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Use the partial derivatives \( f_x(1,1) \) and \( f_y(1,1) \) of \( f(x,y) = \cos(xy) \) to calculate the tangent plane to the graph of \( f \) at the point \( (1,1,\cos(1)) \). Draw the plane and the graph of the function.

Answer :

Final answer:

To calculate the tangent plane, we need to find the partial derivatives fx(1,1) and fy(1,1) of the given function f(x,y)=cos(xy). The partial derivatives fx and fy can be found by differentiating the function with respect to x and y, respectively, while treating the other variable as a constant. Once we have fx(1,1) and fy(1,1), we can use these values to find the equation of the tangent plane.

Explanation:

To calculate the tangent plane to the graph of the function f(x,y) = cos(xy) at the point (1,1,cos1), we need to find the partial derivatives fx(1,1) and fy(1,1). The partial derivative with respect to x, fx, can be found by differentiating the function with respect to x while treating y as a constant. Similarly, the partial derivative with respect to y, fy, can be found by differentiating the function with respect to y while treating x as a constant. Once we have fx(1,1) and fy(1,1), we can use these values to find the equation of the tangent plane.

For the given function, the partial derivatives are:

fx(x,y) = -y sin(xy)

fy(x,y) = -x sin(xy)

Substituting the coordinates (1,1) into the partial derivatives:

fx(1,1) = -1 sin(1) = -sin(1)

fy(1,1) = -1 sin(1) = -sin(1)

So, the tangent plane to the graph of the function f(x,y) = cos(xy) at the point (1,1,cos1) can be written as:

z - cos(1) = (-sin(1))(x - 1) + (-sin(1))(y - 1)

This is the equation of the tangent plane. To draw the plane and the graph of the function, you can plot the points on a 3D graph using the coordinates from the tangent plane equation and the function f(x,y) = cos(xy).

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