Question

Use a tangent plane at (8,2) to approximate the value of the following function at the point. Present your answer accurate to 4 decimal places.

Answer 1

To approximate the value of a function at a given point using a tangent plane, we first find the gradient of the function at that point, and then use the equation of the tangent plane to approximate the value. In this case, we are given the point (8,2).

Step-by-step explanation:

To approximate the value of a function at a given point using a tangent plane, we first find the gradient of the function at that point. In this case, we are given the point (8,2). To find the gradient, we can take the partial derivatives of the function with respect to each variable, x and y. Once we have the gradient, we can use it to find the equation of the tangent plane. The equation of the tangent plane is given by:

z = f(a,b) + f_x(a,b)(x - a) + f_y(a,b)(y - b)

where f(a,b) is the value of the function at the point (a,b), f_x(a,b) and f_y(a,b) are the partial derivatives of the function at the point (a,b), and (x,y) are the variables. Plugging in the values from the given point (8,2) into the equation of the tangent plane, we can approximate the value of the function at that point.