Instruction: Write your answers clearly and show
all relevant work including details.
You may use a calculator and5- digit chopping unless specified otherwise. Note the typo in Question 3(c) corrected on 2/16/2009.
Section 2.3 : 1. (3 pts.) Let f(x) = e2x – x2 + cos x. Use Newton’s method to approximate
a root to the equation f (x) = 0 after 3 iterations, by using the initial approximation
2. (4 pts.) Define the same function f(x) = e2x – x2 + cos x as in Question 1.
approximation to a root for equation f(x) = 0 with = –1.0 and
0.0 after 3
iterations, based on (a) the Secant method; and (b) the method of
false position. Section 2.4: 3. (7 pts.) Define f(x) = x2 ln (1+x) for x > −1.
Show that x = 0 is a zero ofmultiplicity 3 for the function f(x). (Hint: use
Theorem 2.11 and prove that f(0) = f’(0) = f’’(0) = 0.) (b) Use Algorithm 2.3 (Newton’s method)
to compute a root for the equation f(x) = x2 ln (1+x) = 0 using an initial
approximation = 0.5 and an
accuracy to within 10-5.
Redo Part (b) by computing an approximation to the root for the equation f(x) =
x2 ln (1+x) = 0 using an initial approximation
= 0.5 based on the modified
Newton-Raphson method described in Equation 2.11 of the text. Compare this
algorithm ’s speed of convergence to the speed of Newton’s method of Part (a).
4. (4 pts.) Suppose the sequence converges to p as n → ∞ and
n ≥ 1. (a) Show that ; and (b) in general, find a
relationship between for any n ≥ 2. Section 2.5: 5. (4 pts.) Let f(x) = x2 ln (1+x) = 0 as in Question 3. Apply Steffensen’s
Newton’s method and compute approximation
as done in Example 2 of
section (p. 85) of the text. Section 2.6: 6. (8 pts.) Let f(x) = x3 – 9x2 – 9x – 10.
(a) Apply Newton’s method and Horner’s method using synthetic division to
approximate the real root of f(x) = 0 after 3 iterations, using initial
= 8.0. Show your work in each of the steps .
(b) Use Algorithm 2.8 (Muller’s method)
to approximate the two complex roots for f (x) = 0 within 10-5, using the initial
approximations.0, 1, and 2. In addition , find the exact values of the two
complex roots. (using the quadratic formula ).
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