Exercise 16 find the tangent plane to f(x, y) = (x + y) ln(x2 + y2) at the point (1,1) in both standard form and linearized form.

Scalar form:

Linear form:

Step-by-step explanation:

We need the partial derivatives of the function f(x,y):

Notice we need to use product rule: (uv)’ u’v+uv’ where here:

Therefore:

We evaluate them in the given point (1,1):

We also need to evaluate the function f(x,y) at (1,1):

Then we plug the pieces into the formula of the tangent plane:

Here: , so:

We collect everything on the left side to get:

Which is the scalar form of the equation of the plane

Then we distribute to get:

Then simplify by combining like terms:

Finally move the constant to the right side to get the linear form:

in problems involving proportions, we can use cross products to test whether two ratios are equal and form a proportion. to find the cross products of a proportion, we multiply the outer terms, called the extremes, and the middle terms, called the means.

hope it