Lecture 1. Basic theorems of differential calculus
We continue to study the differential calculus of one variable functions
The topic of this lecture: basic theorems of differential calculus.
These theorems arevassociated with the names of great French mathematicians, and it was a long time ago - in the 17th century.
Let's look at each of these theoremes.
The first one - Fermat's theorem (about vanishing of the derivative).
Statement of the theorem.
Suppose that function f satisfies the following conditions: it is differentiable in the segment (a, b) and reaches the highest or lowest value at some point x0 of this segment.
The theorem states that the derivative of the function at this point is zero.
Let us prove it.
Thus, suppose that function f reaches maximum in point x0 of segment (a, b) (the situation on the picture is presented in the video).
Let's calculate the derivative value in point x0.
By hypothesis, the function is differentiable in segment (a, b), therefore, it is differentiable at the point x0.
We calculate by definition the limit of rise over run, if the limit exists, it is equal to the value of one-sided left and right limits.
Let’s note that the numerator of this fraction (increment of the function in point x0) is always less or equal to zero (if it is a strict maximum, then strict inequation).
Now we calculate one-sided limits, to be exact we assess their values.
So, in the first case, when Dx is over zero under the limit sign there is a fraction less or equal to zero, and hence the derivative value is less or equal to zero.
In the second case, when Dx is less zero, the whole fraction takes non-negative value, and thus the derivative value is over or equal to zero.
Thus, number f ¢(x0) is not negative and not positive at the same time, obviously only zero satisfies this condition
This proves the theorem.
What is the geometric sense of Fermat theorem?
Derivative value is zero, it means that the corresponding point in the graph has a horizontal tangent.
So, in the points of maximum and minimum values function graph has horizontal tangents.
Let's note that the converse theorem is not true (sometimes we have such a logical error).
We consider the function f(x)=x3, we take zero point as x0.
Obviously, the derivative value is zero (see. Video).
So, if the derivative value is zero, it does not mean that at this point the function takes the maximum or minimum value, look, this is not happening here.
The following theorem - Rolle theorem. (A zero derivative, which takes equal values at segment endpoints).
Let's pay attention to the conditions of the theorem, we know about function f: it is continuous on segment [a, b] and differentiable at all interior points of the segment.
The theorem conclusion states that if the function values at the endpoints are equal, then at some interior point of segment (a, b) the derivative value is zero.
The proof includes two cases.
First, when in all the other points the function value is the same as that at the endpoints - the function is constant (we know that the derivative of a constant equals zero) so we can take any point of segment (a, b) as x0
The second case is the negation of the first one - the function is not constant, then it takes the maximum and minimum values at different points of the segment.
We remember that the function has the maximum and minimum values, these two numbers, at least one of them, are different from number f(a) or f (b), and by Fermat theorem the derivative at this point is zero.
What is the geometric sense of Rolle theorem?
If the values at the segment endpoints are equal, then at some point of the segment the function has a horizontal tangent at the relevant point of the graaph, perhaps there are more points than one on the graph (see. Picture in the video).
Lagrange theorem (finite increaments).
This is the most important theorem, which we will use proving other facts of differential calculus.
Again, the first facts resemble conditions of the previous theorems.
Thus, the function is continuous on the segment [a, b] and differentiable within this segment, then there is a point x0, in which equation f(b)-f(a) = f ¢(x0)(b-a) is true, it is called Lagrange formula of finite increments .
You’ll ask "Where are the increments?".
Look, (b-a) is the change of the argument on X axis (we can consider it an argument increment), and f(b) -f(a) is the change that the function undergoes (we can assume this a function increment), so these two increments are connected by the formula.
Let’s prove that.
Note points that are endpoints of the graph on segment AB by letters a and b.
Draw a cut line through points a and b and form the equation (see. Video).
Next, we form a new function φ(x), it is the difference between function f (x), which is given, and a linear function (the equation of cut line AB) - two functions, which are differentiable on the segment (a, b), and continuous in the segment [a, b].
It is easy to verify that at the endpoints of segment [a, b] values of function φ(x) are equal to zero, then according to Rolle theorem there exists a point x0, where the derivative is zero.
It remains to see what the derivative of φ in x0 looks like.
So, φ ¢ in point x0 is the left part of this equation, the right value is zero.
It remains to convert that, and we get Lagrange formula of finite increments (see. Video).
This proves the theorem. What is the geometric sense of Lagrange theorem?
Let us transform the obtained formula again to the form f(b)-f(a) = f ¢(x0)(b-a) and express f ¢(x0).
Remember the geometrical meaning of the derivative, f ¢(x0) is the slope of the tangent in point x0, and the right side is the slope of cut line AB.
So what does Lagrange theorem say?
If the theorem conditions are fulfilled, there is a point on the graph where the tangent is parallel to cut line AB, and again there may be more than one point.
The last theorem in this list (the fourth) - Cauchy theorem, which presents the relation of finite increments of two functions f and g.
Again continuity condition on the segment [a, b] are true for them as well as differentiability in the segment (a, b), we add further condition that derivative g¢(x) does not convert to zero anywhere.
Finally Theorem states that there exists a point x0 in segment (a, b) for which the equation is true (see. Video).
We will not prove this theorem, its proof a bit like the proof of Lagrange theorem, and let’s switch to application of the theorems.
Now we’ll apply Cauchy theorem.
What for?
For L'Hospital rule.
We are back to calculation of limits - the most important task of mathematical analysis.
And again, as in a spiral, we are back to the same question, but at a new, higher level.
We get a powerful tool that will allow us to evaluate uncertaintis (0/0), (∞ / ∞) and we will see in the examples that we’ll learn to deal with other uncertainties.
So, the name of mathematitian is L'Hospital and this is again the 17th century.
L'Hopital rule consists of two theorems: the first tells about evaluating uncertainty (0/0), and the second - (∞ / ∞).
Let us consider the first case.
What do we know?
Functions f and g are differentiable in the neighborhood of point a with the exception, perhaps, of point a itself, and it is also known that limits f and g in point a are equal to zero.
If there is a limit of derivatives ratio of these functions at point a, then there is a limit of the ratio of these functions in point a and is equal to A.
Look at the conclusion, when we calculate the ratio limit of f to g, we just have to deal with uncertainty (0/0) (as the numerator limitis equal to zero, and the denominator limit is equal to zero).
Let's look at the proof.
So, under the condition, limits of functions f and g in point a are equal to zero.
We assume (we know nothing about point a) that the values of functions f and g in point a are equal to zero.
This will not influence the calculation of limits.
Let’s calculate one-sided limits.
We do it for the right limit and you’ll do it yourself for the left limit
So, x tends point a on the right, then x> a.
Then segment [a, x] complies conditions of Cauchy theorem
If we tend x to a, then x0, being between a and an x, also tends a.
It remains to calculate the limit (x0 tends to a, so the last limit according to the condition is number A).
L'Hospital's rule to evaluate uncertainty (∞ / ∞).
All the same, only the limits of functions f and g in point a equal to infinity.
We will leave this point without proof, but, of course, we can use this rule.
As a result, we get the formula for calculating limits (see. Video).
So, if you calculate the limit, when facing uncertainties (0/0) or (∞ / ∞), then you can apply L'Hospital's rule, provided that there is the right side of this formula.
Let's calculate the limit (see. Video), where the numerator limit is infinity, the denominator limit is also infinity.
If we divide term by term, it is easy to notice that the limit is equal to one (constant limit is equal to one, (sin x/x) limit is equal to zero).
This is (∞ / ∞), it seems possible to apply L'Hospital's rule.
Let's try.
Let’s calculate for functions f and g a limit ratio of these functions derivatives.
The derivative f = 1 + cos x, the function derivative g =1, but the function limit 1 + cos x does not exist in infinity.
Thus, the ratio limit of f to g is not equal in this case to the ratio limit of these functions derivatives, because this limit does not exist.