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:
–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 2: Find the derivative. Use the chain rule for this particular function.
Step 3: Plug the derivative into the left side of the formula.
Step 4: Plug the function inputs (from the question) and the function’s values into the right side of the mean value theorem formula.
Step 5: Work the right side of the equation, evaluating the function (from the question) at f(2) and f(9).
Step 6: Solve for x, using algebra:
- Multiplying both sides by 3, rewriting as 2⁄3 power.
- 2. Multiplying by (x-1)2⁄3
- Multiplying both sides by 7⁄3
- 4. Raising both sides to 3⁄2
- Adding 1 to both sides
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