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MS-A0211 - Differential and integral calculus 2,  period III, 2023 \hfill María C. Quintana \\
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\textbf{Homework for Week 4.} 
To be submitted for marking before the  \textbf{deadline 8.2.2023}. \\See ``Assignments" section in MyCourses for instructions. Please remember that clear and logical mathematical writing is important.



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	\item (2pts) Use the method of Lagrange multipliers to find the maximum area of a rectangle in the first quadrant with two sides contained in the x and y axes and one corner on the line $3y+x=3$.
	
	
	\item (2pts) Find the points on the circle $x^2+y^2=9$ which are closest to and farthest from the point $(3,4)$.
	
	\item (3pts) Find the absolute extrema (maximum and minimum values) of the function $$f(x,y)=xy+\sqrt{9-x^2-y^2}$$ on the closed disk of radius $3.$ 
	
	\item (3pts) Use the Newton's method to estimate the locations of the points of intersection of the circle $x^2+y^2=9$ and the hyperbola $9x^2-y^2=9$. In this case, it is not difficult to find the solutions of the system of equations exactly. Compute the exact solution and notice that the Newton's method gives an accurate approximate solution very quickly. 

	
	

	
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