Mean Value Theorem
Somewhere between two points, the instantaneous slope equals the average slope.
About this topic
The Mean Value Theorem says that if f is continuous on [a, b] and differentiable on (a, b), there is a point c where the instantaneous rate equals the average rate: f′(c) = (f(b) − f(a))/(b − a). Geometrically, some tangent is parallel to the secant joining the endpoints.
A familiar illustration is a road trip: if you travel 150 km in 2 hours, your average speed is 75 km/h, so at some moment your speedometer must read exactly 75. This is how highway cameras can catch speeders by measuring time between two points. Rolle's theorem is the special case where f(a) = f(b), so some tangent is horizontal.
Example: for f(x) = x² on [1, 3], the average rate is (9 − 1)/2 = 4, and f′(c) = 2c = 4 gives c = 2, inside the interval as promised.
The theorem is mostly a tool for proofs: it shows that a function with zero derivative is constant, that two functions with equal derivatives differ by a constant, and that a positive derivative means the function is increasing. The conditions matter: f(x) = absolute value of x on [−1, 1] has average slope 0 but no point with derivative zero because it is not differentiable at 0.
Key ideas
- Requires continuity and differentiability
- Guarantees a point c
- Connects average and instantaneous rates
Quick check
Use cases: where Mean Value Theorem helps
- Homework: review the key idea behind Mean Value Theorem before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Mean Value Theorem without notes.
- Visual learning: play with the interactive lab to see how Mean Value Theorem changes when you adjust its inputs.
- Teaching a friend: explain Mean Value Theorem in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Mean Value Theorem?
Somewhere between two points, the instantaneous slope equals the average slope.
What will I learn in Mean Value Theorem?
Requires continuity and differentiability. Guarantees a point c. Connects average and instantaneous rates.
Is Mean Value Theorem part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Mean Value Theorem lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Mean Value Theorem?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.