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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.
Smooth Curve - Three Dimensional Space A smooth curve is a curve for which → r' (t) is continuous and → r' (t) ≠ 0 for any t except probably at the endpoints. A helix is a s
Example: Suppose your football team has 10 returning athletes and 4 new members. How many ways can the coach choose one old player and one new one? Solution: There are 10 wa
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explain the formular for finding trigonometry
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Sam''s sport''s equipment sells footballs. They maximized their profitability last year at (6,4) where x represents employees and P(x) represents profitability. Sam noticed that wh
Marginal cost & cost function The cost to produce an additional item is called the marginal cost and as we've illustrated in the above example the marginal cost is approxima
One of the simplest physical situations to imagine of is a falling object. Thus let's consider a falling object along with mass m and derive a differential equation as, when resolv
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