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Okay, so today we're talking about, given a function, how do we graph a derivative?

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This is what we did last class, on Wednesday.

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We're given a function, and we found the derivative.

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And how do we find the derivative?

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by the definition, right, which is the limit as h goes to zero.

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I worked this one out for us and it ends up with this, okay, which is another

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function, okay.

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So this is a quadratic function. The derivative turned out to be a linear function.

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The derivative is the slope of the tangent line.

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We talked about this.

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A slope is a number, not a function.

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So how is this still the slope of the tangent line?

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It depends on the value of x.

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Every single x value, the function has a different slope.

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So you want any slope at any particular point.

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you plug in an x value okay so the derivative is a whole nother function

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This was a quadratic, and the derivative happens to be linear. Given a graph, we

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will attempt to graph its derivative.

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keep giving a graph of some function and we're gonna attempt to graph the

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derivative so here are those two functions the quadratic and then the

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derivatives in red. How are we gonna be given this and come up with this? Why don't

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you

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We always start with the easiest value we know.

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Zero.

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And that's when the slope of the tangent line is zero.

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Right here.

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Anytime you have a minimum or a maximum, when it's a smooth,

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OK, the tangent line is horizontal,

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and the derivative is zero.

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So we're going to start with that.

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And then we find the zeros of the derivative.

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And then you go to the right and the left.

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Soon as we go to the right is it a positive slope or negative slope?

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Positive

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It's just gonna keep getting larger and larger and larger and larger

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So I know this is to the right of whatever number. This is one and a half or

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something

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It's gonna go up

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It's not obvious. It's a straight line

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And for right now it really doesn't matter we know it's positive just keep getting

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larger

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And then we go to the left of that zero or where the slope of the tangent line is

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zero.

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Look, is that a positive slope or negative?

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It's negative, it just keeps getting steeper and steeper.

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And so I know our derivative is a negative number down here, just keep it steeper.

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If we wanted to we could estimate the values of the slope plug them in here and then

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plot

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it but we almost never do that.

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Okay so we just first figure out where the derivative is zero how many times is the

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derivative

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zero?

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So when the has a horizontal tangent line we got one right here one right here and

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one

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There are three horizontal tangents. Wherever we have a horizontal tangent line, the

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derivative is zero.

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zero, the derivative is zero, and the derivative is zero.

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And then between these two zeros of the derivative, just you can put a pencil or you

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can do whatever,

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The tangent line negative or positive

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The slope of the tangent line is negative throughout this interval.

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Here the slope is most negative. Then it is still negative, but becomes less steep

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as it approaches zero.

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so wherever the

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Slope is the steepest

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that's going to be our

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least slope here and

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Just put it some it doesn't matter how down

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If you want to estimate that, I don't know, it's about a slope of negative 2.

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Maybe I should have put it down here at negative 2.

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It doesn't really matter.

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And then we just make a nice smooth curve.

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And then between this 0 and this 0,

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Is the slope of the tangent positive or negative?

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Positive.

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All the way to the next one.

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And somewhere right here is the steepest.

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So I'm going to put some point about here.

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And notice I didn't draw it very high because this is not a very steep value,

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especially

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compared to this.

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And now to the right of that zero the derivative that that local maximum

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Slope is positive negative

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Negative and just keep getting more and more and more and more negative

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So we just

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Put some arrow and then to the left of this

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zero

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So for the tangent line positive,

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The slope of the tangent line is positive, and it keeps getting larger.

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So it just keeps going up and up and up and up and up.

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And that's it.

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That's our best guess as to the graph of the derivative.

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Assuming this original function is a polynomial, what's the least degree this could

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be?

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First how about this?

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Is the degree of this polynomial is an even degree or an odd degree?

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Even on okay now we most of us agree. Why is he even?

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The end behavior

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Both going down to negative infinity.

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Okay. It's some even degree.

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Either x squared, x to the fourth, x to the sixth, x to the eighth.

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So we look at the end behavior. Okay.

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And then we have one, two, three extrema or turning points.

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Where it changes from increasing to decreasing.

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So what does it have to at least be?

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Oh, could it be quadratic?

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No, that's what we know a quadratic look like yeah only has one turning point

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This has three

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Extrema a local min or max so it has to be at least a fourth degrees

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We're good that it could be a sixth degree. We can't tell could be an eighth degree

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but actually at least a fourth degree polynomial.

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And what about the derivative, the Green function?

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Is it an odd degree or even degree?

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Odd.

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It's definitely not linear.

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Could it be cubic?

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It could be cubic.

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It has one, two turning points or two little,

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it could be a cubic, okay?

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And this is always gonna happen with polynomials.

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Whatever the degree of the polynomials,

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the derivative is gonna be one degree less.

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And we'll learn exactly why later.

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So first we figure out where the slope of the tangent line is zero.

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horizontal tangent lines which obviously should be here we put a zero

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we got one and to the right is it negative or positive negative okay it

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just keeps getting more and more and more and more negative and to the left

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positive, steep is getting steeper and steeper.

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It's going to look like this.

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This is a quadratic,

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or could it be a fourth degree problem?

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The derivative of a quadratic is a linear function, so that is why I drew a straight

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line.

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Okay.

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Should have drawn it something like that?

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Yes.

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What type of function is this?

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Definitely not.

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Why not?

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Do polynomials have asymptotes?

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No.

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Okay.

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Polynomials are continuous.

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It's definitely not a polynomial.

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What type of function is this?

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Rational, which is the ratio of two polynomials.

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Are there any parts where the derivative is zero here?

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No.

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So we can't start by doing that.

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So now we just start some other slide.

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Just pick a value.

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This would be the tangent line.

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What's approximately the slope of that line?

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It's about one.

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Right at 45 degree angle has a slope of one.

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Okay.

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So I'm putting it right there at one.

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And then when we go to the right, does it have a positive slope or a negative slope?

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The closer we get to that asymptote, the steeper the slope becomes.

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So in fact that's going to be the derivative.

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and then the farther we go to negative infinity that closer the derivative gets

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The derivative approaches zero farther to the left. It is not the same function as

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the original graph.

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And then right here, what's approximately the slope of that line?

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One, positive one.

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So we need a dot up here at one.

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And then when we go to the right, it's still positive but just keeps getting flatter

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and

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flatter and flatter.

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So positive, but gets very, very, very close to zero.

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And when we go to the left of this x value of two or so, it's positive, but it keeps

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getting steeper and steeper and steeper and steeper and steeper and steeper.

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So this goes up, up, up, up, up, up.

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OK.

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This is also a rational function.

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The derivative of rational functions

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are other rational functions.

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And we'll see why that is later.

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We do the derivative.

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OK, questions on this?

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We go to the right of the zero it's the steepest right, I don't know about here

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Okay right here is the steepest

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So that's gonna be our it's gonna go up to here

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And then the closer we get whatever this is x equals 5 it starts to level off get

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close to zero it's never zero

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And

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And then what's the slope of the tangent line right here at the end point?

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Very important.

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It does not exist.

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Derivatives do not exist at end points.

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Why is that?

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Let's call this number 5.

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The limit as h approaches...

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Let's go to the other definition.

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Approaches 5, f of x minus f of 5, over x minus 5.

196
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Why does this limit not exist?

197
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The limit as x approaches five from the left does exist.

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Okay but the limit as x approaches five from the right.

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Okay, this does not exist there are no values to the right of five

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Okay, so that's why the limit as X approaches five does not exist endpoints are not

201
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differentiable

202
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Very important so it's same thing on the left here. This is positive gets steeper

203
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And then we got to put an open circle on the end point

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Problem 42 has one horizontal tangent. To the right, the slope is always negative

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and becomes steeper.

206
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is always negative and just keeps getting steeper. And again if the function ends,

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there's

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00:15:11,860 --> 00:15:24,100
there it's not differentiable somewhere right here is the steepest

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so it's gonna go up and then get closer and closer and closer to zero okay do 43

210
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Shouldn't get something like that.

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Okay, if you're not understanding where I, or how I do this, please ask.

212
00:15:56,650 --> 00:15:59,310
And again, where I put the steepest point doesn't really matter.

213
00:15:59,870 --> 00:16:01,850
I could have drawn it way up here

214
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We can't really tell because we don't really have numbers to deal with

215
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Okay, we'll talk about this x point in a second.

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00:16:13,640 --> 00:16:18,140
We definitely have a zero here, a zero here, and a zero here.

217
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When we go to the right of this point, where's the steepest point?

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It just keeps getting steeper and steeper and steeper and steeper and steeper until

219
00:16:29,280 --> 00:16:30,660
we hit that sharp point.

220
00:16:34,080 --> 00:16:36,000
So it's just going to get steeper.

221
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Okay I could have drawn it there or I could have drawn it here.

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It doesn't really matter.

223
00:16:41,980 --> 00:16:46,000
Just to the right of that sharp point is the slope positive or negative?

224
00:16:47,320 --> 00:16:48,180
Negative.

225
00:16:48,180 --> 00:16:52,400
And in fact every single point on that line has the same slope.

226
00:16:53,080 --> 00:16:56,380
So we are going to draw a negative, some negative value here.

227
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And it is a horizontal line.

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it's always the same value let's say this is at 7 at x equals 7 that sharp

229
00:17:12,100 --> 00:17:18,740
point what's the derivative okay definitely does not exist and why is

230
00:17:18,740 --> 00:17:29,160
that okay it's definitely a cusp but what makes it not differentiable that we

231
00:17:29,160 --> 00:17:36,380
We need an open circle on each side. Why is the derivative undefined?

232
00:17:38,180 --> 00:17:45,780
yes this is exactly right the derp the definition is the limit as let's do the

233
00:17:45,780 --> 00:17:53,040
other definition X approaches 7 of the difference formula or the slope of the

234
00:17:53,040 --> 00:17:53,940
secant line.

235
00:18:00,260 --> 00:18:05,941
And the reason it's not differentiable, the limit from the left is going to equal

236
00:18:05,941 --> 00:18:06,320
whatever

237
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this number is.

238
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Let us say the left-hand slope is 8; these are illustrative values.

239
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Let's make that.

240
00:18:12,880 --> 00:18:17,600
The limit as X approaches seven from the right

241
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is gonna be whatever this number is.

242
00:18:26,200 --> 00:18:27,960
I'm gonna just call it negative two.

243
00:18:31,100 --> 00:18:34,500
Therefore the limit as X approaches seven.

244
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It does not exist because the limit from the left and the limit from the right not

245
00:18:45,220 --> 00:18:46,780
of the function but of the

246
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We call this the difference quotient

247
00:18:53,240 --> 00:18:55,380
We call this the difference quotient, or the slope of the secant line.

248
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Are different so it does not exist. So the reason sharp points are not

249
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differentiable

250
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It's by the definition of the derivative the derivative or the limit for the left

251
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and the right are not the same therefore

252
00:19:08,440 --> 00:19:18,820
the limit doesn't exist therefore there is no derivative there and it will be

253
00:19:18,820 --> 00:19:19,430
important to state that later on. Okay question on that problem is that the

254
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What function is this?

255
00:19:21,730 --> 00:19:23,350
Is it a cosine?

256
00:19:24,130 --> 00:19:25,550
Looks just like cosine.

257
00:19:27,310 --> 00:19:30,050
In fact I believe it is cosine.

258
00:19:32,470 --> 00:19:33,590
That's right at 2 pi.

259
00:19:35,310 --> 00:19:37,210
Does it have any horizontal tangent lines?

260
00:19:39,070 --> 00:19:40,310
Yeah we got lots of them.

261
00:19:44,410 --> 00:19:46,370
Right here, right here, right here.

262
00:19:47,290 --> 00:19:49,210
So the derivatives are going to have lots of zeros.

263
00:19:49,210 --> 00:19:59,450
one here, one here at pi, at 2 pi, 3 pi, negative pi, negative 2 pi.

264
00:20:04,210 --> 00:20:08,070
Between zero and pi, where is it the steepest?

265
00:20:11,710 --> 00:20:14,650
Exactly here, at pi over 2.

266
00:20:16,170 --> 00:20:21,850
In fact, I gave you grid marks. What is the slope of that line?

267
00:20:27,230 --> 00:20:29,510
Plus one. Negative one.

268
00:20:30,230 --> 00:20:35,410
It goes down one over one. It's exactly negative one.

269
00:20:36,570 --> 00:20:39,630
So we go to negative one, which is right here.

270
00:20:45,070 --> 00:20:56,350
okay and a nice smooth point and then between pi and 2 pi the tangent line the

271
00:20:56,640 --> 00:21:00,200
The derivative looks a lot like a trig function as well.

272
00:21:01,020 --> 00:21:02,520
If it were, what would it be?

273
00:21:03,160 --> 00:21:04,340
It's definitely not sine.

274
00:21:06,220 --> 00:21:10,180
Sine starts at zero and goes up to one at pi over two.

275
00:21:11,460 --> 00:21:18,580
But this is the reflection of sine, so it would be negative sine.

276
00:21:30,040 --> 00:21:34,313
In fact, it turns out, we're going to show this later on, the derivative of cosine

277
00:21:34,313 --> 00:21:34,740
is

278
00:21:34,740 --> 00:21:36,240
Exactly negative sine.

279
00:21:36,870 --> 00:21:39,730
What type of function is this?

280
00:21:41,370 --> 00:21:42,690
Looks like an exponential function.

281
00:21:46,230 --> 00:21:48,890
What is the base of the exponential function?

282
00:21:50,950 --> 00:21:55,010
How do we look at a graph and figure out what the base is?

283
00:21:56,030 --> 00:21:58,490
X equals 1, what is the value?

284
00:22:04,150 --> 00:22:08,590
X equals 1, it's right here.

285
00:22:11,310 --> 00:22:15,950
It's not obvious, but this turns out to be E. What is E?

286
00:22:16,230 --> 00:22:21,170
This is our function, e to the x.

287
00:22:29,670 --> 00:22:31,990
It has no minimums or maximums.

288
00:22:33,210 --> 00:22:35,210
This tangent line will never be zero.

289
00:22:35,210 --> 00:22:37,430
So we just got to go to some other point.

290
00:22:39,890 --> 00:22:43,610
Let's pick a point, let's say, right here.

291
00:22:44,670 --> 00:22:47,450
What is the slope of this line?

292
00:22:50,770 --> 00:22:52,450
OK, it looks like it's one.

293
00:22:52,550 --> 00:22:54,110
In fact, it is exactly one.

294
00:22:55,270 --> 00:22:57,710
I'll put a dot right there at 0 comma 1.

295
00:23:00,290 --> 00:23:00,810
OK.

296
00:23:00,810 --> 00:23:04,810
And then to the right here, it's definitely positive.

297
00:23:06,270 --> 00:23:09,690
And it just keeps getting steeper and steeper and steeper.

298
00:23:11,350 --> 00:23:22,170
Okay, and this is not exactly obvious here, but that is the derivative to the right.

299
00:23:23,950 --> 00:23:28,169
And to the left is still positive but keeps getting flatter and flatter and flatter

300
00:23:28,169 --> 00:23:28,919
and

301
00:23:29,330 --> 00:23:30,230
flatter.

302
00:23:33,790 --> 00:23:34,690
Ok.

303
00:23:36,910 --> 00:23:44,830
And for this function it's not obvious but it turns out the derivative of e to the x

304
00:23:45,590 --> 00:23:47,750
is exactly the same function.

305
00:23:47,950 --> 00:23:49,110
It is e to the x.

306
00:23:49,830 --> 00:23:54,470
Which is kind of crazy and we'll prove that later for you

307
00:23:59,950 --> 00:24:02,210
Super cool a

308
00:24:02,210 --> 00:24:06,110
Derivatives of exponential functions are exponential functions as well.

309
00:24:06,710 --> 00:24:08,930
Sometimes it's exactly the same

310
00:24:08,930 --> 00:24:13,480
The other times it'll be just a slight coefficient different have a different

311
00:24:13,480 --> 00:24:15,930
coefficient. It'll still be an exponential function

312
00:24:16,490 --> 00:24:18,390
Let's assume this goes up forever.

313
00:24:21,610 --> 00:24:23,330
What type of function is this?

314
00:24:27,050 --> 00:24:30,230
Or if I gave you an equation for this...

315
00:24:31,610 --> 00:24:34,330
We have to be some piecewise function.

316
00:24:35,530 --> 00:24:37,330
OK, that's three different pieces.

317
00:24:41,910 --> 00:24:45,110
And we'll talk about what the actual function is in a minute.

318
00:24:47,450 --> 00:24:50,030
It does have a horizontal tangent line right here.

319
00:24:51,590 --> 00:24:54,590
So the derivative has a 0 right there.

320
00:24:56,130 --> 00:24:57,410
The derivative is 0 there.

321
00:24:58,850 --> 00:25:00,250
And then to the right here

322
00:25:00,990 --> 00:25:03,130
Does have a positive negative slope?

323
00:25:04,130 --> 00:25:08,668
negative just keeps getting more and more and more negative and then what happens at

324
00:25:08,668 --> 00:25:09,418
the

325
00:25:11,610 --> 00:25:17,530
What's the derivative of that sharp point does not exist so

326
00:25:20,710 --> 00:25:25,330
It's more and more negative we definitely need to put a hole right there

327
00:25:28,990 --> 00:25:33,510
And instantly to the right, does it have a positive slope or negative slope?

328
00:25:35,110 --> 00:25:39,067
It's not obvious, but this is getting larger, just barely larger and larger and

329
00:25:39,067 --> 00:25:39,817
larger.

330
00:25:40,070 --> 00:25:44,430
So from here, it doesn't exist.

331
00:25:50,330 --> 00:25:52,410
It looks something like that.

332
00:25:56,150 --> 00:26:02,470
To the left here, the positive slope, it keeps me steeper and steeper and steeper.

333
00:26:06,970 --> 00:26:12,350
And then right here, whatever value that is, doesn't exist.

334
00:26:16,570 --> 00:26:19,570
And then instantly has negative slope

335
00:26:33,220 --> 00:26:38,100
Derivatives of piecewise functions are also described piecewise.

336
00:26:41,420 --> 00:26:50,100
In fact, this function, the equation is the absolute value of some quadratic.

337
00:26:50,360 --> 00:26:52,620
I don't know exactly what quadratic it is.

338
00:26:52,620 --> 00:26:54,840
I'm just let me make something up.

339
00:26:55,340 --> 00:26:57,080
x squared minus

340
00:27:00,300 --> 00:27:01,200
plus

341
00:27:03,940 --> 00:27:05,940
three x minus

342
00:27:11,960 --> 00:27:14,220
16 or something like that.

343
00:27:18,020 --> 00:27:21,600
Anytime you see an absolute value, it's a piecewise function.

344
00:27:24,180 --> 00:27:26,480
This actually was some quadratic.

345
00:27:28,440 --> 00:27:34,020
If it was without the absolute value, it'd be exactly this function.

346
00:27:35,920 --> 00:27:40,500
But because of the absolute value, it flipped this to this.

347
00:27:41,740 --> 00:27:46,240
It's kind of interesting.

348
00:27:50,960 --> 00:27:53,980
And again, why is it not differentiable there?

349
00:27:54,960 --> 00:27:57,520
Let's say this is at 4.2 or something.

350
00:27:57,520 --> 00:28:02,400
The left- and right-hand limits of the difference quotient are not equal.

351
00:28:02,670 --> 00:28:04,610
Okay, this is some piecewise function.

352
00:28:05,470 --> 00:28:09,595
It does have extreme, there's a maximum here and a maximum there and a minimum

353
00:28:09,595 --> 00:28:10,345
there.

354
00:28:12,750 --> 00:28:14,330
Is the derivative zero there?

355
00:28:15,830 --> 00:28:17,070
Definitely not, it's sharp.

356
00:28:18,070 --> 00:28:20,030
Okay, sharp points are not differentiable.

357
00:28:20,750 --> 00:28:23,710
The limit from the left of the difference quotient limit from the right are not the

358
00:28:23,710 --> 00:28:24,550
same.

359
00:28:24,550 --> 00:28:25,810
So let's just choose some other point.

360
00:28:26,710 --> 00:28:30,350
Let's say, I don't know, right here.

361
00:28:31,870 --> 00:28:34,450
What is the slope of that tangent line there?

362
00:28:37,270 --> 00:28:39,190
It's whatever the slope of that line is.

363
00:28:39,270 --> 00:28:40,370
What is the slope of that line?

364
00:28:41,870 --> 00:28:42,770
One.

365
00:28:43,570 --> 00:28:45,090
Okay, in fact it's right there.

366
00:28:45,610 --> 00:28:49,070
Everywhere on this point has the same exact slope of the tangent line.

367
00:28:49,570 --> 00:28:52,670
The slope of a tangent line of a line is always the slope of the line.

368
00:28:52,670 --> 00:28:57,710
So everywhere here, it's going to have a slope of one

369
00:28:59,670 --> 00:29:03,550
Sharp points are not differentiable open circle

370
00:29:08,210 --> 00:29:11,850
So that's this piece here the slope of this line is one

371
00:29:15,490 --> 00:29:19,410
Between zero and five the slope of that line is negative one

372
00:29:19,520 --> 00:29:23,660
So it's gonna be like this.

373
00:29:27,280 --> 00:29:29,100
And one again.

374
00:29:31,790 --> 00:29:34,950
This is y equals the cube root of x.

375
00:29:40,910 --> 00:29:44,010
It's the continuous function everywhere.

376
00:29:45,610 --> 00:29:46,890
Domain's all real numbers.

377
00:29:53,150 --> 00:29:55,270
It has no minimum or maximum, right?

378
00:29:56,090 --> 00:29:58,150
So the derivative is going to have no zeros.

379
00:29:58,610 --> 00:30:03,550
I don't know, somewhere right about here, the slope of the tangent line is one.

380
00:30:07,450 --> 00:30:08,850
So I'm going to put a dot here.

381
00:30:09,370 --> 00:30:13,477
Ok, we have a positive slope but it just keeps decreasing and decreasing and

382
00:30:13,477 --> 00:30:14,227
decreasing.

383
00:30:15,290 --> 00:30:17,350
So it's going to look something like this.

384
00:30:19,470 --> 00:30:21,230
And have a horizontal asymptote.

385
00:30:24,590 --> 00:30:26,290
What happens when I get closer to zero?

386
00:30:28,230 --> 00:30:30,190
It's steeper and steeper and steeper.

387
00:30:30,190 --> 00:30:32,530
And right here at zero

388
00:30:33,830 --> 00:30:36,590
The tangent line is vertical

389
00:30:38,170 --> 00:30:41,210
What's the slope of vertical lines

390
00:30:51,030 --> 00:30:53,690
Vertical lines don't have slopes

391
00:30:54,990 --> 00:30:56,090
Okay like

392
00:30:57,330 --> 00:31:02,030
x equals 2 is a vertical line. It does not have a slope.

393
00:31:02,630 --> 00:31:04,730
Vertical lines do not have slopes.

394
00:31:06,070 --> 00:31:09,210
So right here the derivative does not exist

395
00:31:09,870 --> 00:31:11,970
because we have a vertical tangent line.

396
00:31:12,990 --> 00:31:14,410
And the closer you get to 0,

397
00:31:16,010 --> 00:31:18,510
This has a vertical asymptote, the derivative.

398
00:31:21,970 --> 00:31:28,370
And then just to the left of zero, it has a positive, very very very steep slope.

399
00:31:29,270 --> 00:31:31,830
It's going to look something like this.

400
00:31:31,830 --> 00:31:33,830
It has...

401
00:31:39,370 --> 00:31:45,455
At this vertical tangent, the derivative does not exist; its graph has a vertical

402
00:31:45,455 --> 00:31:46,205
asymptote.

403
00:31:46,710 --> 00:31:47,750
vertical asymptote

404
00:31:48,450 --> 00:31:51,390
Okay, it might go to positive, it might go to negative, they both...

405
00:31:51,930 --> 00:31:53,090
which is a semi-circle.

406
00:32:02,990 --> 00:32:04,930
Let's say this is the semi-circle.

407
00:32:06,310 --> 00:32:07,310
We definitely have a zero.

408
00:32:11,190 --> 00:32:12,870
And the slope here is negative.

409
00:32:14,710 --> 00:32:16,650
We're going to have a vertical asymptote here.

410
00:32:17,650 --> 00:32:18,610
It's going to look like this.

411
00:32:19,270 --> 00:32:23,936
Approaching the endpoint, the derivative does not exist at the endpoint, and the

412
00:32:23,936 --> 00:32:25,730
slopes become steeper and steeper.

413
00:32:25,730 --> 00:32:26,650
and steeper and steeper.

414
00:32:26,870 --> 00:32:28,930
It goes all the way to negative infinity.

415
00:32:30,210 --> 00:32:33,310
And this will go all the way to positive infinity.

416
00:32:33,850 --> 00:32:38,801
But, okay, yeah, so that makes sure when you draw the graph of a semicircle, the

417
00:32:38,801 --> 00:32:39,110
closer

418
00:32:39,110 --> 00:32:41,370
we get to this value, the steeper it gets.

419
00:32:41,470 --> 00:32:44,230
You're gonna have vertical asymptotes at those two points.