← MATH 01130 · Calculus I
MATH 01130 · Applications of derivatives
Mean Value Theorem idea
Definition
If f is continuous on [a,b] and differentiable on (a,b), then some c in (a,b) has f′(c)=(f(b)−f(a))/(b−a). Instantaneous rate equals average rate somewhere.
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Rolle’s theorem is the special case f(a)=f(b) ⇒ some c with f′(c)=0.
MVT justifies many “there exists a moment when…” arguments and underpins error estimates later.
Hypotheses matter: corners can break differentiability and void MVT.
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Attempt each problem first, then open the worked solution.
1.Verify MVT hypotheses for f(x)=√x on [0,4] and find c.