How do you compute the gradient of the function #p(x,y)=sqrt(24-4x^2-y^2)# and then evaluate it at the point (-2,1)?

Question

William Abdalla · Accepted Answer

This is conceptually very simple, so the only "difficult" part will be the calculations: if #n# is the number of variabiles of a function (in your case thus #n=2#, then the gradient of the function is a vector of length #n#, whose elements are the derivatives with respect to each variables, so if you have #f(x_1,...,x_n)#, the gradient will be the vector
#({\partial f}/{\partial x_1},..,{\partial f}/{\partial x_n})# So, the gradient of your function #p# will be the 2-dimensional vector
#({\partial p}/{\partial x},{\partial p}/{\partial y})#. Let's calculate the two derivatives: by the chain rule, if we need to derive an expression such #\sqrt{f(x,y)}#, the derivative will be #{f'(x,y)}/{2\sqrt{f(x,y)}}#, where of course #f'(x,y)# is the derivative with respect to the right variable. So, deriving with respect to #x# (which means that we must consider #y# as a constant) , we get #f'(x,y)=-8x#. Deriving with respect to #y#, we get instead #-2y#. So, the gradient vector will be #( {-8x}/{2\sqrt{24-4x^2-y^2}}, {-2y}/{2\sqrt{24-4x^2-y^2}})# By making both terms simpler, we can get #( {-4x}/{\sqrt{24-4x^2-y^2}}, {-y}/{\sqrt{24-4x^2-y^2}})# To evaluate this vector, we only need to plug in the values, if #x=-2# and #y=1#, we get #(8/sqrt(7), -1/sqrt(7) )#

Isaiah Allen · Answer

undefined To compute the gradient of the function p(x, y) = sqrt(24 - 4x^2 - y^2), we need to find the partial derivatives with respect to x and y.

The partial derivative with respect to x (denoted as ∂p/∂x) can be found by differentiating the function with respect to x while treating y as a constant.

∂p/∂x = (-8x) / (2 * sqrt(24 - 4x^2 - y^2))

The partial derivative with respect to y (denoted as ∂p/∂y) can be found by differentiating the function with respect to y while treating x as a constant.

∂p/∂y = (-2y) / (2 * sqrt(24 - 4x^2 - y^2))

To evaluate the gradient at the point (-2, 1), substitute x = -2 and y = 1 into the partial derivatives:

∂p/∂x = (-8 * -2) / (2 * sqrt(24 - 4(-2)^2 - 1^2))
∂p/∂y = (-2 * 1) / (2 * sqrt(24 - 4(-2)^2 - 1^2))

Simplifying these expressions will give you the values of the partial derivatives at the given point.