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Problem 1. Find the maximum and the minimum distance from the origin to the ellipse
x2 + xy + y2 = 3.
Hints: (i) Use x2 + y2 as your objective function; (ii) You can assume that the constraint qualification condition and the second order conditions are satisfied in this problem, as well as in problems 2 and 3.
Problem 2. Maximize f (x, y, z) = yz + xz subject to y2 + z2 = 1 and xz = 3.
Problem 3. (a) Maximize f (x, y) = x2 + y2 subject to 2x + y ≤ 2, x ≥ 0 and y ≥ 0.
(b) Use the Envelope Theorem to estimate the maximal value of the objective function in part
(a) when the first constraint is changed to 2x+ 9/8y ≤ 2, the second constraint is changed to x ≥ 0.1,and the third to y ≥ -0.1.
A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball , determine the number of blue balls in the bag.
A bag of 28 tulip bulbs contains 12 red tulip bulbs,7 purple tulip bulbs and 9 yellow tulip bulbs,. Two bulbs are selected without replacement. Determine, a) The probability t
Determine the equation of the plane that consists of the points P = (1, -2, 0), Q = (3, 1, 4) and R = (0, -1, 2). Solution To write down the equation of plane there is a re
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how do you solve quadratic equations by factoring?
Calculus with Vector Functions In this part we need to talk concisely on derivatives, limits and integrals of vector functions. Like you will see, these behave in a quite pred
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If A, B are acute angles and sinA= cosB, then find the value of A+B. Ans: A + B = 90 o
Definition Assume that f(t) is a piecewise continuous function. The Laplace transform of f(t) is denoted L{ f (t )} and defined by, There is an optional notation for L
which quadrilaterals have only 1 pair of parallel sides
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