MATH20602 Numerical Analysis 1 Problem Sheet 0

Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

MATH20602

Numerical Analysis 1

Problem Sheet 0

Part A is provided with solutions for you to work on. Part B will be worked on in class. Problems marked with * are from previous exam papers.

This week there is no Part A! In this sheet we will revise some of the important theo- rems that we will use in this course.

Reminder of the theorems

Theorem 0.1 (Intermediate value theorem).  Let f be continuous on [a, b]. Then f is bounded on [a, b] and if y satisfies

inf   f (x) < y <  sup  f (x),

xe[a,b]                             xe[a,b]

then there exists ξ ∈ [a, b] such that f (ξ) = y. In particular, the infimum and supremum are achieved.

Theorem 0.2 (Taylor’s theorem).  Let f be afunction on [a, b], such that n derivatives of f exist and are continuous on [a, b]. Assume further that f(n)  is differentiable on (a, b). Let x, x0 be in [a, b]. Then there exists ξ ∈ (a, b) such that

f (x) = f (x0 ) + f\ (x0 )(x _ x0 ) + f\\ (x0 )(x _ x0 )2 + . . .

+  (x _ x0 )n +  (x _ x0 )n+1 .

This variant of Taylor’s theorem has the remainder term expressed in Lagrange’s form.

Theorem 0.3 (Mean value theorem).  Let f be continuous on [a, b] and differentiable on (a, b). Then there exists a number ξ ∈ (a, b) such that

f (b) = f (a) + f\ (ξ)(b _ a).

This can also be written as

f (b) _ f (a)

b _ a     .

Theorem 0.4 (Rolle’s theorem).  Let f be continuous on [a, b] and differentiable on (a, b), and such that f (a) = f (b).  Then there exists a number ξ  ∈ (a, b) such that f\ (ξ) = 0.

Part B

(0.1)

1. Consider the function f : [0, 1] → R given by

f (x) =

Why can we not use the intermediate value theorem to show that there is a root to the equation f (x) = 0?

2. Consider the polynomial

p(x) = 16x3 _ 44x2 + 36x _ 9.

Which of these intervals can we guarantee, using the IVT, contains a root of this polynomial: [_1, 0], [0, 1], [1, 2], [2, 3]?

(0.2)

1. Compute the quadratic Taylor’s approximation p2 (x) of the function f (x)  = exp(x) defined on the interval [0, 1] around x0  = 0. Hint: This means compute the first three terms of the formula found in Taylor’s theorem.

2. By Taylor’s theorem there exists ξ ∈ (0, 1) such that the error in this approxima- tion satisfies                                              \\\

ef (x) _ p2 (x)e =  ex _ 0e3 .

Using this, find a bound on the greatest possible difference between f (x) and p2 (x) on [0, 1]. Hint: What is f\\\ (ξ), and what is its supremum on ξ ∈ (0, 1)? What is the maximum value of ex _ 0e for x ∈ [0, 1]?

(0.3)    Verify the result of the mean value theorem for the following polynomial on the interval [_2, 3]:

p(x) = x2 _ 3x _ 2.

(0.4)    Suppose that a smooth function f : [a, b] → R has n distinct roots (with n > 2), i.e. there are n different values of x such that f (x) = 0. Use Rolle’s theorem to find a minimum number of roots that f\ must have.


2023-03-18