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Question: In many applications, we want to maximize or minimize some quantity subject to a condition. Such constrained optimization problems are solved using Lagrange multipliers in multivariable calculus;
Minimize x2 + y2 while satisfying x + y = 4 using the following steps.
(a) Graph x + y = 4. On the same axes, graph x2 + y2 = 1, x2 + y2 = 4, x2 + y2 = 9.
(b) Explain why the minimum value of x2 + y2 on x + y = 4 occurs at the point at which a graph of x2 +y2 = Constant is tangent to the line x + y = 4.
(c) Using your answer to part (b) and implicit differentiation to find the slope of the circle, find the minimum value of x2 + y2 such that x + y = 4.
A cut-vertex of a graph is a vertex whose removal (along with all edges incident with it) increases the number of connected components of the graph. Describe any circumstances under which a graph with a cut vertex can be Hamiltonian.
Let A ? Hn×n. Show that if there exists B ? Hn×n such that BA = I, then A is invertible and B = A-1.
Suppose that a weapons inspector must inspect each of five different sites twice, visiting one site per day.
If the water level is rising at a rate of 15000 centimeters per minute when the height of the water is 4500 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute.
How many units should the company sell per month in order to break even, assuming that it can sell up to 5,000 units per month at the planned price?
suppose that a body with temp. 85F is discovered at midnight, and that the ambient temp. is 70F. the body is removed quickly to the morgue where the ambient temp. is 40F. after 1 hour the body temp. is found to be 60F. estimate time of death.
MARGINAL COST The total weekly cost (in dollars) incurred by Lincoln Records in pressing x compact discs.
Given that (x3+2)4 – y4+ 7 1n (y)cos x – x = 15
suppose you are going a weekend trip to a city that is d miles away. Develop a model that determines your round-trip gasoline costs. What assumptions or approximations are necessary to treat this model as a deterministic model?
The questions we will answer using optimization are: What dimensions should the box have, and what would be the resulting maximum volume?
Where should Steve cut the wire so that the total area of the circle and square combined is as small as possible? What is this minimal area? What should Steve do if he wants the combined area to be as large as possible?
CORRECTIVE LENS USE According to Mediamark Research, 84 million out of 179 million adults in the United States correct their vision by using prescription.
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