Exercise · ZW-CUBIC-SPLINES-009 · ★★☆ Intermediate
If \(S\) is a natural cubic spline with \(S''(x_0)=0\), what happens to \(c_0\) in the piecewise representation?
Choose one option.
Show answer
- D. \(c_0=0\)
Explanation
Since \(S''(x_0)=2c_0\), the natural condition forces \(c_0=0\).
Review the concept: Cubic splines
A cubic spline \(S\) is a piecewise function, each piece a polynomial of degree three, that interpolates the data \((x_i,y_i)\) and is of class \(C^2\) at the nodes.
\[S_i(x)=a_i+b_i(x-x_i)+c_i(x-x_i)^2+d_i(x-x_i)^3\]
More practice problems: cubic splines
- ZW-CUBIC-SPLINES-001 What degree is each piece of a cubic spline?
- ZW-CUBIC-SPLINES-002 What class of continuity does a cubic spline have at interior nodes?
- ZW-CUBIC-SPLINES-003 Which boundary condition defines the natural spline?
- ZW-CUBIC-SPLINES-004 How many cubic pieces does a spline have over n intervals?
- ZW-CUBIC-SPLINES-005 What type of linear system appears when computing the coefficients c_i?
- ZW-CUBIC-SPLINES-006 A natural spline with n=3 intervals has 4·3=12 unknowns. How many interpolation and continuity equa…
- ZW-CUBIC-SPLINES-007 For the natural spline through (0,0),(1,1),(2,0), what is S''(1) using the c_1 form of the system?
- ZW-CUBIC-SPLINES-008 A clamped spline imposes S'(x_0)=f'(x_0). Which of these statements about it is correct?