Homework #5
1. Use calculus to find the constant Csuch that
|f(x)−P(x)| ≤ C
for all x∈[0,0.3], where P(x) is the Lagrange interpolating polynomial for data with
nodes x0= 0, x1= 0.1, x2= 0.3 and values from the underlying function f(x) = ex+1.
2. Use calculus to find the constant Csuch that
|f(x)−P(x)| ≤ C
for all x∈[0,1], where P(x) is the piecewise linear interpolating polynomial for data
with nodes xj=j/10, j= 0,...,10 and values from the underlying function f(x) =
x2+ 1.
3. Find nsuch that the piecewise linear interpolating polynomial for data with nodes
xj= 2j/n,j= 0, . . . , n and values from the underlying function f(x) = sin xhas
absolute error in [0,2] less than 10−5.
4. Suppose x0= 0, x1= 0.4, x2= 0.7 and f[x2]=6, f [x1, x2] = 10, f[x0, x1, x2] = 50/7.
Find the values of f[x0], f[x1], f [x0, x1].
5. Consider the data points (−2,1),(−1,4),(0,11),(1,16),(2,13),(3,−4).
(a) Write down the divided difference table associated to this data.
(b) Determine from the table the degree of the Lagrange interpolating polynomial
passing through these data points.
6. Write down the the Newton form of the Lagrange interpolating polynomial using x1=
1, x2= 3, x3= 4, x4= 6 and f[x3] = 5, f [x3, x2] = −1, f [x3, x2, x4] = −2, f[x3, x2, x4, x1] =
3.
7. (a) Verify that f[x0, x1, x2] = f[x2, x0, x1].
(b) Also verify that the Newton forms for the Lagrange interpolating polynomial
using the data points with nodes x0=a, x1=band the data points with nodes
in reverse order x0=b, x1=asimplify to the same polynomial.
8. Consider the data points (−2,−1),(0,1),(−1,3).
(a) Write down the Newton interpolatory divided difference form for this data.
(b) Add the data point (1,−1) and write down the new Newton interpolatory divided
difference form.
(c) Evaluate the polynomial of part (b) and its derivative at x= 1/2.
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