Math 128A Problem Set 05

Question 1 Let p be a positive integer and
f (x) = 2x
for 0 ≤ x ≤ 2.
(a) Find a formula for the pth derivative f (p) (x).
(b) For p = 0, 1, 2 find a formula for the polynomial Hp of degree 2p + 1
such that
(k)
Hp (xj ) = f (k) (xj )
for 0 ≤ k ≤ p, 0 ≤ j ≤ 1, x0 = 0, x1 = 2.
(c) For general p prove that
|f (x) − Hp (x)| ≤

2p+2

1
p+1

for 0 ≤ x ≤ 2.
(d) Show that one step of Newton’s method for solving
g(y) = x ln 2 − ln y = 0
starting from y0 = H6 (x) gives y1 = f (x) = 2x to full double precision
accuracy for 0 ≤ x ≤ 2.
Question 2 (See BF p. 192.) For integer k ≥ 4 let
π
π
pk = k sin
Pk = k tan
k
k
√
(a) Show that p4 = 2 2 and P4 = 4.
(b) Show that
P2k =

2pk Pk
pk + P k

p2k =

pk P2k

for k ≥ 4.
(c) Approximate π within 10−4 by computing pk and Pk until Pk − pk <
10−4 .
(d) Use Taylor series to show that
∞

π = pk +

∞

qj k

−2j

Qj k −2j

π = Pk +

j=1

j=1

for some constants qj and Qj .
(e) Use extrapolation with h = 1/k to approximate π within 10−12 .
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