Mean Value Theorem (MVT)

Mean Value Theorem hypothesis: Function \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\).

For each function below, circle intervals when the conditions of the MVT are met. For non-circled intervals, provide a reason how the MVT hypothesis fails.

1. \( f(x) = x^2 \), on the intervals

  1. \( [4, 9] \)
  2. \( [-4, 9] \)
  3. \( [0, 9] \)

2. \( f(x) = \sqrt{x} \), on the intervals

  1. \( [4, 9]\)
  2. \( [-4, 9]\)
  3. \( [0, 9]\)

3. \( f(x) = \frac{1}{x} \), on the intervals

  1. \( [-10, 10]\)
  2. \( [1, 10]\)
  3. \( [-10, 0]\)

4. \( f(x) = x^{2/3} \), on the intervals

  1. \( [-8, 8] \)
  2. \( [0, 8] \)
  3. \( [1, 8] \)

Mean Value Theorem result: Then there exists a number \(c \in (a,b)\) such that

\(f'(c) = \frac{f(b) - f(a)}{b - a}\)

Choose the first valid interval from the previous page.

  1. \(f(x) = x^2\), on the interval _____
  2. \(f(x) = \sqrt{x}\), on the interval _____
  3. \(f(x) = \frac{1}{x}\), on the interval _____
  4. \(f(x) = x^{2/3}\), on the interval _____

Next, find the number \(c\) satisfying the conclusion of the MVT for each of the above functions. Hints:

  1. Calculate the slope of the secant line between the endpoints, \(m_{sec} = \frac{ f(b)-f(a) }{ b-a }\)
  2. Solve for number \(c\) so that \(f'(c) = m_{sec}\) where \(c\) is in the interval \((a, b)\)

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Application: A trucker was given a speeding ticket at a toll booth after handing in her toll booth stub. The stub indicated she had covered 149 miles in 2 hours on a toll road with a speed limit of 65 mph. Was this speeding ticket justified?