Question

MULTIVARIABLE CALC: use the exact value f(P) and the differential df to approximate the value f(Q)... f(x,y)=sqrt(x^2-xy+1), P=(3,2), and Q=(2.98,2.03)

Answer 1

f(P)=2. grad(f)=(d/dx f,d/dy f)=((2x-y)/(2*f),(-x)/(2*f)). grad(f)(P)=(1,-3/4).
f(Q) is approximately equal to f(P)+grad(f)(P).(Q-P)
=2+(1,-3/4).(-.02,.03)
=1.9575

Answer 2

thanks! do you know how to do it if you have f(x,y,z)=sin(-xyz)+z with P=(1,2,) and Q=(1.02,1.97,-.02) i'm stumped with 3 variables

Answer 3

the same process should work with three variables. for the gradient, compute
grad(f)=(d/dx f(x,y,z), d/dy f(x,y,z), d/dz f(x,y,z)). then f(Q) is approximately equal to
f(P)+grad(f)(P).(Q-P)