Question

3-40 (a) Find the local linear approximation L to the specified function f at the designated point P.

(b) Compare the error in approximating f by L at the specified point Q with the distance between P and Q. 38. f(x,y,z)= x+y/y+z ;P(−1,1,1),Q(−0.99,0.99,1.01)
39. f(x,y,z)=xeʸᶻ ;P(1,−1,−1),Q(0.99,−1.01,−0.99) 40. f(x,y,z)=ln(x+yz);P(2,1,−1), Q(2.02,0.97,−1.01)

Answer 1

To find the local linear approximation L for the specified functions at the designated points, we need to find the partial derivatives of the functions with respect to x, y, and z.

Step-by-step explanation:

For question 38, we need to find the local linear approximation L for the function f(x, y, z) = x + y/y + z at the point P(-1, 1, 1). To find L, we first find the partial derivatives of f with respect to x, y, and z:

For question 39, we need to find the local linear approximation L for the function f(x, y, z) = x * e^(y*z) at the point P(1, -1, -1). To find L, we first find the partial derivatives of f with respect to x, y, and z:

For question 40, we need to find the local linear approximation L for the function f(x, y, z) = ln(x + y*z) at the point P(2, 1, -1). To find L, we first find the partial derivatives of f with respect to x, y, and z: