Please estimate R(3.1, -3.1)?

Question

Let R(x, y) be a function of two variables such that R(3, -3)=4 and the gradient R(3, -3)=<-1, 2>.

b) Use the linear approximation or differentials to estimate R(3.1, -3.1).

Answer 1

This is actually a lot more intuitive than it seems. For 1-dimensional derivatives, we have that the best linear approximation of f(x) near x=a is:

 

f(x) ≈ f(a) + f'(a) * (x-a)

 

We have the same thing for higher dimensions:

 

R(x) ≈ R(a) + grad(R)(a) · (x-a)

 

The only difference is that this involves vectors, where R(x) is vector-valued, both x and a are vectors, and the · signifies the dot-product. Using the above equation, we have:

 

R( 3.1, -3.1 ) ≈ R( 3, -3) + grad(R)(3,-3) · (  <3.1, -3.1> - <3,-3> )

≈ 4 + <-1,2> · <0.1, -0.1>

≈ 4 + (-0.1 - 0.2)

≈ 4 - 0.3

≈ 3.7