Professor Dave here, let’s get back to graphing.
I’m sure you got enough of graphing when we learned algebra, but we need to expand
our abilities now that we understand differentiation.
First, as we recall, the derivative of a function at a particular point is equal to the rate
of change in the function at that point.
This fact should help us graph the derivatives of functions, and relate them to the original.
A lot of this will seem like painful mental gymnastics, but it all focuses around one concept.
This is the fact that the value of a function and the rate of change of that function are
totally unrelated.
Let’s say we have this function here, a parabola that opens downwards, with its vertex
at the origin.
Without knowing the equation of the parabola, and without doing any algebra at all, let’s
just use our understanding of derivatives to make a rough sketch of the derivative of
this function.
Now, going from negative infinity to zero, this function is always increasing.
All of the values of the function are negative, but that doesn’t matter, as that’s not
what we are looking at.
We are looking at the rate of change of the function.
The function is always increasing, which means a positive rate of change, so the derivative
must be positive over this entire interval.
The positive rate of change does diminish as we go, however, as we can see from the
decreasing slope of the tangent line, until at the vertex, where it will equal zero, as
this is a local maximum, so if we are graphing the derivative, we should start out very positive,
but then decrease in linear fashion until we hit the origin, because at x equals zero,
the rate of change of the function is zero.
Let’s be sure to understand that it is totally coincidental that both the function and its
derivative are equal to zero at this point.
We could move the original function up or down any amount we want, and that would not
influence the derivative one bit.
What matters is the shape of the curve, or how rapidly the function is increasing or
decreasing, so pulling the function up by some constant won’t affect the derivative at all.
This should also make sense in terms of differentiation, because when we take the derivative of a function,
any constant just disappears, making any vertical shift term irrelevant.
Now continuing from there, after the local maximum, the function begins to decrease,
and it does so at a faster and faster rate.
That means the derivative will be negative from this point onward, and it will be increasingly
negative as we go.
And there we have a sketch of the first derivative.
Now let’s graph the derivative of this derivative, which will be the second derivative of the
original function.
Going left to right, the derivative starts positive, but remember, that doesn’t matter to us.
We are looking at the rate of change, and the derivative is decreasing at a constant
rate, because it is a line with a fixed slope.
That means that the second derivative will be a horizontal line somewhere down here below
the x-axis.
The slope of the first derivative is constant, so the second derivative is constant.
Let’s note that we went from a quadratic function to a linear function to a constant,
and this also makes sense in the context of computing derivatives.
Now let’s realize that this set of graphs perfectly illustrates the motion of, say,
a ball being tossed straight up in the air.
Correlating the position function with this is easy.
It goes straight up, and then it comes straight back down.
The function looks like this because we are moving forward in time, but the value of the
function gives us the position of the ball, and that just comes up and hits zero, which
must be the apex of the trajectory, and then it falls back down.
Velocity, which is the derivative of position, isn’t too hard to rationalize either.
The ball will have a positive velocity as it leaves your hand, by virtue of the force
you impart on it with your arm.
But the moment it leaves your hand, the velocity will decrease, getting slower and slower,
until at the peak, for one single instant, it has a velocity of zero.
Then the velocity becomes negative, as it starts moving in the negative direction, getting
more and more negative as it speeds up in its fall.
And lastly, acceleration is the derivative of velocity, and therefore the second derivative
of position, and because of this, the acceleration graph makes sense too.
The acceleration experienced by the ball is just the acceleration due to the gravitational
pull of the earth, and that is a constant negative value.
It is this gravity that takes the initial velocity of the ball and immediately decreases
it, constantly, until it finally hits the ground a few seconds later.
So it is very important, not just for physics, but even just for doing math, that we can
draw the graphs of derivatives of functions strictly by reasoning this way, and that we
can look at graphs of derivatives to understand things about the original function.
If a function’s derivative is positive, the function is increasing at that point.
If the derivative is negative, the function is decreasing at that point.
We can’t say whether the function itself is positive or negative from this information,
just whether it is increasing or decreasing.
If a derivative crosses the x-axis from above, the function has a local maximum at that point.
If a derivative crosses the x-axis from below, the function has a local minimum at that point.
Everything we just said also applies to a comparison of the second derivative and first
derivative, but we can even look at the second derivative to get information about the original function.
To see how, let’s consider the notion of concavity.
A section of a curve is concave upward if every point on it sits above its tangent line.
We can think of this as a curve that opens upward.
A section of a curve is concave downward if every point on it sits below its tangent line.
We can think of this as a curve that opens downwards.
Well looking at this concave upward section here, the derivative is increasing at an increasing rate.
That means the second derivative must also be positive.
So intervals on a second derivative that are positive correspond with intervals on a function
that are concave up.
Similarly, intervals on a second derivative that are negative must correspond with intervals
on a function that are concave down.
Any time the second derivative is equal to zero, this will correspond to an inflection point.
This is a point where a function changes from concave up to concave down, or vice versa.
This is the case because if the second derivative crosses the x-axis, then the first derivative
must be changing from increasing to decreasing, or from decreasing to increasing, and therefore
the original function must be changing its concavity.
Let’s put this all together with an example.
Say we have the function x cubed minus twelve x plus one.
Let’s find any local maxima and minima, any points of inflection, and then sketch
the function.
First things first, we will need to find the first and second derivatives of the function.
That will be easy, the first derivative is three x squared minus twelve, and the second
derivative is six x.
Now, how can we get the information we need?
Well let’s recall that finding the zeros of the first derivative will give us any maxima
and minima, since these are points on the function where the tangent line has a slope
of zero, meaning a horizontal line.
So let’s set three x squared minus twelve equal to zero, and solve for x.
Add twelve to both sides, divide by three, and take the square root to get plus or minus two for x.
That means that points of interest occur when x equals negative two and two.
Well let’s plug these into the original function to get the y values at these points,
so that we can plot the points.
For negative two, we cube to get negative eight, minus (twelve times negative two),
plus one.
That will give us seventeen, so the point negative two, seventeen, is on the function.
Doing the same thing with positive two, we get eight, minus twenty four, plus one, which
will be negative fifteen, so the point two, negative fifteen, will be on the function.
Let’s also check for an inflection point.
Remember, these will happen anywhere the second derivative equals zero.
Six x equals zero when x equals zero, so there is an inflection point at x equals zero.
Plugging zero into the function, zero minus zero plus one equals one, so the inflection
point is at zero, one.
We are just about ready to sketch this function, let’s just get the end behavior so that
we know where to start and end.
As we learned in algebra, this is a cubic function, so it’s an odd function, and the
first term is positive, so this must necessarily rise up from negative infinity and eventually
extend up to positive infinity.
We can also find the zeros of the function to be even more thorough, but as that is old
material, we will skip that step right now.
So starting all the way down here, we rise up to the first point, which is therefore
a local maximum, meaning we curve back down.
We pass through the inflection point, meaning we change from concave down to concave up,
and eventually get to the local minimum.
Again, this means we must start to curve back up, and then we just extend to positive infinity.
And that’s how we use derivatives to graph this function.
So our understanding of derivatives should make it fairly easy to graph first and second
derivatives of functions simply by looking at a function.
And as we just saw, we can use information about the first and second derivatives to
help us accurately graph higher-degree polynomials with more precision than we were previously able to.
This is a very important concept, so let’s check comprehension.