A proposed interpolant is p(x)=2x^2-x+1 for data (-1,4), (0,1) and (2,7). What is the residual p(2)-7 at x=2?

A data set contains (0,2), (1,3) and (1,5). Why can no polynomial interpolate all three recorded values?

• The two distinct nodes require a quadratic fit.
• A polynomial must give equal outputs at 0 and 1.
• The input 1 is assigned two different outputs.
• The input 0 cannot be used as a polynomial node.

Three data points have distinct x values 0, 1 and 2. The unique degree-limited interpolant has degree at most one less than the number of points. What is that degree limit?

• 1
• 2
• 3

Adding equally spaced interpolation nodes always makes the absolute error smaller at every point between the endpoint nodes.

Let f have a continuous second derivative on a closed interval containing distinct nodes x0,x1 and a non-node x. Let p have degree at most 1 and match f at both nodes. Which formula holds for some c inside the smallest interval containing these points?

• f(x)-p(x) = f''(c)(x-x0)(x-x1)/2
• f(x)-p(x) = f''(c)(x-x0)(x-x1)/3
• f(x)-p(x) = f'(c)(x-x0)(x-x1)/2

The function f(x)=|x| is interpolated at -1 and 1 by p(x)=1. Which requirement of the lesson's linear interpolation error theorem is not met?

• The function is continuous on the whole interval.
• The function has a continuous second derivative.
• The interpolation nodes have distinct x values.
• The interpolant has degree at most one on [-1,1].

For data (-1,2), (1,0) and (2,5), which expression is the Newton interpolant using the nodes in that order?

• 2-(x+1)+2(x+1)(x-1)
• 2+(x+1)+2(x+1)(x-1)
• 2-(x+1)+5(x+1)(x-1)
• 2-2(x+1)+2(x+1)(x-1)

For the points (0,1) and (2,9), what is the first divided difference?

• 2
• 4
• 8

With nodes 0,1,2, the Lagrange basis polynomial for node 0 is L_0(x)=(x-1)(x-2)/2. What is L_0(0.5)? Enter your answer as a decimal.

Two different polynomials, each of degree at most 2, can pass through the same three points with distinct x coordinates.