*Notes to a video lecture on http://www.unizor.com*

__Derivatives - Speed of Change__

Consider a real function

**defined on a set of real numbers and taking real values.**

*f(x)*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

**is defined on some real interval**

*f(x)***(**of its argument

*a,b*)**, and we are analyzing the function's behavior around a particular point**

*x***inside this interval. More precisely, we are interested to know how fast the function changes in the neighborhood of this point**

*x*_{0}**∈**

*x*_{0}**(**.

*a,b*)*Example 1*

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

**and**

*G(h*_{0})**on two different levels**

*G(h*_{1})**and**

*h*_{0}**, we can find the average speed of change of gravity on this interval per unit of elevation:**

*h*_{1}**[**

*G(h*] ∕_{1})−G(h_{0})*(h*_{1}−h_{0})Notice, this is an

*average*rate of change of the gravity force on interval from

**to**

*h*_{0}**, not the rate of change at point**

*h*_{1}**we are interested in, because there is a larger space between these two points for this rate to change.**

*h*_{0 }What is also important is that the further points

**and**

*h*_{0}**are from each other - the further our average rate of change is from the one we are interested in. And, the closer point**

*h*_{1}**is to**

*h*_{1}**- the closer our average rate of change of gravity is to the rate at point**

*h*_{0}**we need to know.**

*h*_{0}An obvious solution is to move point

**as close to**

*h*_{1}**as possible.**

*h*_{0}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

**gets infinitely close to**

*h*_{1}**:**

*h*_{0}*[*

**lim**_{h1→h0}**]/**

*G(h*_{1})−G(h_{0})

*(h*_{1}−h_{0})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

**.**

*h*_{0}*Example 2*

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 -

**and**

*A*_{0}**- and, using the stopwatch, measures the time during which any car moves between these points.**

*A*_{1}Now the

*average*speed on this interval from

**to**

*A*_{0}**will the length of this interval divided by time it took the car to cover it.**

*A*_{1}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

**should be as close to**

*A*_{1}**as possible.**

*A*_{0}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

**and distance from origin**

*t***at each moment in time.**

*D(t)*Point

**was reached at time**

*A*_{0}**and its distance from origin is**

*t*_{0 }**. Point**

*D(t*_{0})**was reached at time**

*A*_{1}**and its distance from origin is**

*t*_{1}**.**

*D(t*_{1})The average speed on interval from

**to**

*A*_{0}**can be expressed as**

*A*_{1}**[**

*D(t*] ∕_{1})−D(t_{0})*(t*_{1}−t_{0})To get the speed at point

**, we have to know the following limit:**

*A*_{0}*[*

**lim**_{t1→t0}**]/**

*D(t*_{1})−D(t_{0})

*(t*_{1}−t_{0})*Example 3*

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

**the level of water was rising with a speed of**

*t*_{0}**meters per hour" (whatever the values of time**

*M***required).**

*t*_{0}To accomplish this, you construct the function

**of the level of water at each moment in time. Considering you have the value of this function for all moments of time**

*L(t)***, you can derive the speed of rising the water at any concrete moment**

*t***using the following procedure.**

*t*_{0}You take the level of water at moment

**, which is**

*t*_{0}**, and at some moment**

*L(t*_{0})**after**

*t*_{1}**, which is**

*t*_{0}**.**

*L(t*_{1})The difference between these two levels of water signifies the rising of its level during a period from

**to**

*t*_{0}**.**

*t*_{1}The ratio

**[**

*L(t*] /_{1})−L(t_{0})*(t*_{1}−t_{0})is an average speed of rising water during this period.

If moment

**is closer to**

*t*_{1}**, this average speed would represent the speed of rising the waters at moment**

*t*_{0}**more precisely, and the closer are these moments - the better would be the evaluation.**

*t*_{0}Exact speed of rising of the level of water at moment

**is, therefore**

*t*_{0}*[*

**lim**_{t1→t0}**] /**

*L(t*_{1})−L(t_{0})

*(t*_{1}−t_{0})*CONCLUSION*

In all the above examples for a given function

**we were interested in the speed of its change at some specific value of the argument**

*f(x)***.**

*x*_{0}This speed of change at

**is defined as**

*x=x*_{0}*[*

**lim**_{x1→x0}**]/**

*f(x*_{1})−f(x_{0})

*(x*_{1}−x_{0})The limit above, if it exists (which might not be the case) is called a

*derivative*of function

**at point**

*f(x)***and denoted as**

*x=x*_{0}**.**

*f'(x*_{0})Since existing of this limit implies its identical value in any trajectory of

**approaching**

*x*_{1 }**, we can reformulate this using the following formula with identical meaning and notation Δ**

*x*_{0}*:*

**x=x**_{1}−x_{0}*lim*

_{Δx→0}[

**Δ**

*f(x*_{0}+**]/Δ**

*x)−f(x*_{0})

*x*which is a more traditional definition of

*derivative*of function

**at point**

*f(x)***.**

*x=x*_{0}
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