2025-11-07 Quiz 6

  1. Which of the following limit statements regarding the first derivative of a smooth function \(f \left( x \right)\) is true?

  1. \(f' \left( x \right) = \lim_{h \rightarrow 0} \frac{f \left( x + h \right) - f \left( x \right)}{h}\)

  2. \(f' \left( x \right) = \lim_{h \rightarrow 0} \frac{f \left( x - h \right) - f \left( x \right)}{-h}\)

  3. \(f' \left( x \right) = \lim_{h \rightarrow 0} \frac{f \left( x + h \right) - f \left( x - h \right)}{2h}\)

  4. All of the above

Answerd

2. A first-order difference approximation of the first derivative is written in the form \(f' \left( x \right) = G \left( h \right) + a h + \mathcal{O} \left( h^2 \right)\), where \(a\) is some constant. Which of the extrapolation formulas below is an \(\mathcal{O} \left( h^2 \right)\) approximation of \(f' \left( x \right)\)?

  1. \(\frac{4 G \left( h / 2 \right) - G \left( h \right)}{3}\)

  2. \(\frac{2 G \left( h / 2 \right) - G \left( h \right)}{3}\)

  3. \(2 G \left( h / 2 \right) - G \left( h \right)\)

  4. None of the above

Answerc

3. Draw a pair of graphs showing why the forward finite difference formula for approximating the first derivative of a function will be incorrect. Do not forget to add labels to your charts. Include a brief description explaining your graphs. Hint: One of your graphs should be the true tangent line of the function at a point.

AnswerSee the discussion on Wikipedia.