Practice question · Multiple choice
The Mean Value Theorem says a differentiable function on [a,b] has some point where the instantaneous rate equals the average rate over the whole interval. Why is that guaranteed rather than merely plausible?
Hints
- Suppose your speed were always strictly below your average speed for the trip. Could you have covered the distance?
- The hypotheses are continuity on the closed interval and differentiability on the open one. Where is each needed?
Show the answer
C. Because always below could not climb far enough, always above too far.
Why
The driving version makes it obvious: cover 120 miles in two hours and if the speedometer never read 60, it was either always below and you fell short, or always above and you overshot. The hypotheses do real work, drop differentiability and |x| on [−1, 1] is a counterexample. It underlies f' > 0 implying increasing, and the error bounds on Taylor approximations.
Practise Taylor Approximation
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