gen-define-the-mean-value-theorem-and-give-one-short-worked-exam-944c

Define the mean value theorem and give one short worked example.

# Mean Value Theorem ## Definition If a function *f* is: - **continuous** on the closed interval [*a*, *b*], and - **differentiable** on the open interval (*a*, *b*) then there exists at least one point *c* in (*a*, *b*) such that: $$f'(c) = \frac{f(b) - f(a)}{b - a}$$ *In words: there's at least one point where the instantaneous rate of change equals the average rate of change.* --- ## Worked Example **Problem:** Verify the MVT for *f(x) = x²* on [1, 3]. **Solution:** 1) Check conditions: - *f(x) = x²* is continuous on [1, 3] ✓ - *f(x) = x²* is differentiable on (1, 3) ✓ 2) Find the average rate of change: $$\frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = \frac{8}{2} = 4$$ 3) Find where *f'(c) = 4*: - *f'(x) = 2x* - Set *2c = 4* - Therefore *c = 2* 4) Verify: *c = 2* is in (1, 3) ✓ **Conclusion:** The MVT is satisfied at *c = 2*.