Mean Value Theorem (MVT)
Read this topicExplanations 4
- In the previous section, we talked about Rolle's theorem, which stated that given a function f(x)f(x)f(x) continuous on [a,b)[a,b)[a,b) and differentiable on (a,b)(a,b)(a,b), if…
By Steven Kwon, Alex Federspiel, Victor Wang
- Steven Kwon has written about the intuition and proof behind the MVT here. But why do we need the Mean Value Theorem (MVT)? Well, the MVT tells us that for any interval [a,b][a,…
By Victor Wang, Moor Xu
- Here is a worksheet to practice your newly learned skill: applying the Generalized Rolle's Theorem (a.k.a Mean Value Theorem). Good luck! P.S. Answers are on the bottom.
By Freya Edholm
- (Video) The Mean Value Theorem by PatrickJMT Here's a video by patrickJMT explaining the Mean Value Theorem and how it is used. The Mean Value Theorem states that some function,…
By Alex Federspiel
Problems 10
- Suppose that fff is a differentiable function. If f(1)=20f(1) = 20f(1)=20 and f′(x)≥4f'(x) \ge 4f′(x)≥4 for 1≤x≤41 \le x \le 41≤x≤4, what is the smallest value that f(4)f(4)f(4)…
- Does there exist a differentiable function f(x)f(x)f(x) such that f(0)=−1f(0) = -1f(0)=−1, f(2)=4f(2) = 4f(2)=4, and f′(x)≤2f'(x) \le 2f′(x)≤2 for all xxx?
- Use the Mean Value Theorem for the function f(x)=x3−2x2+4x−9f(x)=x^3-2x^2+4x-9f(x)=x3−2x2+4x−9 on the interval [e,π][e,\pi][e,π].
- Use the Mean Value Theorem for the function f(x)=4xx2−4f(x)=\frac{4x}{x^2-4}f(x)=x2−44x on the interval [0,10][0,10][0,10].
- Use the Mean Value Theorem for the function f(x)=sin(x)f(x)=\sin(x)f(x)=sin(x) on the interval [0,π][0,\pi][0,π].
- Use the Mean Value Theorem for the function f(x)=1x2−1f(x)=\frac{1}{x^2-1}f(x)=x2−11 on the interval [2,2.1][2,2.1][2,2.1].
- Use the Mean Value Theorem for the function f(x)=x3−6x2+11x−6f(x)=x^3-6x^2+11x-6f(x)=x3−6x2+11x−6 on the interval [1,3][1,3][1,3].
- Use the Mean Value Theorem for the function f(x)=x+4f(x)=\sqrt{x+4}f(x)=x+4 on the interval [−4,4][-4,4][−4,4].
- Use the Mean Value Theorem for the function f(x)=1ex2f(x)=\frac{1}{e^{x^2}}f(x)=ex21 on the interval [−2,2][-2,2][−2,2].
- Use the Mean Value Theorem for the function f(x)=4−x2f(x)=\sqrt{4-x^2}f(x)=4−x2 on the interval [0,2][0,2][0,2].