Question

Use a tangent plane to approximate the value of the following function at the point (7.1, -16.1):

f(x, y) = arctan(y) - 1791
Present your answer accurate to four decimal places.

Answer 1

To approximate the value of the function f(x, y) = arctan(y) - 1791 at the point (7.1, -16.1) using a tangent plane, we need to find the equation of the tangent plane at that point.

Step-by-step explanation:

To approximate the value of the function f(x, y) = arctan(y) - 1791 at the point (7.1, -16.1) using a tangent plane, we need to find the equation of the tangent plane at that point. The equation of a tangent plane can be given by:

z = f(a, b) + fx(a, b)(x - a) + fy(a, b)(y - b)

Substituting the values (a, b) = (7.1, -16.1) into the equation, we can find the value of the function at that point. Let's calculate:

z = arctan(-16.1) - 1791 + 0(x - 7.1) + (1/(1 + (-16.1)2))(-16.1 - (-16.1))(y - (-16.1))

Simplifying the equation, we can find the approximate value of the function at the point (7.1, -16.1).