Exercise · ZW-FINITE-DIFFERENCES-FOR-DERIVATIVES-010 · ★★☆ Intermediate
With \(f(0)=1\), \(f(0.2)=1.44\) and \(f(-0.2)=0.84\), what is the centered approximation of \(f'(0)\) with \(h=0.2\)?
Choose one option.
Show answer
- A. \(1.5\)
Explanation
With the centered formula, \(\frac{1.44-0.84}{2\cdot 0.2}=\frac{0.6}{0.4}=1.5\); the other values come from using the forward or backward formula or dividing by 2 instead of \(2h\).
Review the concept: Finite differences for derivatives
Finite differences are approximations of the derivatives of a function \(f\) from its values at discrete points separated by a step \(h\). They turn differential problems into algebraic systems that a computer can solve.
\[f'(x)\approx \frac{f(x+h)-f(x)}{h}\]
More practice problems: finite differences for derivatives
- ZW-FINITE-DIFFERENCES-FOR-DERIVATIVES-001 What is the forward finite difference formula for f'(x)?
- ZW-FINITE-DIFFERENCES-FOR-DERIVATIVES-002 What is the backward finite difference formula for f'(x)?
- ZW-FINITE-DIFFERENCES-FOR-DERIVATIVES-003 What is the centered finite difference formula for f'(x)?
- ZW-FINITE-DIFFERENCES-FOR-DERIVATIVES-004 In finite differences, what does h represent?
- ZW-FINITE-DIFFERENCES-FOR-DERIVATIVES-005 You have f(1)=2 and f(1.1)=2.5. What is the forward approximation of f'(1) with h=0.1?
- ZW-FINITE-DIFFERENCES-FOR-DERIVATIVES-006 Let f(x)=x^2. Compute the centered approximation of f'(2) with h=0.5.
- ZW-FINITE-DIFFERENCES-FOR-DERIVATIVES-007 If the step h is halved in the centered formula, how does the truncation error change approximately?
- ZW-FINITE-DIFFERENCES-FOR-DERIVATIVES-008 Expanding f(x+h)=f(x)+hf'(x)+(h^2)/(2)f''(x)+cdots, what is the leading term of the error in (f(x+h…