Average or Lagrange value theorem. Exercises solved step by step.

Next I’m going to explain what it says and how to interpret the average value theorem, also known as Lagrange or finite increment theorem. This theorem is explained in the 2nd year of high school when the applications of the deviradas are studied.

We will see what it means step by step and we will apply it in several solved exercises.

Average value theorem

The theorem of the average value says so:

If we have a continuous f(x) function in the closed interval [a,b] (it has to be continuous in x=a and x=b) and derivable in the open interval (a,b) (it doesn’t have to be derivable either in x=a or in x=b), then there is at least one point c, belonging to the open interval (a,b), such that at that point it is verified:

mean value theorem

Also f(a) and f(b) have to be different.

Symbolically, we can express it this way:

theorem of the mean value examples

And what does this mean?

We have a f(x) function that is continuous in [a,b], derivable in (a,b), like this one:

mean value theorem resolved exercises

The points x=a and x=b, belong to the function and we also see that the point x=a, has a value of the function f(a) and the point b has a value of the function f(b) that is different from f(a).

Therefore, this function fulfills the conditions for the theorem of the average value to be fulfilled.

If we draw a line through points A and B:

mean value theorem

The slope of that line has the following formula:

theorem of mean value exercises solved

That corresponds to the slope of a line passing through two points.

What the theorem of the average value says is that if all the previous conditions are fulfilled, which we have seen yes, then there is at least one point c, in which the tangent line at that point is parallel to the line that passes through points A and B:

lagrange theorem

The equation of the slope of the tangent line at a point is equal to that derived from the function at that point. Therefore, at point c, the equation of the slope of the tangent line will be:

lagrange theorem exercises

When two lines are parallel, it means that they have the same slope, so the slope of the tangent line at point c and the slope of the line through A and B are equal and therefore:

ejercicios teorema del valor medio

The average value theorem says that there is at least one point c, which verifies all of the above, or in other words, that there can be more than one point.

In this case, as we see in the graph of the function, we have another point d where the line tangent to the function is parallel to the line passing through A and B:

lagrange exercises solved

Therefore, this point is also fulfilled:

median value theorem exercises

How to apply the theorem of the average value. Exercises solved

Let us now see some examples of how to apply the average value theorem and calculate the c point of the theorem.

Example 1

Calculate the point c that satisfies the average value theorem for the next function in the interval [0.1]:

lagrange theorem solved exercises

First of all, we must check if the conditions are fulfilled so that the theorem of the average value can be applied. We must check if the equation is continuous in [0,1] and derivable in (0,1).

Continuity:

The function is continuous in all R, being a polynomial function, so it will also be continuous in the interval [0,1].

Derivability:

The function is derivable in (0,1) if its derivative is continuous in that interval.

The derivative of the function is:

midpoint theorem

That is continuous in all R being a polynomial function, therefore f(x) is derivable.

It is continuous in [0,1] and derivable in (0,1), therefore, there is a value of c in that interval such that:

mean value theorem for derivatives

Let’s go on to calculate point c of the theorem.

We calculate what the function is worth at the extremes of the interval:

mean value of a function solved exercises

theorem of the mean value exercises

And we calculate f'(c):

teorema valor medio

On the other hand, we calculate f'(c) from f'(x):

mean value exercises

Substituting x for c:

average value of a function exercises solved

We match both results of f'(c) and we are left with an equation that depends on c and where we can clear it and find the value of c they are asking for:

ejercicios teorema valor medio

Example 2

Calculates the c point that satisfies the average value theorem for the next function in the interval [0.4]:

mean value

We have to check that the function is continuous and devirable in that interval. We have a critical point at point x=2, so we are going to study continuity and derivability at that point (both sections are continuous and derivable because they are polynomials).

Continuity:

To see if the function is continuous in x=2, we have to check that its lateral limits and the value of the function in x=2 coincide.

The limit on the left of x=2 is:

theorem mean value resolved exercises

The limit on the right:

ejercicios teorema de lagrange

And the value of the function:

lagrange exercises

The lateral limits and the value of the function in x=2 coincide:

mean value theorem for resolved integrals exercises

So the function is continuous at x=2

Now let’s see if the function is derivable in x=2

For this, we obtain the derivative of f(x):

theorem of the intermediate value exercises resolved

And now we check if f'(x) is continuous at x=2.

The limit on the left is:

medium value theorem exercises

The limit on the right:

lagrange exercises

And the value of f'(x) in x=2 is:

ejercicios resueltos de lagrange

The lateral limits and the value of f'(x) coincide:

ejercicios resueltos lagrange

Therefore f'(x) is continuous for x=2 and f(x) is derivable for x=2.

They fulfill the two obligatory conditions, then the theorem of the average value can be applied and there will be a point c in the interval [0,4] such that:

lagrange theorem definition

We calculate the value of the function at the extremes:

theorem of the intermediate value examples

the midpoint theorem

And we calculate the value of f'(c):

theorem of the mean value solved exercises pdf

On the other hand, we get f'(c), from f'(x), replacing x with c:

teorema de rolle ejercicios resueltos

In the first stretch we get no value of c, but in the second stretch, it depends on c, which we equal to the value of f'(c) previously calculated and we get what c is worth:

theorems of the mean value

Example 3

Find a and b for f(x) to meet the conditions of the average value theorem in [0,2] and calculate the c point that satisfies the average value theorem for that interval:

resolved mean-value theorem exercises

For the conditions of the average value theorem to be met, the function must be continuous and derivable at point x=1.

For it to be continuous, the lateral limits of x=1 must coincide.

The limit on the left is:

lagrange mean value theorem

And the limit on the right is:

intermediate value theorem resolved exercises

As they have to be equal, we match both results, obtaining the following equation

mean value theorem for derivatives

For the function to be derivable, its derivative must be continuous. Therefore, we obtain the derivative of the function:

teorema de rolle ejercicios

And we check its continuity in x=1. For it the lateral limits must coincide:

applications of the mean value theorem

theorem of the mean value of the integral calculation resolved exercises

Matching both results, we obtain directly the value of b:

theorem of the mean value formula

This value of b, we substitute it in the equation obtained previously:

lagrange method solved exercises

From where we clear the value of a:

lagrange exercises

Once the values of a and b have been calculated, we substitute them in the function

lagrange exercises resolved

And we are going to calculate point c of the theorem of the average value.

We already know that the function is continuous and derivable, so we now calculate the value of the function at the extremes of the interval:

mean value theory

theorem of the intermediate value exercises

And now we calculate the value of f'(c):

resolved exercises of the mean value theorem

On the other hand, we obtain f'(c), from f'(x), replacing x with c:

lagrange theorem exercises

The first section depends on c, so we equal it to the value of f'(c) that we have obtained before and we clear the value of c:

mean value theorem for integrals examples

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