First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 1. INTRODUCTION 7
Notation: C n (A) denotes the set of all functions f such that f and its first n derivatives
are continuous on A. If f is only continuous on A, then we write f ∈ C 0 (A). C ∞ (A)
consists of functions that have derivatives of all orders, for example, f(x) =sinx or
f(x) =e x .
The following well-known theorems of Calculus will often be used in the remainder of the
book.
Theorem 4 (Mean value theorem). If f ∈ C 0 [a, b] and f is differentiable on (a, b), then
there exists c ∈ (a, b) such that f ′ (c) = f(b)−f(a)
b−a
.
Theorem 5 (Extreme value theorem). If f ∈ C 0 [a, b] then the function attains a minimum
and maximum value over [a, b]. If f is differentiable on (a, b), then the extreme values occur
either at the endpoints a, b or where f ′ is zero.
Theorem 6 (Intermediate value theorem). If f ∈ C 0 [a, b] and K is any number between
f(a) and f(b), then there exists c ∈ (a, b) with f(c) =K.
Theorem 7 (Taylor’s theorem). Suppose f ∈ C n [a, b] and f (n+1) exists on (a, b), and x 0 ∈
(a, b). Then, for x ∈ (a, b)
f(x) =P n (x)+R n (x)
where P n is the nth order Taylor polynomial
P n (x) =f(x 0 )+f ′ (x 0 )(x − x 0 )+f ′′ (x 0 ) (x − x 0) 2
2!
+ ... + f (n) (x 0 ) (x − x 0) n
n!
and R n is the remainder term
R n (x) =f (n+1) (ξ) (x − x 0) n+1
(n +1)!
for some ξ between x and x 0 .
Example 8. Let f(x) =x cos x − x.
1. Find P 3 (x) about x 0 = π/2 and use it to approximate f(0.8).
2. Compute the exact value for f(0.8), and the error |f(0.8) − P 3 (0.8)|.
3. Use the remainder term R 3 (x) to find an upper bound for the error |f(0.8) − P 3 (0.8)|.
Compare the upper bound with the actual error found in part 2.