Statistics How To

Mean Value Theorem: How to Use It in Easy Steps

Calculus > Mean Value Theorem

The Mean Value Theorem states that if a function is continuous on a closed interval [a,b] and if the function is differentiable on the open interval (a,b) then there is a number c in (a,b) such that:
mean-value-theorem-formula-300x79


–which can be read as “the first derivative at c is equal to the function value at b, minus the function value at a, divided by b-a. What does this mean in English? The theorem can be applied to many real-life situations. For example, if you are driving a car, you could find your average speed, f'(c), by dividing the distance you traveled (f(b)-f(a)) by the time you took (b-a). The Mean Value Theorem states that there must be at least one moment when your speed was equal to your average speed.

Mean Value Theorem Sample Problem

Sample problem: Find a value of c for f(x) = 1 + 3√(x-1) on the interval [2,9] that satisfies the mean value theorem.

Step 1: Check that the function is continuous and differentiable. This particular function — a cubed root — is both differentiable and continuous.

Step 2: Find the derivative. Use the chain rule for this particular function.
mean-value-theorem-derivative


Step 3: Plug the derivative into the left side of the formula.
mean value theorem derivative-21-300x69


Step 4: Plug the function inputs (from the question) and the function’s values into the right side of the mean value theorem formula.
mean-value-theorem-2


Step 5: Work the right side of the equation, evaluating the function (from the question) at f(2) and f(9).
mean-value-theorem-3-300x77


Step 6: Solve for x, using algebra:
mean-value-theorem-algebra-1

  1. Multiplying both sides by 3, rewriting as 23 power.
    mean-value-theorem-algebra-2
  2. 2. Multiplying by (x-1)23
    mean-value-theorem-algebra-3
  3. Multiplying both sides by 73
    mean-value-theorem-algebra-4
  4. 4. Raising both sides to 32
    mean-value-theorem-algebra-5
  5. Adding 1 to both sides
    mean-value-theorem-algebra-6-150x114

A negative can’t be part of the solution (considering that the interval [2,9] is positive) so the point that solves the equation is:
mean-value-theorem-algebra-7

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Mean Value Theorem: How to Use It in Easy Steps was last modified: October 12th, 2017 by Stephanie Glen