1. Page 325 Problem 5

(a) use centered formula of order O()

(b) use centered formula of order O()

(c)

2. Page 364 Problem 3(a)

for

since

so,

suppose

(1) approximate the surface area using the composite trapezoidal rule with M=10.

(2) approximate the surface area using the composite simpson rule with M=5.

3. Page 423 Problem 6

the distance between a point ()and a line is:

the line is

so,

so the distance between any point (x,y) on circle and this line is:

here (x,y) also satisfy

so that

solve

so

We have 4 possible points:

(1.09756, 4.878049)

(1.09756, -4.878049)

(-1.09756, 4.878049)

(-1.09756, -4.878049)

thus, the point on the circle which is farthest from the chord AB is (-1.09756, -4.878049)

4. Page 440 Problem 5

(a)

(1) h=0.2

(2)h=0.1

(b) exact solution

(c) F.G.E

So the F.G.E behaves as expected when h is halved.

5. Page 448 Problem 4

(a)

· h=0.2

· h=0.1

(b) exact solution

(c) F.G.E

So the F.G.E behaves as expected when h is halved.

6. Page 457 Problem 5

(a)

h=0.1

(b)

exact solution is

7. Page 471 Problem 4

(a)

· h=0.2

· h=0.1

(b) exact solution

(c) F.G.E

So the F.G.E behaves as expected when h is halved.

8. Page 494 Problem 8

(a) Reformulate

use substitution , then

the differential equation becomes a system:

with

(b) Programming result:

(Courtesy of Alan T. Saunders)

Enter the left end point of the interval: 0

Enter the right end point of the interval: 2

Enter the number of steps: 40

Input the initial value of : 2

Input the initial value of : 3

(c) (Courtesy of Alan T. Saunders)