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Higher Order Derivatives Using Limits

In class, you have used the limit definition of a derivative to find the derivative of a function. In this quiz, you are going to be using the limit definition to do higher order derivatives instead of using the usual "shortcut" methods.

  1. Just as a review, use the limit definition to find the derivative of \(f(x) = 2x^2 - x\).
  2. Now that you remember the limit definition for \(f'(x)\), find a similiar limit definition for \(f''(x)\) that does NOT have \(f'\) in the answer. That is, your answer should be: \(\(f''(x) = \lim_{h \to 0} \frac{?}{?}\)\)
  3. Show that your limit definition for \(f''(x)\) works for \(f(x) = 2x^2 - x\). (Hint: Lots of algebra here)
  4. Show that your limit definition works for \(f(x) = a_0 + a_1x + a_2x^2 + a_3x^3\), where \(a_n\) is a constant.
  5. Now for even higher order derivatives!
    1. Determine the limit definition for \(f'''(x)\). (Remember that you should not have any derivatives in your limit definition.)
    2. Determine the limit defintion for \(f^{(4)}(x)\). (Hint: Look for patterns)
    3. Determine the limit definition for \(f^{(n)}(x)\).

Solution

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