List of top Calculus Questions

The maximum value of \( f(x, y) = 49 - x^2 - y^2 \) on the line \( x + 3y = 10 \) is __________.

If the function \( f(x, y) = x^2 + xy + y^2 + \frac{1}{x} + \frac{1}{y} \), \( x \neq 0, y \neq 0 \) attains its local minimum value at the point \( (a, b) \), then the value of \( a^3 + b^3 \) is __________ (round off to TWO decimal places).

The surface area of the paraboloid \( z = x^2 + y^2 \) between the planes \( z = 0 \) and \( z = 1 \) is __________ (round off to ONE decimal place).

The rate of change of \( f(x, y, z) = x + x \cos z - y \sin z + y \) at \( P_0(2, -1, 0) \) in the direction from \( P_0(2, -1, 0) \) to \( P_1(0, 1, 2) \) is __________.

The work done by the force \( \mathbf{F} = (x + y) \hat{i} - (x^2 + y^2) \hat{j} \),
where \( \hat{i} \) and \( \hat{j} \) are unit vectors in \( \mathbf{O X} \) and \( \mathbf{O Y} \) directions, respectively, along the upper half of the circle \( x^2 + y^2 = 1 \) from \( (1, 0) \) to \( (-1, 0) \) in the \( xy \)-plane is

f(x,y)=\(\left\{\begin{matrix} 1-cos(\frac{x^2}{y^2}) \\  \sqrt{x^2+y^2} \end{matrix}\right.\) ;if y≠0 & n∈R, then

Number of finite non-isomorphic groups with exactly 3 conjugacy classes_____.