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Hands-On Exercises #6 (11 September 2013)
1.1
A – roots of polynomials
Find the roots of these quadratic polynomials from H-O #1 using the R function
uniroot.
1. f1 (x) = 6x2 − 11x + 4
2. f2 (x) = 4x2 − 13x + 3
3. f3 (x) = 2x2 − x − 6
And also the roots of this cubic:
f4 (x) = x3 − 5x2 − 17x + 21
If the intervals are difficult to determine, recall that the roots of a function
are separated by the roots of its derivatives. So
1. f10 (x) = 12x − 11
2. f20 (x) = 8x − 13
3. f30 (x) = 4x − 1
4. f40 (x) = 3x2 − 10x − 17
1.2
B – tricky root finding
√
The function f5 (x) = x/ x2 + 1 has an obvious root at the origin. Try to
confuse uniroot by giving it intervals that include 0 but get large. That is, try
intervals of the form [−1, B] and try large values of B.
1.3
C – approximation problem
A simple approximation to the square root of a number x in the interval [1/16, 1]
is the function
y(x) = 1.681595 − 1.288973/(.8408065 + x)
√ √
√
Find the extreme points of either (y(x) − x)/ x or (y(x) − x)2 /x.
1.4
D – robust estimator of location
Consider the estimate of location which minimizes
n
X
f (µ) =
|Yi − µ|3/2
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as something between the mean (exponent 2) and median (exponent 1). Write
a function with argument a vector y that creates f (µ) as its result. For the
data Yi , i = 1, . . . , n taking the values 3, 5, 6, 8, 9, 12, 16, 22, minimize f (µ).
jfm, 11 September 2013
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