Test Bank for Numerical Analysis-10th Edition by Richard I. Burden J. Douglas Faires Annette M Burde

Test Bank for Numerical Analysis-10th Edition by Richard I. Burden J. Douglas Faires Annette M Burde

Numerical Analysis 10E Name (Print):
Chapter 01 Sample Exam
1. (10 points) Given the quadratic equation 0.987x
2 + 11.2x + 0.246 = 0 . Find the best approximation to each of the two solutions using 3 digit chopping arithmetic and the appropriate
equations for x1 and x2.
2. (10 points) Given the quadratic equation 0.987x
2 − 11.2x − 0.246 = 0. Find the best approximation to each of the two solutions using 3 digit rounding arithmetic and the appropriate
formulas.
3. (15 points) Let x0 = 0.5. Given
f(x) = −2e
−x + 1/4x
4 −
1
120
x
5 + 2x f0
(x) = 2e
−x + x
3 −
1
24
x
4 + 2
f
00(x) = −2e
−x + 3x
2 −
1
6
x
3
f
000(x) = 2e
−x + 6x −
1
2
x
2
f
(4)(x) = −2e
−x + 6 − x f(5)(x) = 2e
−x − 1
f
(6)(x) = −2e
−x
(a) (5 points) Find the Taylor Polynomial, T3(x), of degree at most 3 for f(x) expanded about
x0.
(b) (5 points) Give the general error formula for f(x) − T3(x) for any x.
(c) (5 points) Find the absolute error in using T3(0.65) to approximate f(0.65).
4. (10 points) Let x0 = 0. Given
f(x) = −2e
−x + 1/4x
4 −
1
120
x
5 + 2x f0
(x) = 2e
−x + x
3 −
1
24
x
4 + 2
f
00(x) = −2e
−x + 3x
2 −
1
6
x
3
f
000(x) = 2e
−x + 6x −
1
2
x
2
f
(4)(x) = −2e
−x + 6 − x f(5)(x) = 2e
−x − 1
f
(6)(x) = −2e
−x
(a) (5 points) Find the Taylor Polynomial, T3(x), of degree at most 3 for f(x) expanded about
x0.
(b) (5 points) Use the error formula to find a bound for the absolute error in approximating
f(0.65) with T3(0.65).
5. (10 points) Let f(x) = x
3 − e
−x
, x0 = 0.5.
(a) (5 points) Find the Taylor Polynomial, T2(x), of degree at most 2 for f(x) expanded about
x0.
(b) (5 points) Evaluate T2(0.8) and compute the actual error |f(0.8) − T2(0.8)|
Numerical Analysis 10E Name (Print):
Chapter 02 Sample Exam
1. (10 points) The equation f(x) = x
2 − 2e
x = 0 has a solution in the interval [-1,1].
(a) (5 points) With p0 = −1 and p1 = 1 calculate p2 using the Secant method.
(b) (5 points) With p2 from part (a) calculate p3 using Newton’s method.
2. (15 points) The equation f(x) = 2 − x
2
sin x = 0 has a solution in the interval [-1,2].
(a) (5 points) Verify that the Bisection method can be applied to the function f(x) on [-1,2].
(b) (5 points) Using the error formula for the Bisection method find the number of iterations
needed for accuracy 0.000001. Do not do the Bisection calculations.
(c) (5 points) Compute p3 for the Bisection method.
3. (15 points) The following refer to the fixed-point problem
(a) (5 points) State the theorem which gives conditions for a fixed-point sequence to converge
to a unique fixed point.
(b) (5 points) Given g(x) = 2 − x
3 + 2x
3
, use the theorem to show that the fixed-point sequence will converge to the unique fixed-point of g for any p0 in [-1,1.1].
(c) (5 points) With p0 = 0.5 generate p3.
4. (10 points) Suppose the function f(x) has a unique zero p in the interval [a, b]. Further,
suppose f
00(x) exists and is continuous on the interval [a,b].
(a) (5 points) Under what conditions will Newton’s Method give a quadratically convergent
sequence to p?
(b) (5 points) Define quadratic convergence.
5. (10 points) Let g(x) = 2 − x
3 + 2x
3
on the interval [-1, 1.1]. Let the initial value be 0 and
compute the result of 2 iterations of Stefffensen’s Method to approximate the solution of x =
g(x).

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