Paul's Online Notes
Paul's Online Notes
Home / Calculus I / Applications of Derivatives / The Mean Value Theorem
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 4-7 : The Mean Value Theorem

For problems 1 & 2 determine all the number(s) c which satisfy the conclusion of Rolle’s Theorem for the given function and interval.

  1. \(f\left( x \right) = {x^2} - 2x - 8\) on \(\left[ { - 1,3} \right]\) Solution
  2. \(g\left( t \right) = 2t - {t^2} - {t^3}\) on \(\left[ { - 2,1} \right]\) Solution

For problems 3 & 4 determine all the number(s) c which satisfy the conclusion of the Mean Value Theorem for the given function and interval.

  1. \(h\left( z \right) = 4{z^3} - 8{z^2} + 7z - 2\) on \(\left[ {2,5} \right]\) Solution
  2. \(A\left( t \right) = 8t + {{\bf{e}}^{ - 3\,t}}\) on \(\left[ { - 2,3} \right]\) Solution
  3. Suppose we know that \(f\left( x \right)\) is continuous and differentiable on the interval \(\left[ { - 7,0} \right]\), that \(f\left( { - 7} \right) = - 3\) and that \(f'\left( x \right) \le 2\). What is the largest possible value for \(f\left( 0 \right)\)? Solution
  4. Show that \(f\left( x \right) = {x^3} - 7{x^2} + 25x + 8\) has exactly one real root. Solution