MA/CS/EGR 537: Assignment 7 Answers/Grading

p. 204, 4 (a) 1 - x + x² - x³ + x⁴ - x⁵

(b) (1 + c)^-7 x⁶, where 0 < |c| < |x| and c and x have the same sign.

-2 goof up trivially
-5 wrong error term

p. 213, 2 (a) P(3) = 1.78

(b) P'(3) = -0.40

(c) I(P) = 5.97

(d) P(4.5) = 4.5

(e) We can derive P(x) = D³ + Cx² + Bx + A by solving the system of linear equations

A	+						=	2.08
A	+	B	+	C	+	D	=	2.02
A	+	2B	+	4C	+	8D	=	2.00
A	+	4B	+	16C	+	64D	=	2.00

which this gives us P(x) = -0.04x³ + 0.14x² - .16x + 2.08.

-2 goof up part trivially
-5 miss part entirely
-2 I(0) wrong

p. 225, 2
f(1.5) = 2.83333, f(1.2) = 2.86667
Lagrange
(a) P₂(x) = .4x² - 1.2x + 3.8
P₂(1.5) = 2.9
P₂(1.2) = 2.936

(b) P₃(x) = .8x³ + 4.8x² - 8.8x + 7.8
P₃(1.5) = 2.7
P₃(1.2) = 2.7696

Hermite
(a) b₀ = 3, b₁ = -1, b₂ = 1, b₃ = -.5, b₄ = .2, and b₅ = -.08
P₂(1.5) = 2.83
P₂(1.2) = 2.8637824

(b) b₀ = 4.5, b₁ = -7, b₂ = 8, b₃ = -8, b₄ = 4, b₅ = -2, b₆ = .8, and b₇ = -.32
P₃(1.5) = 2.82
P₃(1.2) = 2.861013504

-2 close enough
-5 wrong b_i
-10 gave up
-15 did not do Hermite case

p. 235, 2
P₁(x) = 5 - 2x
P₂(x) = 5 - 2.5x + .5x²
P₃(x) = 5 - 2.7x + .8x² - .1x³
P₄(x) = 5 - 2.718x + .8033x² - .118x³ + .003x⁴

P₁(2.5) = 0
P₂(2.5) = 1.875
P₃(2.5) = 1.6875
P₄(2.5) = 1.6846875

-5 close
-12 missing

p. 247, 1
T₄(x) = 2x T₃(x) - T₂(x) = 8x⁴ - 8x² + 1
T₅(x) = 2x T₄(x) - T₃(x) = 16x⁵ - 20x² + 5x

-5 mislabeled
-5 nonsimplified
-10 quite incorrect
-12 missing

p. 248, 8
(a) Using Theorem 4.9,

max | sin(x) - P₃(x) | < [ 2 ( 2³ ) ] / [ 4⁴ 4! ] max | sin⁽⁴⁾(x) | for -1 < x < 1.

Using inspection (i.e., plotting the curves and looking at the differences), we can determine that the largest error occurs at the endpoints and is 9.93043x10^-4.

-2 almost
-2 not simplified
-5 did not write out P₃(x)
-5 wrong nodes
-5 wrong formula for Theorem 4.9
-15 did not try

Cheers,
Craig C. Douglas