Show that ze?-? = cosz has one solution on {2EC:[2]<1}, and the solution is real number.
(2.2) Let a be a real number with 1<a< 2. Put f(x) = Q +r 1+2 (a) Show that f maps (1, 0) into (1, 0). (b) Show that f is a contraction on [1, ) and find its fixed point.
Graph the solution to the inequality on the number line. 1.-3<3
8. Prove that max { x ( 2 -*): x is a real number} < 1 using the MAX/MIN method. 9 Sunoco thot Sand Taro qubooto ful
One characteristic measured about high schools is the percent
free lunch, which is the percentage of the student body that is
eligible for free and reduced-price lunches. The top 100 schools,
grouped according to their percent free lunch, is as follows.
Percent free
lunch (x)
Number of top
100
ranked high schools
46
20
12
10
12
If stratified random sampling with proportional allocation is
used to select a sample of 25 high schools, how many would be
selected...
Exercise 1 Consider the initial-value problem y(t)=1+3940), 25t<3; y(2) = 0. a) Show that the problem has a unique solution. b) Compute (by hand) an approximation of y(3) using the forward Euler method with a step size h = 0.5 (namely perform 2 steps of the method).
7. The vector field F =< 3x2z In y + ze+2 +20, - 3y?, x° In y + ce2 +423 > is conservative. Find a potential function f(x, y, z) such that F=Vf. Y
Show that if 0 < μ < 2-r has a unique relative extreme (max) value for x in (0,1)
If X has a gamma distribution with alfa = 2 and beta = 1, find pl 1.6 < X <2.8) Goma Distidiosa Fox) = f(...) for x 30 Q2. Let X be a continuous random variable with f(x) = 0.5ež for x > 0 and 0 otherwise. Find the expected value of g(x) = ei x <2.8) for xxo Gumma. Distribution F() =fQ, XB) - Ba otherwise
1. Let p be a positive real number with 0 < p < 1, and let ni, N2, N3 be positive integers. Sup- pose that Xı has a Bin(ni,p) distribution, X2 has a Bin(n2, p) distribution, and X3 has a Bin(n3, p) distribution. Show that the random variable Z = X1 + X2 + X3 also has a binomial distribution, and find the parameters of that distribution.
Show that if //AB-1//=E<1,
then//A^-1-B//<=//B//(E/(1-E))
16. Show that if | AB-1|| = < < 1, then 14--BISHBI