Description
1.1 Calculus, Taylor series
Consider the function f(x) = sin(x)
x
.
1. Compute limx→0 f(x) using l’Hˆopital’s rule.
2. Use Taylor’s remainder theorem to get the same result:
(a) Write down P1(x), the first-order Taylor polynomial for sin(x) centered at a = 0.
(b) Write down a good upper bound on the absolute value of the remainder R1(x) = sin(x)−
P1(x), using your knowledge about the derivatives of sin(x). The goal here is to show
that R1(x)/x is negligible.
(c) Express f(x) as f(x) = P1(x)
x +
R1(x)
x
, and compute the limits of the two terms as x → 0.
1.2 Asymptotic notation
Recall the definitions of the asymptotic notations. We will say that f(x) has “order of growth x
α
as x → x0” (where x0 is either some fixed real number or ±∞) if f(x) = Θ(x
α) as x → x0.
1. Consider the functions f(x) = x sin x and g(x) = x. Is f(x) = Θ(g(x)) as x → ∞? Why or
why not? (Hint: As always, you should refer back carefully to the definition of Θ(·).)
2. Suppose that we know that f(x) = x + Θ(x
2
) and g(x) = Θ(x) > 0 as x → 0. Determine the
order of growth of f(x) + g(x).
(This problem is meant to get you comfortable with manipulating asymptotic notation when
it appears in expressions. When I say something like “f(x) = x + Θ(x
2
)”, this means that
there is some function h(x) = Θ(x
2
), and f(x) = x+h(x). That is, the fact that h(x) = Θ(x
2
)
is the only thing you know about h(x).)
3. Suppose that we know that f(x) = e
Θ(x) as x → ∞. Does this imply that f(x) = Θ(e
x
)?
(Hint: Think carefully about the definition of Θ(·), and consider f(x) = e
2x
.)
1-1
1-2 Homework 01: ICSI 401 – Numerical Methods
1.3 Relative versus absolute error
1. Suppose that you are approximating a function g(n) by some function f(n). Suppose, further,
that you know that the absolute error in approximating g(n) by f(n) satisfies |f(n)−g(n)| =
o(1) as n → ∞ (that is, limn→∞ |f(n) − g(n)| = 0). Is it true that the relative error also
decays to 0? If not, come up with functions f(n) and g(n) for which this is not true. (Hint:
Come up with some g(n) and f(n) satisfying g(n) = o(1) and f(n)/g(n) = Θ(1).)
1.4 Matlab warmup/Gentle linear algebra review
1. Complete G&C Chapter 2, Exercise 2.
2. Complete G&C Chapter 2, Exercise 3.
