Mean Value Theorem Calculator with steps | Lagrange's theorem

Mean Value Theorem Calculator

f(x) =

Left Endpoint (a) =
Right Endpoint (b) =


Legend

The Mean Value Theorem Calculator with Steps is an excellent aid to study and understand how to find the value c that satisfies the theorem.

To use the mean value theorem calculator you just have to perform these simple actions:

  1. Enter the function, whose independent variable should be x.
  2. Enter the values of the interval [a,b].
  3. Press the green “Calculate” button, doing so will display the solution explained in detail. To see the solution represented graphically, press the “Show graph” button.

What is the Mean Value Theorem?

Mean Value Theorem or Lagrange’s Theorem states that if a function f(x) is continuous on a closed interval [a, b], there is at least one point (which is in turn differentiable) belonging to the open interval c ∈ (a , b), in which it is fulfilled that:

Mean Value Theorem Formula
Mean Value Theorem formula

Viewed graphically:

graph of the Mean Value Theorem

Since the derivative of a function at a point is the tangent of the angle α formed by the tangent line at that point, the tangents of the angles of the two lines, the secant line through AB and the tangent line at the point (c , f(c)) are parallel and the tangent of both is f'(c).

Mean Value Theorem proof

Starting from the graph shown above, we have that g(x) the secant line to the function f(x) passes through the points (a, f(a)) and (b, f(b)).

Knowing this we can define the equation of the secant line g(x):

$$g(x) – f(a) = [ f(b) – f(a) ] / (b – a) (x-a)$$

$$g(x) = [ f(b) – f(a) ] / (b – a) (x-a) + f(a)$$  

Assuming that the function h(x) is equal to f(x) – g(x) we have the following:

$$h(x) = f(x) – [[ f(b) – f(a) ] / (b – a) (x-a) + f(a)] $$

If we assume that h(a) = h(b) = 0 and h(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b).
Using Rolles’ theorem, there is some x = c in (a,b) such that h'(c) = 0.

For x=c on the open interval (a,b), h'(c) = 0. With this we have

$$h'(c) = f'(c) – [ f(b) – f(a) ] / (b – a) = 0$$

$$\boxed{f'(c) = [ f(b) – f(a) ] / (b – a)}$$

 

How to find numbers that satisfy the Mean Value Theorem

To find c in the Mean Value Theorem you must follow these steps:

  1. Check if we can apply the mean value theorem, for which we must:
    1. Determine if f(x) is continuous on the closed interval [a, b].
    2. Determine if f(x) is differentiable over the open interval } (a, b).
  2. Apply the Mean Value Theorem Formula:
    1. Find the derivative of the function and then evaluate it at c, f'(c).
    2. Introduce all known values into the Mean Value Theorem formula. You will get an equation whose independent variable is c.
    3. Isolate c to get the solution.

Examples

Example 01: Find the value of c, of mean value theorem, when f(x)=x3/4 – 2x in

Since the function f(x) is polynomial, we know that it is continuous and differentiable throughout its domain and, therefore, also in the closed and open intervals required by the theorem. Therefore, there will be at least one point that satisfies the theorem.

We apply the formula of the theorem to know the value of the derivative when x=c.

example of mean value theorem step 1

Knowing the numerical value of the derivative when x=c, we proceed to calculate the derivative of the function.

example of mean value theorem step 2

We equate the previous expression with the numerical value of the derivative in c.

These are the two values of the abscissas where the Mean Value Theorem is satisfied: c1 = -2.08 and at c2 = 2.08.

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