EXAM-Introduction to scientific computing math

Math 551 Homework Assignment 1 Page 1 of 2

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1. (15 points) Recall Taylor’s theorem from Calculus: “Assume a function f(x) that has k + 1 derivatives in an interval [a, b], or simply, f ∈ Ck+1 [a, b] and x0 ∈ [a, b]. Then, for every x ∈ [a, b], ∃ ξ between x0 and x such that

f(x) =

k ∑

n=0

f(n)(x0)

n! (x − x0)

n

︸ ︷︷ ︸

Pk(x)

+ f(k+1)(ξ)

(k + 1)! (x − x0)

k+1

︸ ︷︷ ︸

Rk(x)

, (1)

where Pk(x) is called the kth Taylor polynomial for f around x0 and Rk(x) is called the remainder, or truncation error”. Note that

lim k→∞

Pk(x)

gives the Taylor series for the same function f about x = x0 and also a function f is analytic in (a, b) if the Taylor series equals f ∀x ∈ (a, b). Finally, the Taylor series around x = x0 ≡ 0 is called Maclaurin series.

(a) (5 points) Find P1(x), P2(x) and P3(x) around x0 = 0 if f(x) = x 2 − 4x + 3.

How P3(x) is related to f(x)?

(b) (5 points) Same as part (a) but consider x0 = 1.

(c) (5 points) In general, given a polynomial f(x) with degree m, what can you say about f(x) − Pk(x) for k ≥ m?

2. (10 points) Given the function f(x) = cos x, find both P2(x) and P3(x) about x0 = 0, and use them to approximate cos (0.1). Show that in each case the remainder term provides an upper bound for the true (absolute) error.

3. (10 points) If f(x) = ex, then

(a) (5 points) derive the Maclaurin series of the function f(x) = ex, i.e., the Taylor series about x0 = 0 (write separately Pk(x) and Rk(x)),

(b) (5 points) find a minimum value of k necessary for Pk(x) to approximate f(x) to within 10−6 on the interval [0, 0.5] (here, you must use the remainder term).

Math 551 Homework Assignment 1 Page 2 of 2

4. (25 points) Let f(x) be: f(x) = cosh x + cos x − γ,

where γ is a parameter and takes the values of γ = 0, 1, 2, 3. Make a graph of the function f(x) for each value of γ on the interval [−3, 3] and determine whether f(x) has a root. To do so, you have to check the criteria required by the Intermediate Value Theorem. Then, using the m-file “bisect.m”, approximate the root with absolute tolerance 10−10 for the value of γ that f(x) does have a root. Include a copy of the graph of f(x) for the respective cases and MATLAB output.