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Find the surface integral of the vector field π(π₯,π¦,π§)=(π₯,π¦,π§) over the part of the paraboloid π§=1βπ₯^2βπ¦^2 with π§β₯0 and having normal pointing upwards.
Hint: take π₯ and π¦ as independent parameters.
Evaluate the line integral: β«π(π₯,π¦,π§)Γβ π,
where π(π₯,π¦,π§)=(π¦^2,π₯,π§) and the curve πͺ is described by π=π¦=π^π₯ with π₯β[0,1].
If l, m be the portions of the axes of x and y intercepted by the tangent at any point ( x,y) on the curve (Β x/a )ο»Ώ^2/3+Β Β (Β Β y/b)ο»Ώ^2/3Β Β , show thatΒ ( l^2/a^2) +(m^2/b^2)=1
The drawing below shows a square with side a. A straight line intersects the square and encloses
an area A. The heights x and y on the left and right side (in a distance d from the square) of
the intersecting line can be varied. Assuming that x y and x; y a, nd an expression for
the enclosed area A(x; y) with respect to x and y.
It's time to tidy up your work desk. you are given 27cm^2 of cardboard to build a rectangular box without a lid to store small electronic components. By using the knowledge of partial derivative, determine the maximum volume of this box
Number of turning point of y=(x+4)(x+2)(x-1)(x-3)
Give an example of a function of two variables such that f(0,0) = 0 but f is NOT continuous at (0,0). Explain why the function f is NOT continuous at (0,0).
Give an example of a function of two variables whose first order partial derivatives exist at(0,0) (that is fx(0,0) and fy(0,0) both exist), but f is NOT differentiable at (0,0).Β Explain also why the function is NOT differentiable at (0,0).
A fisherman would like to put up fences in an open rectangular fishpond covering an
area of 10,000 square meters. The longer fences on opposite sides will cost β±600 per
meter while the shorter fences will cost β±150 per meter. What should be the dimensions
of the pond to minimize the cost of the project?
Find the area of the surface cut from the bottom of the paraboloid xΒ²+yΒ²-z=0 by the plane z = 4
