Mean Value Theorem
Some point has tangent slope equal to average slope.
Key idea
Some point has tangent slope equal to average slope.
In depth
If f is continuous on [a, b] and differentiable on (a, b), some c has f′(c) = (f(b) − f(a))/(b − a): at some point the instantaneous rate equals the average rate. If you average 60 mph, you were going exactly 60 mph at some moment.
Worked example
f(x) = x² on [0, 2]: average slope 2, and f′(1) = 2, so c = 1.
Formula
Real-world uses
- Proving that a speed camera’s reading must match the average speed over the road section
Common mistake
Applying it when the function is not differentiable inside the interval.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Mean Value Theorem?
Some point has tangent slope equal to average slope. It is a topic at the expert level in Calculus.
What is the formula for Mean Value Theorem?
The key formula is f′(c) = (f(b)−f(a))/(b−a).
Where is Mean Value Theorem used in real life?
Real-world examples: Proving that a speed camera’s reading must match the average speed over the road section.
What should I learn before Mean Value Theorem?
Make sure you are comfortable with Optimization Problems, the previous topic in Calculus.
What is a common mistake with Mean Value Theorem?
Applying it when the function is not differentiable inside the interval.
How can I practise Mean Value Theorem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Mean Value Theorem?
Move on to Rolle’s Theorem next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.