Hi!
In the rollercoaster video two questions were
raised: how to determine the slope of the
rollercoaster track and how to determine the
speed of the rollercoaster cart.
The funny thing is that both questions can
be answered using differentiation.
Let me show you how!
Let us start with the concept of slope.
How is it defined?
First we look at a straight but inclined
part of the rollercoaster track.
When a cart moves from point P to point Q,
there is a corresponding horizontal displacement
Δx and a vertical displacement Δy.
The slope is simply defined as Δy divided
by Δx.
For example, if Δx equals 10 meter and
Δy equals 5 meter, the slope is 0.5.
Because the line is straight, it does not
matter where we take P and Q.
If we take Q twice as far, Δx will double,
but so will Δy, so the slope remains
the same.
We can even take Q on the other side of P.
Then Δx is negative, say -4 meter.
Δy then equals -2 meter, so the
quotient remains the same.
As you would expect, the slope is a measure
of steepness: if the track is steeper, then
for a fixed Δx, the corresponding 
Δy will be larger, and so the slope will be
larger.
Finally, descent corresponds to negative slope.
If Δx is positive, say
10 meter, the corresponding Δy will be
negative, say -4 meter.
Therefore the slope will be negative, in this
case -0.4.
What if the track is not straight?
Suppose we want to determine the slope at
point P.
If we zoom in at the track near P, the track
will look approximately straight.
So if we take Q close to P, and take the quotient
of the displacements, we get an approximation
of the slope of the track.
To improve the approximation, we move Q closer
to P.
The quotient of Δy and Δx approaches
a limit value.
This value is, by definition, the slope at
P.
Now let us consider the concept of speed.
First, suppose that the speed of the rollercoaster
cart is constant on some part of the track.
In that case, you can determine it by measuring
the distance Δx travelled during a time
interval Δt, and then take the quotient.
For constant speed, it does not matter what
time interval we take.
Now suppose that the speed is not constant,
and we want to determine the speed at time t.
Again we can measure the distance Δx
travelled during a time interval Δt and
take the quotient.
This will give us the average speed on the time
interval, but not the speed at time t.
However, if the time interval is small, the
speed is approximately constant during that
time interval.
So the speed at time t is approximately equal
to Δx divided by Δt.
To make the approximation better, we take
Δt smaller and smaller.
The quotient of Δx and Δt then approaches
some limit value.
This is precisely the speed at time t.
We see that both slope and speed can be determined
by taking the limit of a quotient.
That is exactly what differentiation is.
More precisely, suppose a function f and a
point with x-coordinate equal to ‘a’ are given.
If the x-coordinate changes from ‘a’ to
a + Δx, the function value changes.
The vertical increase is equal to 
f(a+Δx) - f(a), the horizontal increase is
simply equal to Δx.
The quotient of these two is called a difference
quotient.
Now we can take the limit of Δx to zero.
If this limit exists, the number you obtain
is called the derivative of f at x = a.
It is denoted by f’(a) or df/dx(a).
The whole process of determining the derivative
is called differentiation.
If you look at the graph, both the difference
quotient and the derivate have a geometric
interpretation.
The difference quotient is precisely the slope
of the line connecting the points P and Q
on the graph.
Now if we let Δx go to zero, you see
that this connecting line approaches a limiting
position.
The slope of this limiting line is precisely
the derivative of f at ‘a’.
The limiting line itself is called the tangent
line to the graph, at the point P.
It is the unique line with the property that
it passes through the point P and has the
same slope as the graph in that point.
Now, let us return to the original questions:
what is slope and what is speed?
We have seen that both slope and speed can
be interpreted as derivatives: slope as the
derivative of vertical position y as function
of horizontal position x, speed as the derivative
of position x as function of time t. The
definition in terms of the limit of a difference
quotient may seem abstract, but it provides
the basis for many measuring devices.
For example, a bicycle computer can measure
the time it takes to for the wheel to make
one revolution.
In this case, Δt is the time measured,
Δx is the circumference of the wheel.
The current speed is approximated by Δx divided by Δt: the difference quotient.
Usually, this approximation is sufficiently
accurate.
What if you want to calculate derivatives
exactly?
Taking limits every time can be cumbersome.
In the next section, you will learn how to
deal with this..
But first, let’s practice with the concepts!
Good luck!
