Show that You may take for granted that max z€0,2 VI -in
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Do just last three parts
![Problem 2. In this problem, we will practice a bit with numerical integration. We take
the following facts for granted: for any a <b, any f: [a, b] →R, and any n ≥ 1,
and
|RIGHT(ƒ,n) – ª f(x)dx| ≤
-
SIMP(f. n)-
Compute
- [* 1(2)dz < (b − a)²
180n
(S)
where RIGHT(f, n) is the right hand Riemann sum with n equal intervals and SIMP(f, n)
is the approximation of f f(r)dx using Simpson's rule with n equal intervals. Here f(iv)
is the fourth derivative of f.
(i) Suppose that f is a third degree polynomial f(x) = a32³ +₂2² +₁+ao. Argue
that, for any n, a, and b,
You may take for granted that
Justify your answer.
1 (b-a)²
2 11
SIMP(f.n) =
can you determine if
(ii) Show that the same is not true for RIGHT(f, n) by giving an example of a third degree
polynomial f, a natural number n, and endpoints a <b so that
RIGHT(f, n) + [*ª f(x)dx.
Hint: linear functions are third degree polynomials and n = 1 is a natural number...
(iii) For the remaining parts, use = 0, b=2, and
f(x) =
Justify your answer.
- ²1(2)
-max f(iv) (r),
zab
T
2
max f'(x)
z€[a,b]
RIGHT(f, 4) and
SIMP(f, 4).
You may do this by hand, by writing code, or by using an online calculator, but you
must document your computations (either via screenshots or by writing down your
work).
(iv) Show that
f(r)dr.
-4.
max f'(x)| ≤
S
z€0,2
(v) Using the value of RIGHT(f, 4) from (in), as well as (iv) and (L), do you have enough
information to determine if
[√x -
(L)
<1?
(vi) Using the value of SIMP(f, 4) from (iii), as well as (S) and
max f)(x) ≤ 3,
z€ 0,2
[²* 1(x)dx= [²√² ce-4d₂ <1?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b0d42b5-3479-420b-a2e4-bea672ad5500%2Fe556a85d-8603-4928-a4ce-c22d866d4e94%2Fp8srmnd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 2. In this problem, we will practice a bit with numerical integration. We take
the following facts for granted: for any a <b, any f: [a, b] →R, and any n ≥ 1,
and
|RIGHT(ƒ,n) – ª f(x)dx| ≤
-
SIMP(f. n)-
Compute
- [* 1(2)dz < (b − a)²
180n
(S)
where RIGHT(f, n) is the right hand Riemann sum with n equal intervals and SIMP(f, n)
is the approximation of f f(r)dx using Simpson's rule with n equal intervals. Here f(iv)
is the fourth derivative of f.
(i) Suppose that f is a third degree polynomial f(x) = a32³ +₂2² +₁+ao. Argue
that, for any n, a, and b,
You may take for granted that
Justify your answer.
1 (b-a)²
2 11
SIMP(f.n) =
can you determine if
(ii) Show that the same is not true for RIGHT(f, n) by giving an example of a third degree
polynomial f, a natural number n, and endpoints a <b so that
RIGHT(f, n) + [*ª f(x)dx.
Hint: linear functions are third degree polynomials and n = 1 is a natural number...
(iii) For the remaining parts, use = 0, b=2, and
f(x) =
Justify your answer.
- ²1(2)
-max f(iv) (r),
zab
T
2
max f'(x)
z€[a,b]
RIGHT(f, 4) and
SIMP(f, 4).
You may do this by hand, by writing code, or by using an online calculator, but you
must document your computations (either via screenshots or by writing down your
work).
(iv) Show that
f(r)dr.
-4.
max f'(x)| ≤
S
z€0,2
(v) Using the value of RIGHT(f, 4) from (in), as well as (iv) and (L), do you have enough
information to determine if
[√x -
(L)
<1?
(vi) Using the value of SIMP(f, 4) from (iii), as well as (S) and
max f)(x) ≤ 3,
z€ 0,2
[²* 1(x)dx= [²√² ce-4d₂ <1?
Expert Solution
Step 1
“Since you have asked multiple questions, we will solve the first question for you. If you want any specific question to be solved then please specify the question number or post only that question.”
(iv)
To prove
Let
Then its derivative is
We need to show that
Step by step
Solved in 2 steps
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