2.
[4:46] It doesn't really matter.
[4:48] And then we just make a nice smooth curve.
[5:00] And then between this 0 and this 0,
[5:05] Is the slope of the tangent positive or negative?
[5:09] Positive.
[5:10] All the way to the next one.
[5:13] And somewhere right here is the steepest.
[5:19] So I'm going to put some point about here.
[5:37] And notice I didn't draw it very high because this is not a very steep value, especially
[5:42] compared to this.
[5:45] And now to the right of that zero the derivative that that local maximum
[5:51] Slope is positive negative
[5:54] Negative and just keep getting more and more and more and more negative
[5:59] So we just
[6:04] Put some arrow and then to the left of this
[6:09] zero
[6:10] So for the tangent line positive,
[6:12] The slope of the tangent line is positive, and it keeps getting larger.
[6:15] So it just keeps going up and up and up and up and up.
[6:27] And that's it.
[6:28] That's our best guess as to the graph of the derivative.
[6:32] Assuming this original function is a polynomial, what's the least degree this could be?
[6:44] First how about this?
[6:46] Is the degree of this polynomial is an even degree or an odd degree?
[6:51] Even on okay now we most of us agree. Why is he even?
[6:53] The end behavior
[6:56] Both going down to negative infinity.
[6:59] Okay. It's some even degree.
[7:03] Either x squared, x to the fourth, x to the sixth, x to the eighth.
[7:07] So we look at the end behavior. Okay.
[7:10] And then we have one, two, three extrema or turning points.
[7:16] Where it changes from increasing to decreasing.
[7:19] So what does it have to at least be?
[7:22] Oh, could it be quadratic?
[7:26] No, that's what we know a quadratic look like yeah only has one turning point
[7:32] This has three
[7:36] Extrema a local min or max so it has to be at least a fourth degrees
[7:41] We're good that it could be a sixth degree. We can't tell could be an eighth degree
[7:45] but actually at least a fourth degree polynomial.
[7:50] And what about the derivative, the Green function?
[7:54] Is it an odd degree or even degree?
[7:57] Odd.
[8:00] It's definitely not linear.
[8:02] Could it be cubic?
[8:06] It could be cubic.
[8:08] It has one, two turning points or two little,
[8:12] it could be a cubic, okay?
[8:15] And this is always gonna happen with polynomials.
[8:18] Whatever the degree of the polynomials,
[8:20] the derivative is gonna be one degree less.
[8:24] And we'll learn exactly why later.

## 8:26 — Parabolas and a Reciprocal-Type Graph

[8:26] So first we figure out where the slope of the tangent line is zero.
[8:30] horizontal tangent lines which obviously should be here we put a zero
[8:37] we got one and to the right is it negative or positive negative okay it
[8:45] just keeps getting more and more and more and more negative and to the left
[8:52] positive, steep is getting steeper and steeper.
[8:56] It's going to look like this.
[8:59] This is a quadratic,
[9:06] or could it be a fourth degree problem?
[9:08] The derivative of a quadratic is a linear function, so that is why I drew a straight line.
[9:14] Okay.
[9:15] Should have drawn it something like that?
[9:17] Yes.
[9:18] What type of function is this?
[9:23] Definitely not.
[9:24] Why not?
[9:28] Do polynomials have asymptotes?
[9:32] No.
[9:33] Okay.
[9:34] Polynomials are continuous.
[9:36] It's definitely not a polynomial.
[9:38] What type of function is this?
[9:40] Rational, which is the ratio of two polynomials.
[9:47] Are there any parts where the derivative is zero here?
[9:54] No.
[9:56] So we can't start by doing that.
[9:58] So now we just start some other slide.
[10:02] Just pick a value.
[10:02] This would be the tangent line.
[10:05] What's approximately the slope of that line?
[10:09] It's about one.
[10:11] Right at 45 degree angle has a slope of one.
[10:19] Okay.
[10:20] So I'm putting it right there at one.
[10:24] And then when we go to the right, does it have a positive slope or a negative slope?
[10:31] The closer we get to that asymptote, the steeper the slope becomes.
[10:38] So in fact that's going to be the derivative.
[10:43] and then the farther we go to negative infinity that closer the derivative gets
[10:51] The derivative approaches zero farther to the left. It is not the same function as the original graph.
[11:13] And then right here, what's approximately the slope of that line?
[11:18] One, positive one.
[11:21] So we need a dot up here at one.
[11:28] And then when we go to the right, it's still positive but just keeps getting flatter and
[11:36] flatter and flatter.
[11:39] So positive, but gets very, very, very close to zero.
[11:49] And when we go to the left of this x value of two or so, it's positive, but it keeps
[11:54] getting steeper and steeper and steeper and steeper and steeper and steeper.
[12:00] So this goes up, up, up, up, up, up.
[12:05] OK.
[12:10] This is also a rational function.
[12:13] The derivative of rational functions
[12:15] are other rational functions.
[12:25] And we'll see why that is later.
[12:28] We do the derivative.
[12:29] OK, questions on this?

## 12:31 — Endpoints and Restricted Domains

[12:31] We go to the right of the zero it's the steepest right, I don't know about here
[12:39] Okay right here is the steepest
[12:43] So that's gonna be our it's gonna go up to here
[12:47] And then the closer we get whatever this is x equals 5 it starts to level off get close to zero it's never zero
[13:00] And
[13:01] And then what's the slope of the tangent line right here at the end point?
[13:09] Very important.
[13:11] It does not exist.
[13:14] Derivatives do not exist at end points.
[13:16] Why is that?
[13:19] Let's call this number 5.
[13:23] The limit as h approaches...
[13:29] Let's go to the other definition.
[13:32] Approaches 5, f of x minus f of 5, over x minus 5.
[13:44] Why does this limit not exist?
[13:56] The limit as x approaches five from the left does exist.
[14:03] Okay but the limit as x approaches five from the right.
[14:19] Okay, this does not exist there are no values to the right of five
[14:26] Okay, so that's why the limit as X approaches five does not exist endpoints are not differentiable
[14:35] Very important so it's same thing on the left here. This is positive gets steeper
[14:42] And then we got to put an open circle on the end point
[14:46] Problem 42 has one horizontal tangent. To the right, the slope is always negative and becomes steeper.
[14:57] is always negative and just keeps getting steeper. And again if the function ends, there's
[15:11] there it's not differentiable somewhere right here is the steepest
[15:28] so it's gonna go up and then get closer and closer and closer to zero okay do 43
[15:38] Shouldn't get something like that.
[15:46] Okay, if you're not understanding where I, or how I do this, please ask.
[15:56] And again, where I put the steepest point doesn't really matter.
[15:59] I could have drawn it way up here
[16:03] We can't really tell because we don't really have numbers to deal with
[16:07] Okay, we'll talk about this x point in a second.
[16:13] We definitely have a zero here, a zero here, and a zero here.
[16:18] When we go to the right of this point, where's the steepest point?
[16:25] It just keeps getting steeper and steeper and steeper and steeper and steeper until
[16:29] we hit that sharp point.
[16:34] So it's just going to get steeper.
[16:38] Okay I could have drawn it there or I could have drawn it here.
[16:40] It doesn't really matter.

## 16:41 — Corners and One-Sided Slopes

[16:41] Just to the right of that sharp point is the slope positive or negative?
[16:47] Negative.
[16:48] And in fact every single point on that line has the same slope.
[16:53] So we are going to draw a negative, some negative value here.
[16:57] And it is a horizontal line.
[16:59] it's always the same value let's say this is at 7 at x equals 7 that sharp
[17:12] point what's the derivative okay definitely does not exist and why is
[17:18] that okay it's definitely a cusp but what makes it not differentiable that we
[17:29] We need an open circle on each side. Why is the derivative undefined?
[17:38] yes this is exactly right the derp the definition is the limit as let's do the
[17:45] other definition X approaches 7 of the difference formula or the slope of the
[17:53] secant line.
[18:00] And the reason it's not differentiable, the limit from the left is going to equal whatever
[18:06] this number is.
[18:08] Let us say the left-hand slope is 8; these are illustrative values.
[18:10] Let's make that.
[18:12] The limit as X approaches seven from the right
[18:23] is gonna be whatever this number is.
[18:26] I'm gonna just call it negative two.
[18:31] Therefore the limit as X approaches seven.
[18:40] It does not exist because the limit from the left and the limit from the right not of the function but of the
[18:50] We call this the difference quotient
[18:53] We call this the difference quotient, or the slope of the secant line.
[18:55] Are different so it does not exist. So the reason sharp points are not differentiable
[19:03] It's by the definition of the derivative the derivative or the limit for the left and the right are not the same therefore
[19:08] the limit doesn't exist therefore there is no derivative there and it will be
[19:18] important to state that later on. Okay question on that problem is that the

## 19:19 — Cosine Becomes Negative Sine

[19:19] What function is this?
[19:21] Is it a cosine?
[19:24] Looks just like cosine.
[19:27] In fact I believe it is cosine.
[19:32] That's right at 2 pi.
[19:35] Does it have any horizontal tangent lines?
[19:39] Yeah we got lots of them.
[19:44] Right here, right here, right here.
[19:47] So the derivatives are going to have lots of zeros.
[19:49] one here, one here at pi, at 2 pi, 3 pi, negative pi, negative 2 pi.
[20:04] Between zero and pi, where is it the steepest?
[20:11] Exactly here, at pi over 2.
[20:16] In fact, I gave you grid marks. What is the slope of that line?
[20:27] Plus one. Negative one.
[20:30] It goes down one over one. It's exactly negative one.
[20:36] So we go to negative one, which is right here.
[20:45] okay and a nice smooth point and then between pi and 2 pi the tangent line the
[20:56] The derivative looks a lot like a trig function as well.
[21:01] If it were, what would it be?
[21:03] It's definitely not sine.
[21:06] Sine starts at zero and goes up to one at pi over two.
[21:11] But this is the reflection of sine, so it would be negative sine.
[21:30] In fact, it turns out, we're going to show this later on, the derivative of cosine is
[21:34] Exactly negative sine.

## 21:36 — The Exponential That Matches Its Derivative

[21:36] What type of function is this?
[21:41] Looks like an exponential function.
[21:46] What is the base of the exponential function?
[21:50] How do we look at a graph and figure out what the base is?
[21:56] X equals 1, what is the value?
[22:04] X equals 1, it's right here.
[22:11] It's not obvious, but this turns out to be E. What is E?
[22:16] This is our function, e to the x.
[22:29] It has no minimums or maximums.
[22:33] This tangent line will never be zero.
[22:35] So we just got to go to some other point.
[22:39] Let's pick a point, let's say, right here.
[22:44] What is the slope of this line?
[22:50] OK, it looks like it's one.
[22:52] In fact, it is exactly one.
[22:55] I'll put a dot right there at 0 comma 1.
[23:00] OK.
[23:00] And then to the right here, it's definitely positive.
[23:06] And it just keeps getting steeper and steeper and steeper.
[23:11] Okay, and this is not exactly obvious here, but that is the derivative to the right.
[23:23] And to the left is still positive but keeps getting flatter and flatter and flatter and
[23:29] flatter.
[23:33] Ok.
[23:36] And for this function it's not obvious but it turns out the derivative of e to the x
[23:45] is exactly the same function.
[23:47] It is e to the x.
[23:49] Which is kind of crazy and we'll prove that later for you
[23:59] Super cool a
[24:02] Derivatives of exponential functions are exponential functions as well.
[24:06] Sometimes it's exactly the same
[24:08] The other times it'll be just a slight coefficient different have a different coefficient. It'll still be an exponential function

## 24:16 — Absolute Value Creates Corners

[24:16] Let's assume this goes up forever.
[24:21] What type of function is this?
[24:27] Or if I gave you an equation for this...
[24:31] We have to be some piecewise function.
[24:35] OK, that's three different pieces.
[24:41] And we'll talk about what the actual function is in a minute.
[24:47] It does have a horizontal tangent line right here.
[24:51] So the derivative has a 0 right there.
[24:56] The derivative is 0 there.
[24:58] And then to the right here
[25:00] Does have a positive negative slope?
[25:04] negative just keeps getting more and more and more negative and then what happens at the
[25:11] What's the derivative of that sharp point does not exist so
[25:20] It's more and more negative we definitely need to put a hole right there
[25:28] And instantly to the right, does it have a positive slope or negative slope?
[25:35] It's not obvious, but this is getting larger, just barely larger and larger and larger.
[25:40] So from here, it doesn't exist.
[25:50] It looks something like that.
[25:56] To the left here, the positive slope, it keeps me steeper and steeper and steeper.
[26:06] And then right here, whatever value that is, doesn't exist.
[26:16] And then instantly has negative slope
[26:33] Derivatives of piecewise functions are also described piecewise.
[26:41] In fact, this function, the equation is the absolute value of some quadratic.
[26:50] I don't know exactly what quadratic it is.
[26:52] I'm just let me make something up.
[26:55] x squared minus
[27:00] plus
[27:03] three x minus
[27:11] 16 or something like that.
[27:18] Anytime you see an absolute value, it's a piecewise function.
[27:24] This actually was some quadratic.
[27:28] If it was without the absolute value, it'd be exactly this function.
[27:35] But because of the absolute value, it flipped this to this.
[27:41] It's kind of interesting.
[27:50] And again, why is it not differentiable there?
[27:54] Let's say this is at 4.2 or something.
[27:57] The left- and right-hand limits of the difference quotient are not equal.

## 28:02 — Piecewise-Linear Graphs

[28:02] Okay, this is some piecewise function.
[28:05] It does have extreme, there's a maximum here and a maximum there and a minimum there.
[28:12] Is the derivative zero there?
[28:15] Definitely not, it's sharp.
[28:18] Okay, sharp points are not differentiable.
[28:20] The limit from the left of the difference quotient limit from the right are not the
[28:23] same.
[28:24] So let's just choose some other point.
[28:26] Let's say, I don't know, right here.
[28:31] What is the slope of that tangent line there?
[28:37] It's whatever the slope of that line is.
[28:39] What is the slope of that line?
[28:41] One.
[28:43] Okay, in fact it's right there.
[28:45] Everywhere on this point has the same exact slope of the tangent line.
[28:49] The slope of a tangent line of a line is always the slope of the line.
[28:52] So everywhere here, it's going to have a slope of one
[28:59] Sharp points are not differentiable open circle
[29:08] So that's this piece here the slope of this line is one
[29:15] Between zero and five the slope of that line is negative one
[29:19] So it's gonna be like this.
[29:27] And one again.

## 29:31 — Cube Roots and Vertical Tangents

[29:31] This is y equals the cube root of x.
[29:40] It's the continuous function everywhere.
[29:45] Domain's all real numbers.
[29:53] It has no minimum or maximum, right?
[29:56] So the derivative is going to have no zeros.
[29:58] I don't know, somewhere right about here, the slope of the tangent line is one.
[30:07] So I'm going to put a dot here.
[30:09] Ok, we have a positive slope but it just keeps decreasing and decreasing and decreasing.
[30:15] So it's going to look something like this.
[30:19] And have a horizontal asymptote.
[30:24] What happens when I get closer to zero?
[30:28] It's steeper and steeper and steeper.
[30:30] And right here at zero
[30:33] The tangent line is vertical
[30:38] What's the slope of vertical lines
[30:51] Vertical lines don't have slopes
[30:54] Okay like
[30:57] x equals 2 is a vertical line. It does not have a slope.
[31:02] Vertical lines do not have slopes.
[31:06] So right here the derivative does not exist
[31:09] because we have a vertical tangent line.
[31:12] And the closer you get to 0,
[31:16] This has a vertical asymptote, the derivative.
[31:21] And then just to the left of zero, it has a positive, very very very steep slope.
[31:29] It's going to look something like this.
[31:31] It has...
[31:39] At this vertical tangent, the derivative does not exist; its graph has a vertical asymptote.
[31:46] vertical asymptote
[31:48] Okay, it might go to positive, it might go to negative, they both...

## 31:51 — Sketching a Semicircle Derivative

[31:51] which is a semi-circle.
[32:02] Let's say this is the semi-circle.
[32:06] We definitely have a zero.
[32:11] And the slope here is negative.
[32:14] We're going to have a vertical asymptote here.
[32:17] It's going to look like this.
[32:19] Approaching the endpoint, the derivative does not exist at the endpoint, and the slopes become steeper and steeper.
[32:25] and steeper and steeper.
[32:26] It goes all the way to negative infinity.
[32:30] And this will go all the way to positive infinity.
[32:33] But, okay, yeah, so that makes sure when you draw the graph of a semicircle, the closer
[32:39] we get to this value, the steeper it gets.
[32:41] You're gonna have vertical asymptotes at those two points.