Find the linear approximation of the f(x,y)= x/(x+y) at the point (1,4) .

Question

Find the linear approximation of the f(x,y)= x/(x+y) at the point (1,4).

Answer 1

Here's an outline of the necessary steps (the order is somewhat flexible):

1) Evaluate the function at the given point; the result is 1/5 in this case.

2) Compute the partial derivative with respect to x (pretending that y is a constant). You'll need to do this one using the Quotient Rule, giving a result of y/(x+y)^2 after simplification.

3) Evaluate this at the given point; the result is 4/25 here.

4) Compute the partial derivative with respect to x (pretending that y is a constant). You'll can also do this one using the Quotient Rule (but that's not the only way), giving a result of -x/(x+y)^2 after simplification.

5) Evaluate at the given point; the result is -1/25 in this case.

6) Put everything into the Tangent Plane/Linearization Formula:

T(x,y) = 1/5 + 4/25*(x - 1) - 1/25*(y-4)

For those who have trouble remembering that formula, it's much like the one-variable version from Calc 1:

L(x) = f(x0) + f'(x0)*(x - x0)

with f' replaced by the partial derivative with respect to x, and an additional similar term for y.