The Mean-Value Theorem for Derivatives

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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

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