The Mean-Value Theorem for Derivatives

1. The Mean-Value Theorem for Derivatives states:
If f(x) is continuous function on the (closed and finite) interval [a, b] and differentiable on (a, b), then
f(b) − f(a)
b − a
= f
0
(ξ), for some ξ ∈ (a, b).
Let f(x) = √
x and [a, b] = [0, 4]. Find the point(s) ξ specified by the theorem. Find the second Taylor
polynomial P2(x) for this function about x0 = 1. Give an upper bound on the absolute value of interpolating
error.
2. Show that the following inequalities hold for any vector x:
1

n
||x||2 ≤ ||x||∞ ≤ ||x||2 ≤ ||x||1 ≤

n||x||2 ≤ n||x||∞.
Hint: Use the Cauchy-Schwartz inequality.
3. Consider the sample version of the Principal Component Analysis: Y = (X − 1x
T
)G, where X (n × p) is
the data matrix, 1 is a vector of length n consisting of ones, and G is the orthogonal matrix containing the
standardized eigenvectors corresponding to the eigenvalues l1 ≥ · · · ≥ lp of S, the sample matrix of X.
(a) Prove that the columns of Y have mean zero;
(b) Prove that the sample variance of the ith column of Y equals li
;
(c) Prove that the sample correlation between any two columns of Y is zero.
4. Given the following histogram (r = Gray level, n = number of occurrences),
r 0 1/7 2/7 3/7 4/7 5/7 6/7 1
n 400 700 800 900 500 400 196 200
(a) Perform histogram equalization;
(b) Perform histogram matching using the specified histogram in the following table (r = Gray level, p =
probability of occurrences):
r 0 1/7 2/7 3/7 4/7 5/7 6/7 1
p 0.05 0.05 0.1 0.1 0.15 0.2 0.25 0.1
1
5. Explain why the Sobel and the Prewitt matrices compute horizontal and vertical gradients. Which of the two
masks evaluate vertical gradients?
6. Derive the normal equations for the c
∗ = [c

1
, c

2
]
that minimize the approximation error in the least square sense
c
∗ = argmax
c
(
X
N
n=1
[fn − F(xn, c)]2
)
in case of F(x, c) = c1e
c2x
. Are the normal equations still linear?
2

Are you looking for a similar paper or any other quality academic essay? Then look no further. Our research paper writing service is what you require. Our team of experienced writers is on standby to deliver to you an original paper as per your specified instructions with zero plagiarism guaranteed. This is the perfect way you can prepare your own unique academic paper and score the grades you deserve.

Use the order calculator below and get ordering with idealtermpapers.com now! Contact our live support team for any assistance or inquiry.

Type of paper Academic level Subject area
Number of pages Paper urgency Cost per page:
 Total:

Purchase Guarantee

Why ORDER at IdealTermPapers.com?

  • Educated and experienced writers.
  • Quality, Professionalism and experience.
  • Original Content writing.
  • Best customer support.
  • Affordable Pricing on orders.
  • Thorough research.
  • Ontime delivery of finished work.
  • 100% plagiarism free papers.

Reasonable Prices

  • To get the best quality papers isn’t cheap so don’t trust extremely low prices.
  • We can’t claim that we have unreasonably low prices because low prices equal to low quality.
  • Our prices are good and they balance with the quality of our work.
  • We have a Moneyback guarantee.

Original and Quality work

  • Our writers are professionals and they write your paper from scratch and we don’t encourage copy pasting.
  • All writers are assessed and they have to pass our standards for them to work with us.
  • Plagiarism is an offence and it’s never tolerated in our company.

Native Writers plus Researchers

  • Our writers are qualified and excellent and will guarantee the best performance in your order.
  • Our team has writers who have master's and PhD qualifications who can handle any assignment
  • We have the best standards in essay writing.

We have been in business for over 7 syears

  • We have always served our customers from all over the world and they have continued to order with us.
  • We value our customers since they have trusted us to do their assignments.
  • We are competent in our writing gained from experience over the years
  • Our company has 24/7 Live Support.

You will get

  •  Custom Admission Essay written by competent professional English writers.
  •  Free revisions according to our revision policy if required
  •  Paper format:  275 words per page, Times New Roman font and size 12, doublespaced text and1 inch margin
  •  On time delivery and direct order download
  •  Privacy guaranteed

We can help you:

  •  acquire a comprehensive professional presentation.
  •  get a unique and remarkable content as per your instructions.
  •  Get an additional portion that can be included to your existing presentation;
  •  turn your work in to an eyecatching presentation with well communicated ideas.
  •  improve your presentation to acquire the best professional standards.