Practice English Speaking&Listening with: Definition of derivative - Differentiation - Mathematics- Pre-university Calculus - TU Delft

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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

x and a vertical displacement y.

The slope is simply defined as y divided by x.

For example, if x equals 10 meter and y 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, x will double, but so will y, so the slope remains

the same.

We can even take Q on the other side of P.

Then x is negative, say -4 meter.

y 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 x, the corresponding y will be larger, and so the slope will be

larger.

Finally, descent corresponds to negative slope.

If x is positive, say 10 meter, the corresponding y 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 y and x 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 x travelled during a time

interval t, 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 x travelled during a time interval t 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 x divided by t.

To make the approximation better, we take t smaller and smaller.

The quotient of x and t 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 + x, the function value changes.

The vertical increase is equal to f(a+x) - f(a), the horizontal increase is

simply equal to x.

The quotient of these two is called a difference quotient.

Now we can take the limit of x 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 x 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, t is the time measured, x is the circumference of the wheel.

The current speed is approximated by x divided by t: 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, lets practice with the concepts!

Good luck!

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