Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

2 CHAPTER 1. INTRODUCTION
f(x) =
n∑ f (k) (c) (x − c) k
f(x) −
k!
k=0
n∑
∣ f(x) − f (k) (c) (x − c) k
k! ∣
k=0
n∑ f (k) (c) (x − c) k
k=0
k!
= f (n+1) (ξ) (x − c) n+1
(n + 1)!
∣ f (n+1) (ξ) ∣ |x − c|
n+1
=
.
(n + 1)!
+ f (n+1) (ξ) (x − c) n+1
(n + 1)!
On the left hand side is the difference between f(x) and its approximation by Taylor’s series.
We will then use our knowledge about f (n+1) (ξ) on the interval [a, b] to find some constant M such
that
n∑
∣ f(x) − f (k) (c) (x − c) k
k! ∣
k=0
=
∣
∣f (n+1) (ξ) ∣ ∣ |x − c| n+1
(n + 1)!
≤ M |x − c| n+1 .
Example Problem 1.2. Find an approximation for f(x) = sin x, expanded about c = 0, using
n = 3.
Solution: Solving for f (k) is fairly easy for this function. We find that
so
f(x) = sin x = sin(0) + cos(0) x
1!
= x − x3
6
+
sin(ξ) x4
+ ,
24
( )∣ ∣ sin x − x − x3 ∣∣∣
=
6
− sin(0) x2
2!
+
− cos(0) x3
3!
sin(ξ) x 4
∣ 24 ∣ ≤ x4
24 ,
+
sin(ξ) x4
4!
because |sin(ξ)| ≤ 1.
⊣
Example Problem 1.3. Apply Taylor’s Theorem for the case n = 1.
Solution: Taylor’s Theorem for n = 1 states: Given a function, f(x) with a continuous derivative
on [a, b], then
f(x) = f(c) + f ′ (ξ)(x − c)
for some ξ between x, c when x, c are in [a, b].
This is the Mean Value Theorem. As a one-liner, the MVT says that at some time during a trip,
your velocity is the same as your average velocity for the trip.
⊣
Example Problem 1.4. Apply Taylor’s Theorem to expand f(x) = x 3 − 21x 2 + 17 around c = 1.
Solution: Simple calculus gives us
f (0) (x) = x 3 − 21x 2 + 17,
f (1) (x) = 3x 2 − 42x,
f (2) (x) = 6x − 42,
f (3) (x) = 6,
f (k) (x) = 0.