Calculus > Mean Value Theorem
You may want to read this article first: What is the Intermediate Value Theorem?
What is the Mean Value Theorem?
The Mean Value Theorem (MVT) states that if the following two statements are true:
- A function is continuous on a closed interval [a,b], and
- If the function is differentiable on the open interval (a,b),
The Common Sense Explanation
The “mean” in mean value theorem refers to the average rate of change of the function. It’s basic idea is: given a set of values in a set range, one of those points will equal the average. This is best explained with a specific example.
Let’s say you travel from your house to work, varying your speed between 40 and 50 mph. The speedometer needle will fluctuate between 40 and 50, and let’s say you average 54 mph. As the needle moves from 40 to 50, it has to pass this point at least once. I picked 54 mph arbitrarily, but you could pick any number between 40 and 50 (i.e. the “closed interval”) and the needle would have to pass that point.
Mean Value Theorem Example Problem
Example 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. Then rewriting as 2⁄3 power.
- Multiplying by (x-1)2⁄3.
- Multiplying both sides by 7⁄3.
- Raising both sides to 3⁄2.
- Adding 1 to both sides.
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