Tuesday, October 4, 2016
Unizor - Derivatives - Definition
Notes to a video lecture on http://www.unizor.com
Derivatives - Speed of Change
Consider a real function f(x)defined on a set of real numbers and taking real values.
It represents a dependency of some numerical value on certain input parameter. For example, temperature as a function of time or distance the car went as a function of amount of fuel spent, or the force of gravity as a function of an elevation above the surface of the Earth etc.
Let's assume further that this function f(x) is defined on some real interval (a,b) of its argument x, and we are analyzing the function's behavior around a particular point x0 inside this interval. More precisely, we are interested to know how fast the function changes in the neighborhood of this point x0∈(a,b).
Back to one of our examples, consider the force of gravity on different levels of elevation. We can measure this force on any level and, comparing these forces G(h0) and G(h1) on two different levels h0 and h1, we can find the average speed of change of gravity on this interval per unit of elevation:
[G(h1)−G(h0)] ∕ (h1−h0)
Notice, this is an average rate of change of the gravity force on interval from h0 to h1, not the rate of change at point h0 we are interested in, because there is a larger space between these two points for this rate to change.
What is also important is that the further points h0 and h1 are from each other - the further our average rate of change is from the one we are interested in. And, the closer point h1 is to h0- the closer our average rate of change of gravity is to the rate at point h0 we need to know.
An obvious solution is to move point h1 as close to h0 as possible.
Mathematical solution is to know the function of gravity force as it depends on the elevation at any elevation level and calculate the limit of the average rate of change as point h1 gets infinitely close to h0:limh1→h0[G(h1)−G(h0)]/(h1−h0)
If this limit exists (and we should not assume it always exists for any function), it can be considered as a true rate of change at point h0.
Consider another practical problem
Assume, you are a policeman, who is measuring the speed of the passing cars in order to enforce some speed restrictions in the area.
Assume further that the only tool you have is the stopwatch.
To perform his job, policemen marks two points on the road - A0 and A1 - and, using the stopwatch, measures the time during which any car moves between these points.
Now the average speed on this interval from A0 to A1 will the length of this interval divided by time it took the car to cover it.
But what if the car moved faster than speed limit in the beginning of this interval and slower at the end, so the average speed was below maximum? To detect this, point A1 should be as close to A0 as possible.
Mathematical approach to this problem is as follows.
Let's start with some point on the road as an origin and mark each point with a distance from this origin.
Let's consider a function associated with the movement of a car - exact time t and distance from origin D(t) at each moment in time.
Point A0 was reached at time t0 and its distance from origin is D(t0). Point A1 was reached at time t1 and its distance from origin is D(t1).
The average speed on interval from A0 to A1 can be expressed as
[D(t1)−D(t0)] ∕ (t1−t0)
To get the speed at point A0, we have to know the following limit:
Assume, you are a reporter who has to report the news about the flood. What you need to report is something like this: "At time t0 the level of water was rising with a speed of M meters per hour" (whatever the values of time t0 required).
To accomplish this, you construct the function L(t) of the level of water at each moment in time. Considering you have the value of this function for all moments of time t, you can derive the speed of rising the water at any concrete moment t0 using the following procedure.
You take the level of water at moment t0, which is L(t0), and at some moment t1 after t0, which is L(t1).
The difference between these two levels of water signifies the rising of its level during a period from t0 to t1.
[L(t1)−L(t0)] / (t1−t0)
is an average speed of rising water during this period.
If moment t1 is closer to t0, this average speed would represent the speed of rising the waters at moment t0 more precisely, and the closer are these moments - the better would be the evaluation.
Exact speed of rising of the level of water at moment t0 is, therefore
limt1→t0[L(t1)−L(t0)] / (t1−t0)
In all the above examples for a given function f(x) we were interested in the speed of its change at some specific value of the argument x0.
This speed of change at x=x0 is defined as
The limit above, if it exists (which might not be the case) is called a derivative of function f(x) at point x=x0 and denoted as f'(x0).
Since existing of this limit implies its identical value in any trajectory of x1 approaching x0, we can reformulate this using the following formula with identical meaning and notation Δx=x1−x0:
which is a more traditional definition of derivative of function f(x) at point x=x0.